RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
THIERRY DE PAUW AND ROBERT HARDT
Contents
1. Introduction
2. Preliminaries: The Metric Space X and the Group G
3. The Group Rm (X; G) of m Dimensional Rectifiable G Chains in X
3.1. Rectifiable Sets in a Metric Space
3.2. 0 Dimensional Rectifiable G Chains
3.3. m Dimensional Parameterized G Chains
3.4. Restriction and Sum
3.5. Push-forward
3.6. Two Alternate Characterizations of Rectifiable G Chains
3.7. Slicing
3.8. Relations Between Operations
4. The Group F0 (X; G) of 0 Dimensional Flat G Chains in X
4.1. G-Oriented Lipschitz Curves
4.2. The Boundary of a 1 Dimensional Lipschitz G Chain
4.3. The Flat Norm F and Flat Completion F0 (X; G) of L0 (X; G)
4.4. Some Relations Between R0 (X; G) and F0 (X; G)
5. The Group Fm (X; G) of m Dimensional Flat G Chains in X
5.1. The Boundary of an m + 1 Dimensional Lipschitz G Chain
5.2. The Flat Norm F and Flat Completion Fm (X; G) of Lm (X; G)
5.3. Flat Chains in Finite Dimensional Spaces
5.4. The Support of a Flat Chain
5.5. Some Relations between Rm (X; G) and Fm (X; G)
5.6. A Slice-null Flat Chain is Zero
5.7. The Slicing Mass of a Rectifiable Chain and its Comparability to M
6. The Slicing Mass of a Flat Chain
6.1. The Borel Regular Measure µ̂T
6.2. The Restriction of a Finite Mass T to a µ̂T Measurable Set A
7. The Group M0 (X; G) of 0 Dimensional Flat Chains of Finite Mass
7.1. The G−valued Borel Measure ΨT
7.2. Rectifiability
8. The Group Mm (X; G) of m Dimensional Flat Chains of Finite Mass
8.1. Bounding the Total Variation of the Slice of a Flat Chain
8.2. Rectifiability
9. Virtual Flat Chains
References
The research of the second author was partially supported by NSF grant DMS-0604605.
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2
THIERRY DE PAUW AND ROBERT HARDT
1. Introduction
Various classes of chains and cochains are often introduced on a space X not only
to generate a standard homology or cohomology theory but also to exhibit relations
with other structures on X. For example, H. Federer and W. H. Fleming [12] introduced rectifiable currents (with integer coefficients) to give a Plateau-problem
existence theory for m dimensional chains in a Riemannian manifold X of least
mass among those having a given boundary or lying in a given integral homology
class. [12] shows that such rectifiable chains having rectifiable boundaries provide
the standard integral homology of X in case X is a smooth compact Riemannian
manifold or more generally Lipschitz Euclidean neighborhood retract. In [13] W.H.
Fleming also considered rectifiable chains in Euclidean space, but now those having
a finite coefficient group G. Unlike with Z or R coefficients, such a G chain may
not be identified as a current acting linearly on a space of differential forms. Such
G chains nevertheless do allow for different Plateau-problem solutions. For example, there is a Möbius band type minimal surface in R3 that supports a unique 2
dimensional chain with coefficients in Z/2Z which is characterized by having least
area among all such mod 2 chains with the same boundary circle.
In 1999-2000, there were two important extensions of these works. B. White [22]
obtained the main rectifiability results in a Euclidean space X while generalizing
the chain coefficient group of [13] to be any complete normed abelian group which
does not contain any nonconstant Lipschitz curve. On the other hand, L. Ambrosio
and B. Kirchheim [3] generalized the ambient space. [3] provided basic definitions,
properties, and rectifiability theorems of [12] for currents in a general (weakly separable) metric space X. In the present paper we provide new definitions and results
to handle simultaneously both generalizations. We work in a general complete metric space X with chains having coefficients in any complete normed abelian group.
For the main theorem in §8.2 that a finite mass flat chain is rectifiable, we also
assume the coefficient group G satisfy the no-Lipschitz-curve condition as B. White
has shown that this condition on G is also necessary for such rectifiability. We also
just found the interesting preprints [5] by L. Ambrosio and S. Wenger and [2] by
L. Ambrosio and M. Katz which treat the rectifiability for flat chains mod p in
Banach spaces and metric spaces as well as relations to isoperimetric inequalities
and filling radius.
The concept of an H m rectifiable (more precisely H m measurable, (H m , m)
rectifiable) subset of a metric space and its relation to Hausdorff measure H m has
been well-developed in [17], [4], and [3]. This allows us to develop the notion of an
m dimensional rectifiable G chain by adding to an H m rectifiable set M , an H m
measurable orientation-multiplicity suitable (as explained in §3.6) for the group
G. A rectifiable G chain is initially defined as an equivalence class of a locally
Lipschitz parameterized G chain. The mass M of a rectifiable G chain is the
integral of the density with respect to H m , and the group of rectifiable G chains
may, as in previous works (see §3.6(1)-(4)), be alternately characterized as the mass
completion of more elementary chains such as finite sums of Lipschitz simplices.
Flat chains, as first conceived by H. Whitney [24], are more subtle and require
the completion with respect to some weaker flat norm on polyhedral or Lipschitz
chains. For our definition of flat norm, we wish to allow the possible existence
of zero dimensional, infinite mass flat chains in totally disconnected spaces. For
example, a Cantor space X ⊂ R contains no nonconstant Lipschitz curve. Thus
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
3
if L0 (X; G) denotes the elementary group of finite sums of G multiples of point
masses in X, then the completion of L0 (X; G), using the flat norm defined with
boundaries of Lipschitz chains in X, becomes simply the mass completion. This
does not include, for example, the chain corresponding to the standard (infinite
mass) flat chain in R given by the sum of the boundaries of the complementary
open intervals. To resolve this and also to obtain a flat norm that is invariant
under isometric embedding, we define in §4.3 and §5.2 the flat norm on rectifiable
chains by allowing boundaries of chains from larger spaces acccessible by isometric
embedding. As noted in §5.3, our resulting flat chains still coincide exactly with
the previous notions in finite dimensional Euclidean space studied by Federer and
Fleming [12], Fleming [13], and White [22].
On 0 dimensional rectifiable chains, the mass norm is, by Theorem 4.4.1, lower
semicontinuous under flat convergence. So we may extend the definition of mass
to 0 dimensional flat chains by approximation, as in [13] and [22] and continue to
have flat lower semicontinuity of mass. As in [22], we find in §7.1, 7.2 that a 0
dimensional flat chain of finite mass is equivalent to a G-valued Borel measures and
that such a measure is purely atomic whenever the group G satisfies the additional
no-Lipschitz-curve condition of White. Thus, here a 0 dimensional flat chain of
finite mass is rectifiable. Our proof, modifying [22], involves a new construction,
for any atom-free Borel measure µ on X, of a µ measurable f : X → [0, 1] so that
µ(f −1 {t}) = 0 for all t ∈ [0, 1].
Flat chains of dimension m ≥ 1, which are defined similarly by flat completion
of Lipschitz chains, offer many new problems. These we address, as in [22] and
[3], by using their 0 dimensional slices by Lipschitz maps into Rm . First we verify
in §5.6 that a flat chain is, in fact, determined by such slices. We prove this by
reducing this result, through several steps, to cases where X is special, including
where X is a separable Banach space satisfying the metric approximation property,
to eventually where X is finite-dimensional, and finally ordinary Euclidean space.
In this case, the theorem has been established by White [22, Theorem 3.2].
For general m, the notion of mass presents another difficulty because the Hausdorff measure mass M described above is unknown to be lower semicontinuous
under flat convergence, even in a general finite dimensional Banach space. In §5.7
we use slicing to define, for rectifiable chains, another mass M̂, which is lower semicontinuous and still satisfies a universal comparability, (2m)−m M ≤ M̂ ≤ M. We
extend, again by approximation, M̂, to m dimensional flat chains. Note that using
either M or M̂ for rectifiable chains gives comparable flat norms and precisely the
same spaces of flat chains and flat chains of finite mass. But M̂ has the advantage
of being lower semicontinuous. The papers [3] and [18] use the observation of [16]
that the slice of a normal current by a fixed Lipschitz map defines a (metric-spacevalued) BV function. In Theorem 8.1.1, we give another proof which works for a
general flat G chain T having finite mass and finite boundary mass. For a Lipschitz
map p : X → Rm , the atoms in the slices of T by p, define, by an argument of [3]
and [16], an H m rectifiable set. The chain T may be restricted, without change, to
the union of countably many such H m rectifiable sets corresponding to countably
many such Lipschitz maps. Then approximate bilipschitz parameterizations of the
resulting H m rectifiable set and the fact (Theorem 5.4.2(1)) that the push-forward
of a flat chain by a Lipschitz map is determined by the values on the support of
the chain finally give in §8.2 the desired rectifiability results.
4
THIERRY DE PAUW AND ROBERT HARDT
In the last section §9, we briefly introduce the alternate notion of a virtual flat
chain in a metric space X which is defined by allowing pull-backs of flat chains
from the larger space `∞ (X). This may provide the space X with an even larger
class of chains. For example, the usual Koch curve K, as a metric space, contains
many nontrivial virtual flat 1 dimensional chains (all of infinite mass) even though
F1 (X; G), being a completion of L1 (X; G) = {0}, must also be {0}. In contrast to
flat chains, virtual flat chains here give the standard homology to the topological
circle K.
There are several other important related topics which we do not pursue here.
First there are relevant compactness theorems and existence theorems for Plateautype mass-minimizing problems. These require discussion of additional conditions
on the space X and the group G. See the results and open problems in [3] and the
work [18] of U. Lang on currents in metric spaces. In case G equals R or Z our results
do not necessarily apply to general metric space currents in X of [3] or [18], only
to those that are rectifiable or lie in our flat completion of the rectifiable currents.
Also see the paper [1] of T. Adams concerning flat chains in Banach space and the
existence of mass minimizers. There are also questions about the relation between
normal currents, polyhedral approximation, and flat convergence. In general an
analogue of the deformation theorem for Rn [10, 4.2.9], is unavailable. Scans were
introduced in [14] and [15] to describe bubbling limits of smooth mappings and
in [7] and [8] to describe certain Plateau problem solutions. Although Theorem
5.6.1 implies that the scan obtained by slicing a flat chain is unique, there remain
numerous problems about more general classes of scans. Finally there are questions
in a metric space about the relation between flat chains and flat cochains. The latter
were also introduced by H. Whitney in [23] and [24]. See also the paper [11] of H.
Federer, and the thesis [S] of M. Snipes. Similarly one would like to understand,
in various spaces, the relation between normal currents and charges, as introduced
and studied in [20] and [9] by L. Moonens, W. Pfeffer, and the first author.
2. Preliminaries: The Metric Space X and the Group G
Recall that for a map φ between metric spaces X and X̃
distX̃ (φ(x), φ(y))
∈ [0, ∞] ,
Lip φ :=
sup
distX (x, y)
x,y∈X,x6=y
and φ is called Lipschitz in case Lip φ < ∞. Moreover,
φ is an isometic embedding
if Lip φ = 1, φ is injective, and Lip φ−1 φ(X) = 1. Recall also that, for any set
Y,
`∞ (Y ) = {f : Y → R : kf k`∞ (Y ) < ∞ } where
kf k`∞ (Y ) := sup |f (y)| ,
y∈Y
is a Banach space.
From now on, let X denote a fixed complete metric space. We will repeatedly
use the following two well-known elementary observations.
Lemma 2.0.1. (1) The metric space X admits an isometric embedding into `∞ (X).
(2) Any Lipschitz map φ of any subset W of X into `∞ (Y ) admits a Lipschitz
extension ψ : X → `∞ (Y ) so that Lip ψ = Lip φ.
Proof. For (1) one may check that, for any fixed x0 ∈ X, the map
Φ : X → `∞ (X) , Φ(x) = dist(·, x) − dist(·, x0 ) ,
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
5
provides such an isometric embedding.
For (2), one may define, as in [10, 2.10.44],
[ψ(x)](y) =
for x ∈ X and y ∈ Y .
inf { [φ(w)](y) + (Lip φ) distX (x, w) }
w∈W
A norm k k on a group G is a map k k : G → [0, ∞) such that
(1) k − gk = kgk
(2) kg + hk ≤ kgk + khk
(3) kgk = 0 ⇔ g = 0G (the zero element) .
Then the formula distk k (g, h) = kg − hk defines a metric on G, and one says the
normed group (G, k k) is complete if this metric is complete. In any case, the metric
space completion Ḡ of G is again a group and the norm k k extends to a norm k k
on Ḡ making (Ḡ, k k) a complete normed group. One may view Ḡ as the group of
all (absolutely) k k convergent series, that is,
∞
I
X
X
kgi k < ∞ ,
gi = 0 for some gi ∈ G with
g ∈ Ḡ ⇐⇒ lim g −
I→∞ i=1
I=1
P∞
in which case, we say that g equals the k k convergent sum i=1 gi and kgk =
PI
limI→∞ k I=1 gi k. It is easy to verify that:
Lemma 2.0.2. If L : H → K is a Lipschitz homomorphism between normed
groups, then:
(1) L admits a unique continuous extension L̄ : H̄ → K̄.
(2) L̄ is a homomorphism.
(3) Lip L̄ = Lip L.
P∞
P∞
(4) For any k kH convergent series
i=1 L(hi ) is
P∞ i=1 hi ∈ H, the series
k kK convergent and equals L̄ ( i=1 hi ).
In this paper (G, k k) will denote a fixed complete normed abelian group, which
we will continue to indicate additively. Notationally it is convenient to use the
usual Z action on G so that, for example, (1)g is g and (−1)g is the inverse −g. We
will also consider many normed groups of chains with coefficients in G, as well as
their completions, and the above Lemma will be repeatedly used to extend chain
homomorphisms to these completions. Note, in particular, that formula (4) for
L̄(h) is well-defined independent of the k k convergent series representation of h.
3. The Group Rm (X; G) of m Dimensional Rectifiable G Chains in X
3.1. Rectifiable Sets in a Metric Space. Letting H m denote m dimensional
Hausdorff measure, we observe the following consequence of [17, Lemma 4]:
Lemma 3.1.1. For any Lipschitz map f from a bounded Lebesgue measurable
subset A of Rm into X, there are disjoint compact subsets Ki of A so that each
restriction f Ki is bilipschitz, each pair of images f (Ki ), f (Kj ) either coincide
∞
or are disjoint, and H m (f (A ∼ ∪∞
i=1 Ki )) = 0. In particular, the union ∪i=1 f (Ki )
of all images covers H m almost all of f (A).
Proof. From the area formula [17] we recall that Y = {y ∈ X : f −1 {y} is infinite}
has H m measure zero. Following [10, 3.2.22,3.2.23] and [17, Lemma 4], we may
choose disjoint Lebesgue measurable subsets Ci of A, so that each restriction f Ci
6
THIERRY DE PAUW AND ROBERT HARDT
m
(f (Z)) = 0. By subtracting appropriate
is bilipschitz and Z = A ∼ ∪∞
i=1 Ci has H
subsets, we may assume that the Ci are disjoint and do not intersect f −1 f (Z) .
For every finite subset J of positive integers, let
\
[
DI =
f (Ci ) ∼
f (Cj ) and I = {I ⊂ N : DI 6= ∅} .
i∈I
j ∈I
/
∞
Then {DI : I ∈ I } is a partition
of ∪i=1 f (Ci ). Also, for each pair i ∈ I ∈ I ,
−1
the restriction f Ci ∩ f (DI ) is a bilipschitz map onto DI . Fixing a compact
subset EI of DI with H m (EI ) > 12 H m (CI ), we deduce that f maps each set
Ki,I = Ai ∩ f −1 (EI ) in a bilipschitz fashion onto EI whenever i ∈ I. In particular,
Ki,I is compact, and two images f (Ki,I ) = EI and f (Kj,J ) = EJ either coincide
or are disjoint. Note that
!
[
1 X
m
H m (DI )
H
f (A) ∼ Y ∼ f (Z) ∼
EI
<
2
I∈I
I∈I
!
∞
1 m [
1
f (Ci ) = H m (f (A)) .
= H
2
2
i=1
We can now repeat this argument with A replaced by
!
A1
=
A∼f
−1
Y ∪ f (Z) ∪
[
EI
I∈I
to find suitable compact sets whose images partially cover at least 21 the Hausdorff
measure of f (A1 ). Continuing, we obtain the desired covering of H m almost all of
f (A).
Recalling [10, 3.2.14], [17], [4], and [3], we say an H m measurable subset M of
the metric space X is H m rectifiable if H m (M ∼ ∪∞
i=1 Mi ) = 0 where each Mi is
the Lipschitz image of a bounded subset of Rm . (In the terminology of [10, 3.2.14],
M is both H m measurable and (H m , m) rectifiable.) Using Lemma 3.1.1, we may
in fact insure that H m (M ∼ γ ( ∪∞
i=1 Ai )) = 0 where the Ai are disjoint compact
subsets of Rm , and the restricted maps γi := γ Ai are bilipschitz with disjoint
images in M . We will call such a γ : ∪∞
i=1 Ai → M ⊂ X a locally bilipschitz almost
parameterization of M . The reason for the cumbersome terminology is that the
map γ does not parameterize all of M (it may miss a set of H m measure zero),
and the entire map γ is itself not bilipschitz because the family of images γ(Ai ) may
even fail to be locally finite. Nevertheless these images can be somewhat “measuretheoretically separated” in the following sense: For H m almost every point x ∈
γ(Ai ), the Hausdorff measure on the remaining sets H m ∪∞
i6=j=1 γ(Aj ) has m
dimensional density 0 at x while H m γ(Ai ) has density 1 at x, and H m Ai =
L m Ai has density 1 at γi−1 (x).
3.2. 0 Dimensional Rectifiable G Chains. For a ∈ X and g ∈ G, we let g[[a]]
denote the G valued atomic measure
(
g
in case a ∈ E ,
(g[[a]]) (E) =
0G in case a ∈ X ∼ E
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
7
for E ⊂ X. Then
(
L0 (X; G) =
)
X
ga [[a]] : A is a f inite subset of X, ga ∈ G
a∈A
P
P
is an abelian group with norm defined by M
a∈A ga [[a]] =
a∈A kga k. The
group of 0 dimensional rectifiable G chains
(
)
X
X
R0 (X; G) =
ga [[a]] : A is a countable subset of X, ga ∈ G,
kga k < ∞ ,
a∈A
a∈A
is then the M completion of L0 (X; G). Each T ∈ R0 (X; G) is a purely atomic G
valued measure with associated finite positive measure µT where
X
µT (E) :=
kga k for E ⊂ X .
a∈A∩E
The augmentation homomorphism
!
χ : R0 (X, G) → G ,
χ
X
ga [[a]]
a∈A
=
X
ga
a∈A
satisfies
χ(P ) ≤ M(P ) and χ(Φ# P ) = χ(P ) for any isometric embedding Φ : X → X̃ . (1)
3.3. m Dimensional Parameterized G Chains. Higher dimensional rectifiable
G chains are more complicated and admit several descriptions. We first use parameterizations.
An m dimensional parameterized G chain is a triple
[γ, Ai , g]
where γ : ∪∞
i=1 Ai → M ⊂ X is a locally bilipschitz almost parameterization as
above in §3.1, g : ∪∞
i=1 Ai → G is measurable (with respect to the Lebesgue measure
on ∪∞
A
and
the
k k topology on G), and
i=1 i
∞ Z
X
g ◦ γ −1 dH m < ∞ .
(2)
i
i=1
γ(Ai )
Suppose [γ̃, Ãj , g̃] is another m dimensional parameterized G chain in X with
γ̃j := γ̃ Ãj . We say that
[γ̃, Ãj , g̃]
≈
[γ, Ai , g]
if
∞
∞
H m γ(∪∞
= 0 = H m γ̃(∪∞
i=1 Ai ) ∼ γ̃(∪j=1 Ãj )
j=1 Ãj ) ∼ γ(∪i=1 Ai )
(3)
and, H m a.e on each overlap γ(Ai ) ∩ γ̃(Ãj ),
g ◦ γi−1 = σi,j g̃ ◦ γ̃j−1
(4)
where
σi,j = sgn det D(γi−1 ◦ γ̃j ) ◦ γ̃j−1
= sgn det D(γ̃j−1 ◦ γi ) ◦ γi−1
∈ {−1, 1} .
8
THIERRY DE PAUW AND ROBERT HARDT
One readily verifies that ≈ defines an equivalence relation on m dimensional parameterized G chains. For transitivity, one uses the fact that sgn det(κ ◦ λ) =
(sgn(det κ)) (sgn(det λ)). An equivalence class
[[γ, Ai , g]]
is called an m dimensional rectifiable G chain in X, and the family of all these is
denoted
Rm (X; G) .
Since (3) and (4) imply that
[β ◦ γ̃, Ãj , ◦g̃]
≈
[β ◦ γ, Ai , g]
for any bilipschitz map β : M → Y , we have the well-defined bilipschitz pushforward
β# [[γ, Ai , g]] := [β ◦ γ, Ai , g] ∈ Rm (Y ; G) ,
(5)
to be used in §3.5 below.
For T = [[γ, Ai , g]] ∈ Rm (X; G), we see from (2) and (4) that the function
θT : X → R+ , given by
(
kg γi−1 (x) k in case x ∈ γi (Ai )
θT (x) =
(6)
0
in case x ∈ X ∼ ∪∞
i=1 γi (Ai ) ,
is well-defined H m a.e. and is H m integrable. It follows that the set
MT = {x ∈ X : θT (x) > 0 }
is an H m rectifiable subset of X that is well-defined up to a set of H m measure
zero. Also the integral
Z
µT (E) =
θT dH m for H m measurable E ⊂ X ,
E
defines a Borel regular measure on X with the mass M(T ) := µT (X) < ∞.
Using the Jacobian Jm γi and the area formula of [17], we readily find that
µT (E) =
∞ Z
X
i=1
kgi (x)k(Jm γi )(x) dL m x .
Ai ∩γ −1 (E)
Then we deduce from (5) that |Jm β| ≤ (Lip β)m ; hence,2029
M(β# T ) ≤ (Lip β)m M(T ) .
(7)
Note that
M(T ) = 0
⇔
θT = 0 H m a.e.
⇔
g = 0G L m a.e. .
(8)
Under this condition, which is independent of the representation, we simply say
T = 0.
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
9
3.4. Restriction and Sum. A 0 dimensional rectifiable G chain in X may, as a
G valued atomic measure, be restricted to any subset U of X, i.e.,
X
X
ga [[a]] U =
ga [[a]] ∈ R0 (X; G) .
a∈A
a∈A∩U
For T ∈ Rm (X; G) with m ≥ 1, the measure µT is clearly absolute continuous
with respect to H m . For any compact K ⊂ X and representation T = [[γ, Ai , g]]
as above, the restriction
T
K
[[γ γ −1 (K), Ai ∩ γ −1 (K), g]]
=
is well-defined, independent of the choice of representation of T . Since H m M
is Radon, we may, for a general H m measurable subset U of X, choose disjoint
compact subsets Kj ⊂ U ∩ M with H m U ∩ M ∼ ∪∞
j=1 Kj = 0 and define
T
U
=
−1
[[γ γ −1 (∪∞
(Kj ), g]] ,
j=1 Kj ), Ai ∩ γ
and verify this is also independent of the choice of Kj . Note that
µT
U
= µT
U .
Suppose next S = [[β, Bj , h]] ∈ Rm (X; G) is another m dimensional rectifiable
G chain. To define the sum S + T , we first choose a locally bilipschitz almost
parameterizations
∞
∞
η : ∪∞
k=1 Ck → (∪i=1 γ(Ai )) ∪ ∪j=1 β(Bj ) .
and then let
Ci,j,k

−1

for i, j ≥ 1 ,
 Ck ∩ η (γ(Ai ) ∩ β(Bj ))
=
Ck ∩ η −1 γ(Ai ) ∼ ∪∞
β(B
)
for i ≥ 1, j = 0 ,
j
j=1


−1
∞
Ck ∩ η (β(Bj ) ∼ ∪i=1 γ(Ai )) for i = 0, j ≥ 1 .
and ηi,j,k = ηk Ci,j,k . Now let
S + T = [[η, Ci,j,k , f ]] where H m a.e. on Ci,j,k ,
(9)

sgn det(Dγi−1 ) ◦ η) g γi−1 ◦ η)



 + sgn det(Dβ −1 ) ◦ η)h β −1 ◦ η) for i, j ≥ 1 ,
j
j
f =
−1
−1

sgn
det(Dγ
)
◦
η)
g
γ
◦
η)
for i ≥ 1, j = 0 ,

i
i


−1
−1
sgn det(Dβj ) ◦ η) h βj ◦ η) for i = 0, j ≥ 1 .
m
∞
∞
Thus H
η ∪i=0 ∪∞
j=0 ∪k=1 Ci,j,k ∼ (MS ∪ MT ) = 0, and we deduce from (4)
that the sum S + T is well-defined and that Rm (X; G) is consequently an abelian
group.
Moreover, the mass M is a norm on Rm (X; G) by (8) and the fact that k k
is a norm on G. Also since the countable union of H m rectifiable P
sets is again
∞
H m rectifiable, we may
reason
as
above
to
conclude
that
any
series
i=1 Ti with
P∞
Ti ∈ Rm (X; G) and i=1 M(Ti ) < ∞, is M convergent to a rectifiable G chain. In
particular,
Rm (X; G) is M complete .
For the special top-dimensional case of m dimensional chains in X = Rm , inclusion maps into Rm give parameterizations, and we may simplify some notations.
For a single compact A ⊂ Rm and an integrable function g : A → G, we abbreviate
[[A, g]] = [[ιA , A, g]]
(10)
10
THIERRY DE PAUW AND ROBERT HARDT
where ιA : A → Rm is the inclusion map of A into Rm . More generally for any
Lebesgue measurable subset E of Rm and g ∈ L1 (E, G), we see that
[[E, g]]
∞
X
is well − defined as
[[Ai , g Ai ]]
i=1
for any disjoint family {Ai } of compact subsets of E with L m (E ∼ ∪∞
i=1 Ai ) = 0.
Conversely, we deduce from dominated convergence that any T ∈ Rm (Rm ; G) has
the form T = [[E, g]] for some L n measurable E ⊂ Rm and g ∈ L1 (Rm , G).
By approximating g in L1 by simple functions, it is not difficult to obtain an M
convergent representation
[[E, g]] =
∞
X
∞
X
gi [[∆i ]] :=
i=1
[[∆i , gi ]]
(11)
i=1
for some constants gi ∈ G and nondegenerate closed m simplices ∆i in Rm . For
ε > 0, we may also control the overlapping so that
∞
X
m
(12)
kgi kH (∆i ) < ε .
M(T ) −
i=1
3.5. Push-forward. We now describe the push-forward of a rectifiable G chain in
X by a Lipschitz map φ from X to a metric space Y . A zero dimensional rectifiable
G chain in X is simply pushed forward as a G valued measure so that
!
X
X
X
X
φ#
ga [[a]] =
ga [[φ(a)]] =
ga [[b]] .
a∈A
a∈A
b∈φ(A)
a∈φ−1 (b)
In particular, M(φ# T ) ≤ M(T ) whenever T ∈ R0 (X; G).
For T ∈ Rm (X; G) with m ≥ 1, we already defined φ# T in (5) in case the map
φ is bilipschitz. For a general Lipschitz map and T = [[γ, Ai , g]] ∈ Rm (X; G), we
may apply Lemma 3.1.1 to f = φ ◦ γ to obtain disjoint compact subsets Kj of
A = ∪∞
i=1 Ai so that each restriction (φ ◦ γ)j := (φ ◦ γ) Kj is bilipschitz and let
φ# T
=
∞
X
(φ ◦ γ)j# [[Kj , g]]
=
j=1
∞
X
[[(φ ◦ γ)j , Kj , g]] .
j=1
By using (4) and common subdivisions, one checks that φ# T is well-defined independent of the representation T = [[γ, Ai , g]] and of the choice of Kj .
We now see that a parametric representation of a rectifiable G chain gives it as
a sum of push-forwards of m dimensional rectifiable G chains in Rm ,
T = [[γ, Ai , g]] =
∞
X
γ# [[Ai , g]] .
i=1
Lemma 3.5.1. For any Lipschitz maps φ : X → Y and ψ : Y → Z between metric
spaces and T ∈ Rm (X; G),
(ψ ◦ φ)# T
= ψ# (φ# T ) .
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
11
Proof. Suppose Kj are again chosen by applying Lemma 3.1.1 to φ ◦ γ and that
similarly disjoint compact sets Lk are found by applying Lemma 3.1.1 to ψ ◦ φ ◦ γ.
Since




∞
∞
∞
∞
[
[
[
[
H m (ψ ◦ φ ◦ γ)( Ai ∼
Kj ) ≤ (Lip ψ)m H m (φ ◦ γ)( Ai ∼
Kj )
i=1
vanishes, we see that
(ψ ◦ φ)# T =
∞
X
j=1
∪∞
k=1 Lk
i=1
is H
m
almost contained in
(ψ ◦ φ ◦ γ)k# [[Lk , g]] =
∞ X
∞
X
∪∞
i=1 Kj .
j=1
Thus
(ψ ◦ φ ◦ γ)j,k# [[Kj ∩ Lk , g]] .
k=1 j=1
k=1
On the other hand, since φ# T = [[(φ ◦ γ) ∪∞
j=1 Kj , g]] and


!
∞
∞
∞
∞
[
[
[
[
Kj ∼
Lk  ⊂ (ψ ◦ φ ◦ γ)
(ψ ◦ φ ◦ γ) 
Ai ∼
Lk
j=1
also has vanishing H
m
i=1
k=1
k=1
measure, we find, using (7) and (5), that
ψ# (φ# T ) =
=
=
∞
X
j=1
∞
X
ψ# (φ ◦ γ)j# [[Kj , g]]
(ψ ◦ φ ◦ γ)j# [[Kj , g]]
j=1
∞
∞ X
X
(ψ ◦ φ ◦ γ)j,k# [[Kj ∩ Lk , g]] .
k=1 j=1
3.6. Two Alternate Characterizations of Rectifiable G Chains. The group
Lm (X; G) =
I
X
γi# gi [[∆i ]] : I < ∞, gi ∈ G, ∆i is a closed m simplex in Rm ,
i=1
and γi : ∆i → X is Lipschitz for i = 1, . . . , I .
of m dimensional Lipschitz G chains is a subgroup of Rm (X; G). Using (11) and
(12), we readily deduce that Lm (X; G) is M dense in Rm (X; G), so that
Rm (X; G) is the M completion of Lm (X; G) .
(13)
In [10, 4.1.28(4)(5)], an integer-multiplicity rectifiable current in Rn is obtained
by starting with an H m rectifiable set M and then giving, at H m a.e. point
x ∈ M , a positive integer density θ(x) and a choice ξ(x) of orientation to the
approximate tangent space Tanm (H m M, x) of M at x. RThe density function θ
and orientation ξ are required to be H m measurable and M kθk dH m < ∞. For
a G chain with a general G, the density, which should now be G valued, cannot
be normalized to be “positive” and handled separately from the orientation. We
overcome this problem by keeping the two notions together and defining a G-valued
orientation of a vector-space V as an equivalence class
gα
12
THIERRY DE PAUW AND ROBERT HARDT
of pair (g, α) consisting of an element g ∈ G and an orientation α of V where
(g, α) ∼ (h, β)
if and only if either g = h and α = β
or g = −h and α = −β .
As a point x varies over an H m rectifiable set M , we need the notion that a
choice of G-valued orientation for the varying subspace Tanm (H m M, x) is H m
measurable and integrable in x. In case U is a Lebesgue measurable subset of Rm
(which has the standard constant orientation e1 ∧ · · · ∧ em ) an integrable G-valued
orientation simply has the form ge1 ∧ · · · ∧ em corresponding to a function g ∈
L1 (U, G). Since H m measurability and integrability are invariant under bilipschitz
transformation, we see that such a g gives a H m integrable G-valued orientation to
(the approximate tangent spaces of) any bilipschitz image γ(U ) in X, at least when
X is a finite-dimensional Banach space Z. Local bilipschitz almost parameterization
then gives the notion for any H m rectifiable subset M of Z. Since orientation
transforms under a linear isomorphism according to the sign of the determinant,
the key observation is that:
Two parametric G chains (γ, Ai , g) and (γ̃, Ãj g̃) are equivalent as in (3)(4) if and
only if, except for a set of H m measure zero, g and g̃ induce equal G-valued ori∞
entations on the H m rectifiable image ∪∞
i=1 γ(Ai ) = ∪j=1 γ̃(Ãj ).
A similar notion of H m integrable G-valued orientation works for H m rectifiable subsets of an arbitrary Banach space Y . In fact, to discuss properly H m
measurability and employ the above parametric description, one needs to fix a
topology for the Grassman space G(Y, m) of m dimensional subspaces of Y . Here,
since the space of linear embeddings GL(Rm , Y ) has the norm (or equivalently
weak) topology, the Grassman space, being the quotient
G(Y, m) = GL(Rm , Y )/GL(Rm , Rm ) ,
under the right action of GL(Rm , Rm ), has the quotient topology. One notes that
this notion of H m measurability and integrability of a G-valued orientation is
invariant under isometry between H m rectifiable subsets of Banach spaces. Finally,
for a general metric space X, one uses the isometric embedding Lemma 2.0.1(1)
to get a well-defined notion of a H m integrable G-valued orientation for H m
rectifiable subsets of X. We appeal to the same key observation as above to deduce
that
An m dimensional rectif iable G chain T ∈ Rm (X; G) corresponds to a H m
rectif iable subset M of X with a H m integrable G − valued orientation. (14)
In particular, since one can “multiply” a Z orientation by an element of G to
obtain a G orientation, one has, for any rectifiable Z-chain T ∈ Rm (X; Z) and
g ∈ G, a corresponding rectifiable G chain
g T ∈ Rm (X; G) .
(15)
The above characterizations shows that the definition of a rectifiable G chain in
X includes previously studied notions:
(1) In case X = Rn , G = R (respectively, Z), and kgk = |g|, Rm (X; G) equals
the real (respectively, integer-multiplicity) rectifiable currents of FedererFleming [12].
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
13
(2) In case G = R (respectively, Z), kgk = |g|, Rm (X; G) equals the real
(respectively, integer-multiplicity) rectifiable (metric space) currents of
Ambrosio-Kirchheim [3] or of U. Lang [18].
(3) In case X = Rn , Rm (X; G) equals the rectifiable G chains of B. White [W].
(4) In case G = Z, X = Rn , and kgk = |g|α for some α ∈ (0, 1], Rm (X; G)
equals the rectifiable scans with Mα finite of the authors [7].
3.7. Slicing. For an H m rectifiable subset M of X and a Lipschitz f : X → Rk
with k ≤ m, we deduce from [3] and [10, 3.2.22] that, for L k a.e. y ∈ Rk , the set
M ∩ f −1 {y} is H m−k rectifiable. Assuming M = MT for some T ∈ Rm (X; G),
m
we see that
and infer from the coarea formula of [3, Theorem
R θT is H integrable
9.4] that M ∩f −1 { ∗ } θT dH m−k is L k measurable with
Z Z
Z
m−k
k
θT dH
dL y =
Jk (f M )(x) θT (x) dH m x
Rk M ∩f −1 {y}
M
(16)
Z
k
m
k
≤ C(Lipf )
θT dH
= C(Lipf ) M(T ) < ∞ .
M
Assuming T is determined, as in (14), by a G-valued orientation g of MT , the slice
will, for L k a.e. y, be a G chain
h T, f, y i ∈ Rm−k (X; G)
obtained by assigning, at H m−k a.e. x ∈ M ∩ f −1 {y}, a G orientation ±g(x),
where the ± sign will be determined below by the behavior of f near x.
We will now essentially determine this ± sign, at H m−k a.e. x ∈ M ∩f −1 {y}, by
giving a parametric description of the slice h T, f, y i. Suppose that T = [[γ, Ai , g]] as
k
k
−1
above with M = ∪∞
{y} is
i=1 γ(Ai ). Then, for L a.e. y ∈ R , the fiber N = (f ◦γ)
m−k
H
rectifiable, and we may choose a locally bilipschitz almost parameterization
η : ∪∞
j=1 Bj → N and let ηj = η Bj . Thus,
η
γ
f
−1
k
∪∞
{y} ⊂ ∪∞
j=1 Bj −→ (f ◦ γ)
i=1 Ai −→ M −→ R .
As in [10, 2.10.44], we may find a Lipschitz extension F of f ◦ γ to all of Rm . Then,
by the coarea formula [10, 3.2.11], we have, for L k a.e. y ∈ Rk and H m−k a.e.
x ∈ F −1 {y}, F is differentiable at x and rank DF (x) = k so that the orthogonal
projection πx of Rm onto ker DF (x) has rank m − k. We define the slice
h T, f, yi = [[ γ ◦ η, Bj , sgn det DF (x), Dηj−1 ηj (·) ◦ πx h ]] ,
(17)
where h(x) = (g ◦γ ◦η)(x) for a.e. x ∈ Bj and j = 1, 2, . . . . Using (4), we check that
à → M or η̃ : ∪∞
B̃ → N
choosing some other almost parameterizations γ̃ : ∪∞
ĩ=1 ĩ
k̃=1 k̃
gives an equivalent parametric G chain. Thus h T, f, yi is well-defined. Note that
this definition also gives the M convergent representation
h T, f, yi =
∞
X
γ# h [[Ai , g]], f ◦ γ, y i ,
(18)
i=1
which expresses the slice of a chain in X in terms of push-forwards of slices of chains
in Rm .
14
THIERRY DE PAUW AND ROBERT HARDT
Note that, in case m = k, one has, for L k almost all y ∈ Rk that the rank k
condition holds at every point of the countable set M ∩ f −1 {y} and that
X
h T, f, yi =
(sgn det[D(f ◦ γ)(a)]) g(a)[[γ(a)]] .
(19)
a∈(f ◦γ)−1 {y}
3.8. Relations Between Operations.
Theorem 3.8.1. Suppose that S, T ∈ Rm (X; G), φ is a Lipschitz map of X
into a metric space X̃, U and V are H m measurable subsets of X and X̃, k ∈
{1, · · · , m}, f : X → Rk , f˜ : X̃ → Rk , and ψ : Rk → Rk are Lipschitz, and, in
case ` ∈ {1, · · · , m − k}, g : X → R` is Lipschitz.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(S + T ) U = S U + T U .
h S + T, f, y i = h S, f, y i + h T, f, y i for almost all y ∈ Rk .
M(S + T ) ≤ M(S) + M(T ).
φ# (S + T ) = φ# S + φ# T . (φ# T ) V = φ# T φ−1 (V ) .
h T U , f, y i = h T, f, y i U for almost all y ∈ Rk .
M(T U ) ≤ M(T ).
m
M(φ
R # T ) ≤ (Lip φ) M(T ) . k
Mh T, f, y i dy ≤ C(Lip f ) M(T ).
Rk
h φ# T, f˜, y i = φ# h T, f˜ ◦ φ, y i for a.e. y ∈ Rk .
hT, ψ ◦ f, ψ(y) i = sgn[det Dψ(y)]h T, f, y i for almost all y ∈ Rk .
h hT, f, yi, g, z i = h T, (f, g), (y, z) i = (−1)k+` h hT, g, zif, y i for almost all
(y, z) ∈ Rk × R` .
Proof. Statements (1)-(7) follow immediately from the definitions in the previous
four subsections and the subadditivity of k k.
For (8), (9), and (10), we assume that T = [[γ, Ai , g]] with M = ∪∞
i=1 γ(Ai ) choose
Kj as in §3.5,
To obtain (8), we choose Kj as in §3.5, note that φ γ(Kj ) is bilipschitz, and
deduce from (7) that
∞
X
M(φ# T ) ≤
M (φ# [[γ, Kj , g]]) ≤ (Lipφ)m
j=1
∞
X
M[[γ, Kj , g]] = (Lipφ)m M(T ) .
j=1
The formula MhT, f, yi = M ∩f −1 {y} θT dH m−k and [3] imply the measurability
of MhT, f, · i, and
P∞ conclusion (9) follows from the inequality (16). Note that (9)
implies that if i=1 Si is a M convergent series of G chains, then
Z
Z X
∞
∞
X
k
M
Si , f, y dL y ≤
MhSi , f, yi dL k y
R
Rk
=
Rk i=1
i=1
∞ Z
X
i=1
MhSi , f, yi dL k y ≤ C(Lip f )k
Rk
∞
X
M(Si ) < ∞ ,
i=1
which gives, for L k a.e. y ∈ Rk , the relation
∞
X
i=1
Si , f, y
=
∞
X
i=1
hSi , f, yi .
(20)
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
15
For (10), we again chose Kj as in §3.5 and now apply Lemma 3.5.1 and equations
(18) and (20) to see that
hφ# T, f˜, yi = h
∞
X
φ# γ# [[Ki , g]], f˜, y
j=1
=
∞ D
X
(φ ◦ γ)# [[Ki , g]], f˜, y
j=1
= φ#
E
=
∞
X
φ# γ# h([[Ki , g]], f˜ ◦ φ ◦ γ, yi
j=1
∞
X
hγ# [[Ki , g]], f˜ ◦ φ, yi = φ# hT, f˜ ◦ φ, yi .
j=1
Conclusions (11) and (12) are derived from the definition in equation (17).
Corollary 3.8.2. The operations + , U , and φ# are MPcontinuous. If Ti , T ∈
∞
Rm (X; G) and Ti M converges
to T fast enough
so that
i=1 M(Ti − T ) < ∞,
P
∞
then, for a.e. y ∈ Rk ,
M
h
T
−
T,
f,
y
i
<
∞
,
hence,
i
i=1
M h Ti , f, y i − h T, f, y i → 0 as i → ∞ .
Proof. The first conclusion follows from Theorem 3.8.1(3)(7)(8) and Lemma 2.0.2
while the second conclusion follows from Theorem 3.8.1(9) and the fact that
Z X
∞
∞ Z
X
M Ti − T, f, y dL k y =
M Ti − T, f, y dL k y .
Rk i=1
i=1
Rk
Corollary 3.8.3. Suppose S ∈ Rm (X; G). Then
S = 0 ⇔ For all Lipschitz p : X → Rm , h S, p, y i = 0 for almost all y ∈ Rm .
Proof. The implication ⇒ is clear. To verify ⇐, we assume S 6= 0 and S = [[γ, Ai , g]]
where we may assume that g 6= 0G a.e. on A1 . Then we take p :√X → Rm to be,
as in Lemma 2.0.1(2) a Lipschitz extension of γ1−1 (with Lip p ≤ m Lip γ −1 ) and
use Theorem 3.8.1(6)(10) to see that
p# h S, p, y i γ1 (K1 ) = p# h γ# [[A1 , g]], id ◦ p , y i
= h [[A1 , g]], id, y i = g[[y]] 6= 0 for a.e. y ∈ A1 .
4. The Group F0 (X; G) of 0 Dimensional Flat G Chains in X
In §4.3 below, we will define a group F0 (X; G) strictly containing R0 (X; G) in
which to realize
∂S
for any 1 dimensional rectifiable G chain S ∈ R1 (X; G). Such flat G chains have
been treated in Euclidean space by W. H. Fleming [13] and B. White [22] and in a
Banach space by T. Adams [1].
First we begin our discussion of ∂S with the elementary case where S is a G
multiple of a Lipschitz curve or a finite sum of such curves. Although, a general
complete metric space X may fail to have any nontrivial Lipschitz curves, a Banach
space containing an isometric copy of X will contain such curves. The results of
16
THIERRY DE PAUW AND ROBERT HARDT
the next two subsections thus will be useful for describing in §4.3 the “flat” norm
F on L0 (X; G) whose completion will give the flat chains F0 (X; G).
4.1. G-Oriented Lipschitz Curves. For a closed interval [a, b] ⊂ R and an element g ∈ G, we now abbreviate the G oriented interval
g[[a, b]] = [[[a, b], g]] .
Here our use of lower case letters for the endpoints of the interval should help avoid
confusion with our previous notation [[E, g]]. With a Lipschitz map γ : [a, b] → X,
we then get the G-oriented Lipschitz curve
γ# g [[a, b]] ∈ R1 (X; G) .
Then
M γ# g [[a, b]]
Z
≤ kgk
b
|γ 0 (t)| dt
a
where the metric differential [17] |γ 0 (t)| = limh→0 h−1 dist γ(t + h), γ(t) for a.e.
t ∈ [a, b]. We also have the following elementary slicing formula.
Theorem 4.1.1. For Lipschitz maps γ : [a, b] → X and u : X → R and almost
every number r ∈ (u ◦ γ)([a, b]),
(u ◦ γ)−1 {r} = {t1 , t2 , · · · , tI } where a < t1 < t2 < · · · < tI < b ,
and the slice h γ# g[[a, b]] , u, r i equals
 PI
i+1
g[[γ(ti )]] with Iodd

i=1 (−1)


 PI (−1)i g[[γ(t )]] with Iodd
i
Pi=1
I
i+1

(−1)
g[[γ(t
i )]] with Ieven


 Pi=1
I
i
i=1 (−1) g[[γ(ti )]] with Ieven
in
in
in
in
case
case
case
case
u
u
u
u
γ(a) < r
γ(a) > r
γ(a) < r
γ(a) > r
<u
>u
>u
<u
γ(b)
γ(b)
γ(b)
γ(b)
,
,
,
.
Proof. The function f = u◦γ : [a, b] → R is Lipschitz. For almost all r 6= f (a), f (b),
the set f −1 {r} is, by [10, 2.10.11], finite and so has the form {t1 , t2 , · · · , tI } as
above. We may also assume, by [10, 3.2.22], that each f 0 (ti ) exists and is nonzero
for i = 1, 2, · · · , j.
In crossing each ti , f (t) − r changes sign. Also the numbers sgnf 0 (ti ) alternate
as i increases by the Intermediate Value Theorem. For each i = 1, 2, · · · , I, we
compute
D(u M ) γ(ti ) # (gγ 0 (ti )) = Df (ti )# (g e1 ) = sgnf 0 (ti ) g e1 .
Thus, by (19),
h γ# g[[a, b]] , u, r i =
I
X
sgnf 0 (ti ) g [[γ(ti )]] .
i=1
0
In the first and third cases, sgn f (t1 ) > 0, hence sgn f 0 (ti ) = (−1)i+1 , while in
the second and fourth cases, sgn f 0 (t1 ) < 0, hence sgn f 0 (ti ) = (−1)i . Then in the
first two cases I must be odd because sgn f 0 (tI ) > 0, while, in the last two cases I
must be even because sgn f 0 (tI ) < 0.
Corollary 4.1.2.
χ [(g[[γ(b)]] − g[[γ(a)]])
{u < r}] = χ (h γ# g[[a, b]] , u, r i) .
(21)
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
17
Proof. Using Theorem 4.1.1, one readily checks that in each of the four cases, both
sides are the same, respectively, g, −g, 0G , 0G .
Theorem 4.1.3. For any h ∈ G, closed interval [c, d] ⊂ R, and Lipschitz curve
η : [c, d] → R,
η# h[[c, d]] = sgn η(d) − η(c) h [[η(c), η(d)]] .
(22)
Proof. By the push-forward formula §3.5, η# h[[c, d]] = [[η([c, d]), g]] where, for a.e.
r ∈ R,


X
g(r) = 
sgn η 0 (t)  h .
t∈η −1 {r}
As in the proof of Theorem 4.1.1 with u = η

+h in case



 −h in case
g(r) =

0
in case



0
in case
and γ = idR , we find that
η(c) < r
η(c) > r
η(c) < r
η(c) > r
< η(d)
> η(d)
> η(d)
< η(d)
,
,
,
,
and (22) follows.
4.2. The Boundary of a 1 Dimensional Lipschitz G Chain. By §3.6,
( I
)
X
L1 (X; G) =
γi# gi [[ai , bi ]] : I < ∞, gi ∈ G , γi : [ai , bi ] → X is Lipschitz .
i=1
PI
Theorem 4.2.1. For any Q =
i=1 γi# gi [[ai , bi ]] ∈ L1 (X; G) as above, the
boundary ∂Q is well-defined by the formula
∂Q =
I
X
gi [[γi (bi )]] − gi [[γ(ai )]] .
(23)
i=1
PJ
Proof. Suppose we had another such representation Q =
j=1 ηj# hj [[cj , dj ]]
corresponding to hj ∈ G and Lipschitz curves ηj : [cj , dj ] → X. We wish to show
that the difference
P
=
I
X
i=1
gi [[γi (bi )]] − gi [[γ(ai )]] −
J
X
hj [[ηj (dj )]] − hj [[ηj (cj )]]
j=1
vanishes. For this purpose, note that P is in the form P =
where A is the set of all endpoints,
P
a∈A
fa [[a]] with fa ∈ G
A = {γ1 (a1 ), γ1 (b1 ), · · · , γI (aI ), γI (bI )} ∪ {η1 (c1 ), η1 (d1 ), · · · , ηJ (cJ ), ηJ (dJ )} .
For each a ∈ A, let ua (x) = dist(x, a) for x ∈ X. Then, for almost all
positive r < dist(a, A ∼ {a}), we deduce from (21) and the additivity properties of
18
THIERRY DE PAUW AND ROBERT HARDT
χ, slicing, and restriction that
{x : ua (x) < r})
fa = χ (P
I
X
=χ
!
(gi [[γi (bi )]] − gi [[γi (ai )]])
{x : ua (x) < r}
i=1

J
X
− χ
( hj [[ηj (dj )]] − hj [[ηj (cj )]])

{x : ua (x) < r}
j=1
=
I
X
!
χ h γi# gi [[ai , bi ]] , ua , r i


J
X
− 
χh ηj# hj [[cj , dj ]] , ua , r i
i=1
= χh
I
X
j=1
γi# gi [[ai , bi ]] −
i=1
J
X
ηj# hj [[cj , dj ]] , ua , r i = χ (h0, ua , ri) = 0G .
j=1
Thus P = 0, and ∂Q is well-defined by (23).
Theorem 4.2.2. Suppose Q ∈ L1 (X; G).
(1)
(2)
(3)
(4)
χ(∂Q) = 0.
∂(Q + R) = ∂Q + ∂R for any R ∈ L1 (X; G).
∂(φ# Q) = φ# ∂Q for any Lipschitz map φ : X → X̃.
∂(Q {x : u(x) < r}) − (∂Q) {x : u(x) < r} = h Q, u, r i for any
Lipschitz u : X → R and a.e. r ∈ R.
Proof. Conclusions (1) and (2) are obvious. For (3), we may, by (2) and Theorem
3.8.1(4), assume that Q is a single G-oriented curve γ# g[[a, b]] and then deduce that
∂φ# (γ# g[[a, b]]) = ∂(φ ◦ γ)# g[[a, b]] = g[[φ γ(b) ]] − g[[φ γ(a) ]]
= φ# (g[[γ(b)]]) − φ# (g[[γ(a)]]) = φ# ∂(γ# g[[a, b]]) .
Similarly for (4) we again assume, by (2) and Theorem 3.8.1(1), that Q = γ# g[[a, b]].
As in Theorem 4.1.1, Q {x : u(x) < r} decomposes into finitely many G-oriented
curves. We readily compute that ∂(Q {x : u(x) < r}) − (∂Q) {x : u(x) < r}
equals
 PI
i+1
g[[γ(ti )]] + g[[γ(b)]]
in case u γ(a) < r < u γ(b) ,

i=0 (−1)

 PI+1

(−1)i g[[γ(ti )]] − g[[γ(a)]]
in case u γ(a) > r > u γ(b) ,
Pi=1
I+1
i+1

g[[γ(ti )]] + g[[γ(b)]] − g[[γ(a)]] in case u γ(a) < r > u γ(b) ,

i=0 (−1)

 PI
i
in case u γ(a) > r < u γ(b) ,
i=1 (−1) g[[γ(ti )]] − 0
which is, by Theorem 4.1.1, precisely the slice h Q, u, r i.
PI
Proposition 4.2.3. Suppose P = i=0 gi [[ ai ]] where the gi are nonzero elements
of G and the ai are distinct points in X. For any Q ∈ L1 (X; G),
M(P − ∂Q) + M(Q) ≥
1
kg0 k min{1, dist(a1 , a0 ), dist(a2 , a0 ), · · · , dist(aI , a0 ) } .
2
(24)
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
19
Proof. First we consider the special case X = R and I = 1.
Here P = g0 [[a0 ]] + g1 [[a1 ]] for a0 , a1 ∈ R. Suppose a0 < a1 . By replacing Q by
π# Q where π is the retraction of R onto [a0 , a1 ] and noting that
M(P − ∂π# Q) + M(π# Q) = Mπ# (P − ∂Q) + M(π# Q) ≤ M(P − ∂Q) + M(Q)
because Lip π = 1, we may assume that
Q =
J
X
hj [[bj−1 , bj ]] where hj ∈ G and a0 = b0 < b1 < b2 < · · · < bJ = a1 ,
j=1
Since, by the subadditivity of k k and | |,
M(P − ∂Q) + M(Q)


J
J
X
X
= kg0 − h1 k +
khj−1 − hj k + khJ − (−g1 )k +
khj k|bj − bj−1 |
j=2
j=1
≥ kg0 − hk k + khk − (−g1 )k + khk k|a1 − a0 | ,
where khk k = min{kh1 |, · · · , khJ |}, we may assume that J = 1 so that
Q = h[[a0 , a1 ]] for some h ∈ G .
For this Q, we now verify that
M(P − ∂Q) + M(Q) = kg1 − hk + kh + g0 k + khk|a1 − a0 |
1
max{kg0 k, kg1 k} min{1, |a1 − a0 |}
≥
2
by considering the two cases where either khk > 12 max{kg0 k, kg1 k} or not. Finally
note that we get the same result when a1 < a0 . Thus (24) is true in case X = R
and I = 1.
For the general case, we observe that Lip u0 ≤ 1 where,
u0 : X → R , u0 (x) = dist(x, a0 ) for x ∈ X .
By Theorem 4.1.3, u0# Q is a finite sum of G-oriented intervals in [0, ∞). Thus, by
Theorem 4.2.2(4),
∂(u0# Q
B (a0 , r)) − (∂u0# Q)
B (a0 , r) = < u0# Q, u0 , r > = h[[r]]
for some h ∈ G. For a.e. positive r < min{1, dist(a1 , a0 ), · · · , dist(aI , a0 ) } we may
apply the special case of (24) to
P̃ = (ua0 # P )
B (0, r) + h[[r]] = g0 [[0]] + h[[r]] ,
Q̃ = (u0# Q)
B (0, r)
Since Lip ua = 1, we conclude that
M(P − ∂Q) + M(Q)
+ M u0# (Q) B (0, r)
1
kg0 k min{1, r} .
= M(P̃ − ∂ Q̃) + M(Q̃) ≥
2
Finally we may choose r arbitrarily close to min{1, dist(a1 , a0 ), · · · , dist(aI , a0 ) }
to complete the proof.
≥ M u0# (P − ∂Q)
B (0, r)
20
THIERRY DE PAUW AND ROBERT HARDT
4.3. The Flat Norm F and Flat Completion F0 (X; G) of L0 (X; G). In [13],
[22], and [1], which treat chains in linear spaces, the flat norm is defined using
boundaries of polyhedral chains. In case the metric space X is Lipschitz path
connected, then for 0 dimensional chains in X, it would be reasonable to similarly
work with boundaries of Lipschitz curves in X. However here, to obtain a nontrivial
notion of 0 dimensional flat chains in a totally disconnected X (such as the Cantor
set) or in an X without nonconstant Lipschitz curves (such as the Koch snowflake),
we recall Lemma 2.0.1(1) and use boundaries of 1-chains from larger spaces by
defining, for any P ∈ P0 (X; G), the flat norm
F (P ) = inf{M(Φ# P −∂Q) + M(Q) : Φ is an isometric embedding
of X into a Banach space Y and Q ∈ L1 (Y ; G) } .
(25)
(Note that the above collection of nonnegative real numbers is a indexed by a family
of pairs Φ, Q which itself is not a set. Nevertheless, as a subcollection of nonnegative
real numbers, it still has a well-defined infimum. See [6].)
In the def inition of F (P ) one may take the inf imum over `∞ spaces. (26)
In fact, if P, Y, Φ, Q are as in (25) and Ψ : Y → `∞ (Y ) is an isometric embedding
as in Lemma 2.0.1(1), then Ψ ◦ Φ : X → `∞ (Y ) is an isometric embedding, Ψ# Q ∈
L1 (`∞ (Y ); G), and
M( (Ψ ◦ Φ)# P − ∂Ψ# Q) + M(Ψ# Q) = M Ψ# (Φ# P − ∂Q) + M(Ψ# Q)
= M(Φ# P − ∂Q) + M(Q) .
Also, for P ∈ Lm (Z; G) where Z = `∞ (A) for some A,
F (P ) = inf{M(P − ∂Q) + M(Q) : Q ∈ L1 (Z; G) } .
(27)
To prove the inequality ≥, we choose, for ε > 0, an isometric embedding Φ : Z → Y
and a Q ∈ L1 (Y ; G) so that F (P ) + ε > M(Φ# P − ∂Q) + M(Q). By Lemma
2.0.1(2), there is an extension g : Y → Z of Φ−1 with Lip g ≤ 1. Thus
M(P − ∂g# Q) + M(g# Q) = Mg# (Φ# P − ∂Q) + M(g# Q)
≤ M (Φ# P − ∂Q) + M(Q)
<
F (P ) + ε .
Theorem 4.3.1. F is a norm on L0 (X; G).
Proof. We easily check that F (−P ) = F (P ) and F (0) = 0. Moreover, if P =
PI
i=0 gi [[ai ]] 6= 0 with the gi being nonzero elements of G and the ai being distinct points of X, then we may apply Proposition 4.2.3 to the 0 chain Φ# (P ) =
PI
i=0 gi [[Φ(ai )]] ∈ L0 (Y ; G), to see that,
1
kg0 k min{1, kΦ(a1 ) − Φ(a0 )k, · · · , kΦ(aI ) − Φ(a0 )k } ,
2
1
=
kg0 k min{1, dist(a1 , a0 ), · · · , dist(aI , a0 ) } > 0 .
2
F (P ) ≥
Finally, for the subadditivity, assume P± ∈ L0 (X; G) and, for ε > 0 use (26) to
choose isometric embeddings Φ± : X → `∞ (A± ) and Q ∈ L1 (`∞ (A± ); G) so that
F (P± ) + ε > M(Φ±# P± − ∂Q± ) + M(Q± ) .
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
21
The map Φ+ ◦ Φ−1
− , being an isometric embedding of Φ− (X), admits, by Lemma
2.0.1(2), an extension h : `∞ (A− ) → `∞ (A+ ) with Lip h = 1. Thus,
F (P− + P+ ) ≤ M [Φ+# (P− + P+ ) − ∂h# (Q− + Q+ )] + M [h# (Q− + Q+ )]
≤ M [h# (Φ−# P− − ∂Q− )] + M(Q− ) + M(Φ+# P + −∂Q+ ) + M(∂Q+ )
= M(Φ−# P− − ∂Q− ) + M(Q− ) + M(Φ+# P + −∂Q+ ) + M(∂Q+ )
≤ F (P− ) + F (P+ ) + 2ε .
Theorem 4.3.2. Suppose P ∈ L0 (X; G).
(1) F (P ) ≤ M(P ), and F (∂Q) ≤ M(Q) for Q ∈ L1 (X; G).
(2) χ(P ) ≤ F (P ).
(3) For any Lipschitz map φ : X → X̃, F (φ# P ) ≤ max{1, Lip φ}F (P ).
Moreover, F (φ# P ) = F (P ) in case φ is an isometric embedding.
(4) For any Lipschitz map u : X → R, the restriction P {u < · } is F
Rb
measurable and a F (P {u < r}) dr ≤ (b − a + Lip u)F (P ).
Proof. Conclusion (1) is clear from (25).
Conclusion (2) also follows from (25) after using Theorem 4.2.2)(1) and (1) to
see that
χ(P ) = χ(Φ# P ) − 0 = χ(Φ# P ) − χ(∂Q) = χ(Φ# P − ∂Q) ≤ M(Φ# P − ∂Q) .
To verify (3), let Φ̃ : X̃ → `∞ (X̃) be an isometric embedding as in Lemma
2.0.1(1) and choose, for ε > 0, a Banach space Y , an isometric embedding Φ : X̃ →
Y and a Lipschitz chain Q ∈ L1 (Y ; G) so that
M(Φ# P − ∂Q) + M(Q) < F (P ) + ε .
By Lemma 2.0.1(2), the Lipschitz map Φ̃◦φ◦Φ−1 , which is defined on Φ(X), admits
a Lipschitz extension ψ : Y → `∞ (X̃) with Lip ψ =Lip φ. Then ψ# Q ∈ L1 (`∞ (X̃)),
and we now deduce
F (φ# P ) ≤ M[(Φ̃# ◦ φ)# P ) − ∂ψ# Q] + M(ψ# Q)
= M[ψ# (Φ# P − ∂Q)] + M(ψ# Q)
≤ M(Φ# P − ∂Q) + (Lip φ)M(Q)
< max{1, Lip φ}(F (P ) + ε) ,
and let ε ↓ 0 to get the desired inequality. In case φ is itself an isometric embedding
and δ > 0, we may choose an isometric embedding Ψ̃ of X̃ to a Banach space Ỹ
and a Lipschitz chain Q̃ ∈ L1 (Ỹ ; G) so that
M Ψ̃# (φ# P ) − ∂ Q̃ + M(Q̃) < F (φ# P ) + δ .
Since Ψ̃ ◦ φ is an isometric embedding, we conclude from Lemma 3.5.1 that
F (P ) < F (φ# P ) + δ ≤ max{1, Lip φ}F (P ) + δ = F (P ) + δ
and let δ ↓ 0.
For (4), note that P {u < · } is locally constant off the finite set u(spt P ), so
that, in particular, P {u < · } is F measurable. It follows that the real-valued
22
THIERRY DE PAUW AND ROBERT HARDT
function F (P {u < · }) is also measurable. From Theorem 4.2.2(4) we have, for
a.e. r ∈ R, the estimate
F (P
{u < r})
≤ M Φ# (P {u < r}) − ∂(Q {u < r} + M(Q {u < r} )
= M Φ# P − ∂Q
{u < r} − h Q, u, r i + M(Q {u < r} )
≤ M(Φ# P − ∂Q) + M(Q) + Mh Q, u, r i .
Integrating over r, using Theorem 3.8.1(9), and the definition (25) now gives the
inequality in (4).
We now define the group of 0 dimensional flat chains
completion (as in §2) of L0 (X; G).
F0 (X; G)
as the F
Theorem 4.3.3.
(1) The total multiplicity homomorphism χ : L0 (X; G) → G
admits a unique continuous extension χ : (F0 (X; G), M) → (G, k k), and
Lip χ = 1.
(2) For any Lipschitz map φ : X → X̃, the push-forward homomorphism
φ# : L0 (X; G) → L0 (X̃; G) admits
a unique continuous extension φ# :
(F0 (X; G), F ) → F0 (X̃; G), F , and Lip φ# ≤ max{1, Lip φ}. Moreover, φ# is an isometric embedding in case φ is an isometric embedding.
(3) For any Lipschitz map u : X → R, the sublevels-restriction homomorphism
Tu : L0 (X; G) → L1loc (R, (L0 (X; G), F )) ,
Tu (P )(r) = P
{u < r}
for a.e. r ∈ R ,
admits a unique continuous extension
Tu : F0 (X; G) → L1loc (R , (F0 (X; G), F )) ,
Rb
and a F (Tu (T )(r)) dr ≤ (b − a + Lip u)F (T ) for −∞ < a < b < ∞.
Letting T {u <Pr} := Tu (T )(r), one has, for any F convergent repre∞
sentation T =
∈ L0 (X; G), and a.e. r ∈ R, the F
i=1 Pi with Pi P
∞
convergent formula T {u < r} = i=1 Pi {u < r}.
Proof. By Theorem 4.3.2 (2)(3)(4), χ and φ# are Lipschitz homomorphisms from
(L0 (X; G), F ) to (G, k k) and to (L0 (X̃; G), respectively, and we may apply Lemma
2.0.2 to obtain the desired extensions and Lipschitz estimates. Also, for an isometric
embedding φ : X → X̃ and T ∈ F0 (X; G), one may use Theorem 4.3.2(3) to deduce
that F (φ# T ) = F (T ).
For fixed −∞ < a < b < ∞, we may also restrict the images of Tu to each
finite interval [a, b] to obtain similarly the desired continuous extension Tu mapping
F0 (X; G) to L1loc (R , (L0 (X; G), F )). Since
Z bX
∞ Z b
∞
X
F (Pi {u < r}) dr
F (Pi {u < r}) dr =
a i=I
i=I
a
≤ (a + b + Lip u)
∞
X
F (Pi )
→ 0
as
I → ∞,
i=I
we obtain the desired pointwise convergence for a.e. r ∈ R.
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
23
Theorem 4.3.4. The homomorphism ∂ : L1 (X; G) → L0 (X; G) admits a unique
continuous extension ∂ : (R1 (X; G), M) → (F0 (X; G), F ). For S ∈ R1 (X; G), one
has:
(1) F (∂S) ≤ M(S),
(2) χ(∂S) P
= 0.
P∞
∞
(3) ∂S = i−1 ∂Si is F convergent for any M convergent sum S = i−1 Si
with each Si ∈ R1 (X; G),
(4) ∂(φ# S) = φ# ∂S for any Lipschitz map φ : X → X̃.
(5) ∂(S {x : u(x) < r}) − (∂S) {x : u(x) < r} = h S, u, r i for any Lipschitz u : X → R and a.e. r ∈ R.
Proof. Since the definition (25) of F clearly shows that F (∂Q) ≤ M(Q) for Q ∈
L1 (X; G), the existence of the continuous extension and (1) again follow from
the extension Lemma 2.0.2. The definition of F and Theorem 4.2.2 imply that
conclusions (2), (3), (4), and (5) hold in case S ∈ L1 (X; G). For rectifiable S, (2)
follows from the continuity of Theorem 4.3.3(1), and (3) follows from the above
continuity of ∂ on R1 (X; G). Conclusion (4) also uses the continuity of ∂ as well as
the mass continuity (Theorem 3.8.1(8)) of φ# on R1 (X; G) and the flat continuity
(Theorem 4.3.3(2)) of φ# on F0 (X; G). Finally (5) uses the above continuity of ∂
as well as Theorem 4.2.2(4), Theorem 3.8.1(7)(9) and Theorem 4.3.3(2).
4.4. Some Relations Between R0 (X; G) and F0 (X; G). Clearly F
P≤ M. Thus,
I
for T ∈ R0 (X; G), the number F (T ) is well-defined as limI→∞ F
P
for
i
i=1
P∞
any mass convergent representation T =
G). Also,
i=1 Pi with Pi ∈ L0 (X; P
∞
with such T, Pi we have a well-defined associated F convergent sum i=1 Pi ∈
F0 (X; G). It is less obvious that this association is injective or that this new
F does not vanish on some nonzero rectifiable G chain. Both of these facts are
consequences of the following important lower semicontinuity result:
Theorem 4.4.1. If Tk , T ∈ R0 (X; g) and limk→∞ F (Tk − T ) = 0, then
M(T ) ≤ lim inf M(Tk ) .
k→∞
Proof. Suppose, for contradiction, that M(T ) − lim inf k→∞ M(Tk ) > P
ε for some
∞
positive number ε. Starting with an M convergent
representation
T
=
i=1 gi [[ai ]]
P∞
with the ai distinct, we first choose I so that
kg
k
<
ε/8
and
then set
i
i=I+1
s = 21 min1≤i<j≤I {1, dist( ai , aj )}.
We then fix a sufficiently large integer k so that
sε
ε
F (T − Tk ) <
and M(Tk ) +
< M(T ) .
(28)
24
24
PJ
With J > I chosen sufficiently large, the polyhedral chain P = i=1 gi [[ai ]] satisfies
F (T − P ) ≤ M(T − P ) =
∞
X
kgi k <
i=J+1
sε
.
24
(29)
Similarly we choose Pk ∈ L0 (X; G) so that Tk so that
F (Tk − Pk ) ≤ M(Tk − Pk ) <
sε
.
24
Combining (28) with (29) and (30) gives
sε
ε
F (P − Pk ) <
and M(Pk ) +
< M(P ) .
8
8
(30)
(31)
24
THIERRY DE PAUW AND ROBERT HARDT
By definition of F , there is an isometric embedding Φ of X into a Banach space Y
and Q ∈ L1 (Y ; G) so that
sε
M Φ# (P − Pk ) − ∂Q + M(Q) <
.
(32)
8
With ui (y) = ky − Φ(ai )k, we may, by Theorem 3.8.1(9), choose positive ri < s so
that the slices h Q, ui , ri i ∈ L0 (Y ; G) with
I
X
ε
.
8
Mh Q, ui , ri i <
i=1
Letting P̃i = Φ# (P − Pk )
I
X
kχ(P̃i )k =
i=1
I
X
B (ai , ri )
(33)
, we deduce from §3.2 and (31) that
kχ(P
B (ai , ri )) − χ(Pk
kχ(P
B (ai , ri ))k −
B (ai , ri ))k
i=1
≥
I
X
i=1
≥
I
X
I
X
kχ(Pk
B (ai , ri ))k
i=1
k gi + χ (P − gi [[ai ]])
B (ai , ri ) k − M(Pk )
i=1
I X
gi +
≥
i=1 ≥
I
X
kgi k −
i=1
≥
J
X
gj − M(Pk )
I<i≤J,aj ∈B(ai ,ri )
X
J
X
(34)
kgj k − M(Pk )
j=I+1
kgi k − 2
i=1
J
X
kgj k − M(Pk )
j=I+1
≥ M(P ) − M(Pk ) − 2
∞
X
kgj k
>
j=I+1
3ε
.
8
On the other hand, from Theorem 4.2.2(4), we also have the formula
P̃i = Φ# (P − Pk ) − ∂Q B (Φ(ai ), ri ) + ∂(Q B (Φ(ai ), ri )) − h Q, ui , ri i ,
which, along with (32), Theorem 4.2.2(1), and (33), implies that
I
X
kχ(P̃i )k ≤ M Φ# (P − Pk ) − ∂Q + Mh Q, ui , ri i <
i=1
ε
,
4
which contradicts (34) and completes the proof.
Corollary 4.4.2.
If T ∈ R0 (X; G) and F (T ) = 0 , then T = 0 . Thus
(R0 (X; G), M) ⊂ (F0 (X; G), F )
is injective and continuous.
Proof. Take all Tk = 0 in Theorem 4.4.1 to see that M(T ) = 0, hence, T = 0.
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
25
5. The Group Fm (X; G) of m Dimensional Flat G Chains in X
Here we will define the group Fm (Y ; G) of flat G chains, and generalize the
definitions and statements of §4.
5.1. The Boundary of an m + 1 Dimensional Lipschitz G Chain. Suppose
that ∆ is a nondegenerate closed simplex in Rm+1 . As in (10) and §3.6, the standard
orientation e1 ∧· · ·∧em+1 gives the Z chain [[∆]]. In simplicial theory or equivalently
in the theory of currents [10, 4.1.11], the boundary ∂[[∆]] is an m dimensional Z
chain given as the sum of oriented m dimensional faces of K where the orientation
of a face F is given as the ∗νF where νF is the unit exterior normal vector to ∆ on
F . Following (15), we now have the formula
∂ (g [[∆]]) = g ∂[[∆]]
for the boundary of a G oriented m + 1 simplex in Rm+1 , which, for m = 0, is
consistent with our previous definitions. Moreover, for any Lipschitz map q : ∆ →
Rm , one has the boundary slice formula
∂h g [[∆]] , q , y i = (−1)m h∂ g [[∆]] , q , y i
(35)
for almost all y ∈ Rm−1 . [13] establishes this for flat (hence, rectifiable) currents
where G = Z, g = 1. Since the rectifiable current slice coincides, by [10, 4.3.8],
with the slice of the Z chains, as defined in §3.7, and since
gh∂[[∆]], q, y i = h g ∂[[∆]], q, y i ,
gh[[∆]], q, y i = h g [[∆]], q, y i ,
the boundary slice formula is true for a general G.
It is now easy to define the boundary of an m + 1 dimensional Lipschitz chain.
PI
Theorem 5.1.1. For any Lipschitz chain Q = i=1 γi# gi [[∆i ]] ∈ Lm+1 (X; G) ,
a Lipschitz chain boundary ∂Q ∈ Lm (X; G) is well-defined by the formula
∂Q =
I
X
γi# gi ∂[[∆i ]] .
i=1
Moreover,
(1) ∂(∂Q) = 0 for m ≥ 1.
(2) ∂(φ# Q) = φ# (∂Q) for any Lipschitz map φ : X → X̃.
Proof. For any Lipschitz map p : X → Rm and almost all y ∈ Rm , we may apply
(35) with ∆ = ∆i and q = p ◦ γi to see that
∂h[[g Mi ]] , p ◦ γi , y i = (−1)m h ∂g [[Mi ]] , p ◦ γi , y i .
This, along with Theorem 3.8.1(2)(10) and Theorem 4.2.2(3), implies that, for
almost all y,
h
I
X
γi# gi ∂[[Mi ]] , p, y i =
i=1
I
X
γi# h gi ∂[[Mi ]] , p ◦ γi , y i
i=1
I
X
= (−1) ∂γi# h
gi [[Mi ]] , p ◦ γi , y i = (−1)m ∂h Q , p , y i ,
m
i=1
which is independent of the representation of Q. The well-definedness now follows
from Corollary 3.8.3.
26
THIERRY DE PAUW AND ROBERT HARDT
Equation (1) follows by additivity from the well-known calculation (using either
simplicial theory or currents) that ∂(∂[[∆]]) = 0 for any Z oriented simplex [[∆]].
Finally equation(2) follows from Theorem 3.8.1(4), Lemma 3.5.1 and the above
definition of boundary:
!
I
I
I
X
X
X
∂φ#
γi# gi [[∆i ]] = ∂
φ# (γi# gi [[∆i ]]) = ∂
(φ ◦ γ)ii# gi [[∆i ]]
i=1
i=1
=
I
X
i=1
(φ ◦ γi )# gi ∂[[∆i ]] =
i=1
= φ#
I
X
φ# (γi# gi ∂[[∆i ]])
i=1
I
X
!
γi# gi ∂[[∆i ]]
= φ# ∂
i=1
I
X
!
γi# gi [[∆i ]]
.
i=1
5.2. The Flat Norm F and Flat Completion Fm (X; G) of Lm (X; G). As in
§4.3, we define, for any Lipschitz chain P ∈ Lm (X; G), the flat norm
F (P ) = inf{M(Φ# P − ∂Q) + M(Q) : Φ is an isometric embedding
of X into a Banach space Y and Q ∈ Lm+1 (Y ; G) } .
(36)
Again, exactly as in (26) and (27),
F (P ) = inf{M(Φ# P −∂Q) + M(Q) : Φ is an isometric embedding
of X into an `∞ space Y and Q ∈ L1 (Y ; G) } ,
(37)
and, for P ∈ Lm (Z; G) where Z = `∞ (A) for some set A,
F (P ) = inf{M(P − ∂Q) + M(Q) : Q ∈ Lm+1 (Z; G) } .
(38)
Theorem 5.2.1. Suppose P ∈ Lm (X; G) with m ≥ 1.
(1) F (∂P ) ≤ F (P ) ≤ M(P ).
(2) F (φ# P ) ≤ max{(Lip φ)m , (Lip φ)m+1 }F (P ) for any Lipschitz map
φ : X → X̃. Also, F (φ# P ) = F (P ) in case φ is an isometric embedding.
(3) For any Lipschitz u : X → R, the restriction P {u < · } is F measurable,
and
Z
b
F (P
{u < r}) dr ≤ (b − a + Lip u)F (P ) .
a
(4) For any Lipschitz p : X → Rk , the slice h P, p, · i is F measurable, and
Z
F h P, p, y i dy ≤ k k/2 (Lip p)k F (P ) .
Rk
Proof. The proofs of (1) and (2) are similar to those of Theorem 4.3.2(1)(3) using
the fact that
Φ# ∂P = ∂Φ# P = ∂(Φ# P − ∂Q)
for any isometric embedding Φ : X → Y and Q ∈ Lm+1 (Y, G).
In (3) the F measurability of the restriction P {u < · } may be deduced by
noting that, off of some countable subset of R, this restriction is M continuous, and
consequently F continuous by conclusion (1). The integral estimate of (3) follows
as in the proof of Theorem 4.3.2(4).
To verify the measurability of the slice in (4), we first consider an m dimensional
polyhedral chain P̃ in Rm and a piece-wise affine map p̃ : Rm → Rk . By subdivision,
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
27
PI
we may assume that P̃ is a finite sum i=1 gi [[∆i ]] of G-oriented m simplices in
Rm and the restriction of p̃ to each ∆i as well as to each face of ∆i , is affine (of
constant rank). Then we have two possibilities:
In case rank (p̃ ∆i ) < k , L k (p̃(∆i )) = 0 so that h [[∆i ]], p̃, · i = 0 a.e.
In case rank (p̃ ∆i ) = k and y, z ∈ p̃(∆i ), then for every G oriented face
g[[∆]] of g[[∆i ]] with rank (p̃ ∆) ≥ k, we may, by induction on dim ∆, choose a g
orientation for the set ∆ ∩ p̃−1 [y, z] to obtain a polyhedral chain g[[∆]] ∩ p̃−1 [[y, z]]
which satisfies the chain formula
hg[[∆]], p̃, zi − hg[[∆]], p̃, yi = ∂ g[[∆]] ∩ p̃−1 [[y, z]] − (g∂[[∆]]) ∩ p̃−1 [[y, z]]
and mass estimate
M g[[∆]] ∩ p̃−1 [[y, z]]
≤ kgkH (dim ∆)−1 (∆ ∩ p̃−1 [y, z]) ≤ C(∆)kgk |y − z| .
For ∆ = ∆i , these facts show that the slice h [[∆i ]], p̃, · i is F Lipschitz on p̃(∆i ).
Since this slice also vanishes off of p̃(∆i ), it is clearly F measurable. The measurPI
ability of hP̃ , p̃, · i = i=1 hgi [[∆i ]], p̃, · i now follows.
Second for a general Lipschitz p : Rm → Rk , we may find a sequence of piecewise
affine p̃j : Rm → Rk which approach p uniformly on spt P̃ = ∪Ii=1 ∆i . The homotopy
slice formula of [10, 4.3.9] shows the pointwise a.e. F convergence of hP̃ , p̃j , · i to
hP̃ , p, · i, so that, in particular, hP̃ , p, · i is F measurable.
For the general Lipschitz chain P in X, we may, as in (18), express the slice
hP, p, · i as a Lipschitz push-forward of the slice of a polyhedral chain P̃ in Rm .
Since Lipschitz push-forwarding is, by (2), F continuous, we finally conclude that
hP, p, · i is F measurable.
For the integral estimate in (4), we observe that, for any isometric embedding Φ
of X into a Banach space
Y , we may choose a Lipschitz extension q : Y → Rk of
√
p ◦ Φ−1 with Lip q ≤ k Lip p and deduce from Theorem 3.8.1(10)(9)
Z
M Φ# h P, p, y i) − ∂h (−1)k Q, q, yi + Mh (−1)k Q, q, y i dy
Z
≤
M hΦ# P − ∂Q, q, yi + M(h Q, q, y i) dy
≤ k k/2 (Lip p)k M(Φ# P − ∂Q) + M(Q) .
Theorem 5.2.2. F is a norm on Lm (X; G).
Proof. For m = 0, this was shown in Theorem 4.3.1. For m ≥ 1, the relations
F (−P ) = F (P ) and F (0) = 0, are again clear, and the subadditivity of F follows
as in the the proof in Theorem 4.3.1.
Finally if P ∈ Lm (X; G) and F (P ) = 0, then, for any Lipschitz p : X → Rm ,
hP, p, ri = 0 for a.e. r ∈ R by 5.2.1(4). But then P = 0 by Corollary 3.8.3.
The group of flat chains
Fm (X; G)
is defined to be the F completion (as in §2) of the Lipschitz chains Lm (X; G).
Theorem 5.2.3.
(1) For any Lipschitz map φ : X → X̃, the push-forward
homomorphism φ# : Lm (X; G) → Lm (X̃; G) admits a unique continuous extension φ# : (Fm (X; G), F ) →
Fm (X̃; G), F , and Lip φ# ≤
28
THIERRY DE PAUW AND ROBERT HARDT
max{(Lip φ)m , (Lip φ)m+1 }. Moreover, φ# is an isometric embedding in
case φ is an isometric embedding.
(2) For any Lipschitz map u : X → R, the sublevels-restriction homomorphism
Tu : Lm (X; G) → L1loc (R , (Lm (X; G), F )) ,
Tu (P )(r) = P {u < r} for a.e. r ∈ R ,
admits a unique continuous extension
Tu : Fm (X; G) → L1loc (R , (Fm (X; G), F )) ,
Rb
and a F (Tu (T )(r)) dr ≤ (b − a + Lip u)F (T ) for −∞ < a < b < ∞.
Letting T {u <Pr} := Tu (T )(r), one has, for any F convergent repre∞
sentation T = i=1 Pi with Pi ∈P
Lm (X; G), and a.e. r ∈ [a.b], the F
∞
convergent formula T {u < r} = i=1 Pi {u < r}.
(3) For any Lipschitz map f : X → Rk , the slice homomorphism
Sf : Lm (X; G) → L1 Rk , (Lm−k (X; G), F ) ,
Sf (P )(y) = hP, f, yi for a.e. y ∈ Rk ,
admits a unique continuous extension
Sf : Fm (X; G) → L1 Rk , (Fm−k (X; G), F )
with Lip Sf ≤ (Lip f )k . Letting hT,
f, yi := Sf (T )(y) , one has, for any
P∞
F convergent representation T = i=1 Pi with
Pi ∈ Lm (X; G), and a.e.
P∞
y ∈ Rk , the F convergent formula hT, f, yi = i=1 hPi , f, yi.
Proof. The proof follows from Lemma 2.0.2 using the estimates of Theorem 5.2.1(2)
(3)(4) just as in the proof of Theorem 4.3.3.
Theorem 5.2.4. The homomorphism ∂ : Lm+1 (X; G) → Lm (X; G) admits a
unique continuous extension ∂ : (Fm+1 (X; G), F ) → (Fm (X; G), F ). For S ∈
Fm+1 (X; G), one has:
(1) F (∂S) ≤ F (S),
(2) χ(∂S) = 0 in case m = 0.
(3) ∂ 2 S = 0 in case m ≥ 1.
(4) For any Lipschitz φ : X → X̃, ∂(φ# S) = φ# ∂S.
(5) For any Lipschitz u : X → R, ∂(S {u < r}) − (∂S) {u < r} = h S, u, r i
for a.e. r ∈ R.
(6) For any Lipschitz p : X → Rk with k ≤ m, ∂h S, p, y i = (−1)k h ∂S, p, y i
for a.e. y ∈ Rk .
(7) For any Lipschitz q : X̃ → R` , h φ# S, q, z i = φ# h S, q◦φ, z i for a.e. z ∈ R` .
˜ (ψ ◦ φ) S = ψ (φ S).
(8) For any Lipschitz ψ : X̃ → X̃,
#
#
#
Proof. Here, as in the proof of Theorem 4.3.4, conclusions (1), (2), (3), (4), (7)
and (8) follow by continuity from the corresponding relations for Lipschitz chains
established in Theorem 5.2.1(1), Theorem 4.3.4(2), Theorem 5.1.1(1)(2), Theorem
3.8.1(10), and Theorem 3.5.1.
For conclusion (6), we similarly see, by the F continuity of the slice in Theorem
5.2.3(3) and the above F continuity of the the boundary, that we need only prove
(6) assuming S itself belongs to Pm (X, G). We have already done this in Theorem
35 for the very special case where k = m − 1 and T = g[[M ]] for some g ∈ G and m
simplex M .
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
For the case k = m − 1 and S =
and (3) to deduce
∂hS, p, yi =
I
X
PI
i=1
∂h γi# gi [[Mi ]], p, y i =
i=1
=
I
X
=
γi# gi [[Mi ]] ∈ Lm (X, G), we may use (7)
I
X
∂γi# h gi [[Mi ]], p ◦ γi , y i
i=1
γi# ∂h gi [[Mi ]], p ◦ γi , y i =
i=1
I
X
29
I
X
(−1)k γi# h ∂gi [[Mi ]], p ◦ γi , y i
i=1
(−1)k h γi# ∂gi [[Mi ]], p, y i =
i=1
I
X
(−1)k h ∂γi# gi [[Mi ]], p, y i = (−1)k h ∂S, p, y i.
i=1
In the remaining case where k > m − 1 and S ∈ Lm (X, G), we may take an
arbitrary Lipschitz q : X → Rk−m+1 and use the k = m − 1 case and Theorem
3.8.1(12) to see that,
h ∂hS, p, yi, q, z i = (−1)m−k−1 ∂h hS, p, yi, q, z i = (−1)m−k−1 ∂h S, (p, q), (y, z) i
= (−1)m−k−1+k+m−k−1 h ∂S, (p, q), (y, z) i = h (−1)k h∂S, p, yi, q, z i
for a.e. (y, z) ∈ Rk × Rk−m+1 . Conclusion (6) then follows from Corollary 3.8.3.
Finally to prove (5), it suffices as above to assume that S ∈ Lm (X; R). Here
we have already treated in Theorem 4.2.2(4) the case m = 0. Assume now that
m > 0, and consider any Lipschitz map q : X → Rm . Then, by (6), Theorem
3.8.1(2)(6)(11), Fubini’s Theorem, and Theorem 4.2.2(4) (applied to Q = h S, q, y i),
we have, for a.e. r ∈ R, that, for a.e. y ∈ Rm ,
∂(S {u < r}) − (∂S) {u < r} − h S, u, r i , q , y
= h ∂(S
m
{u < r}), q, y i − h (∂S)
= (−1) ∂h S
m
{u < r}, q, y i − h h S, u, r i, q, yi
{u < r}, q, y i − h (∂S), q, y i
= (−1) [∂ (h S, q, y i
= 0.
{u < r} − h S, (u, q), (r, y) i
{u < r}) − (∂h S, q, y i)
{u < r} − h h S, q, y i, u, r i]
So conclusion (5) now follows from Corollary 3.8.3.
5.3. Flat Chains in Finite Dimensional Spaces. In finite dimensional Euclidean space, Federer and Fleming, [12], Fleming [13], and White [22] have consecutively described groups of flat chains with coefficients in respectively, R or Z, a
finite group, and a general normed abelian group. The consistency of these notions
has been discussed in these works. Here we verify that our group of flat chains
Fm (X; G) defined above in §5.2 also coincides with each of these previous notions.
The definition of flat chain in [W] follows that of W. Fleming [WF] (who was
motivated by Whitney[24]). Here one essentially starts with the formula
E (P ) = inf{M(P − ∂S) + M(S) : S is a polyhedral G chain in Rn } .
(39)
for a polyhedral G−chain P in R . One obtains the group Em (R ; G) of flat chains
as the E -completion of the group Lm (Y ; G) of polyhedral G chains. Various operations, including slicing, carry over, as in Theorem 5.2.3, under this completion.
However note that defining the push-forward of an E flat chain T ∈ Em (Rn ; G) under a Lipschitz map γ to a Banach space must first involve piece-wise affine approximation of the map because one is here completing polyhedral rather than Lipschitz
chains. In fact the proof of this approximation shows that any Lipschitz chain is
n
n
30
THIERRY DE PAUW AND ROBERT HARDT
an E limit of polyhedral chains. Brian White[21] uses the resulting E flat chains in
his beautiful new Deformation Theorem. To show that Em (Rn ; G) = Fm (Rn ; G),
we will first verify the alternate formula for E defined using Lipschitz chains:
E (P ) = inf{M(P − ∂S) + M(S) : S ∈ Lm+1 (Rn ; G)} .
(40)
Here we may, as in the proof of Theorem 5.5.1, verify the nontrivial inequality ≤
by approximating S and P − ∂S in E norm by polyhedral chains. The formula (40)
continues to hold for P itself being a Lipschitz chain, and it will suffice to show
that E is comparable to F on Lm (Rn ; G). First we use the standard “identity”
isomorphism between the Euclidean and `∞ norms:
I : (Rn , | |) → `∞ {1, · · · , n}, Ψ(x1 , · · · , xn )(i) = xi ,
√
which has Lip I = 1, Lip I −1 = 2. Thus, by Theorem 5.2.1(2),
2−
n+1
2
F (P ) ≤ F (I# P ) ≤ F (P ) ,
(41)
and, by (38),
F (I# P ) = inf{M(I# P − ∂Q) + M(Q) : Q ∈ Lm+1 (`∞ {1, · · · , n}; G) } . (42)
On the other hand, by (40) and Theorem 3.8.1(8),
2−
m+1
2
E (P ) ≤ inf{M(I# P −∂Q)+M(Q) : Q ∈ Lm+1 (`∞ {1, · · · , n}; G)} ≤ E (P ),
which, along with (41) and (42), establishes the desired equivalence of E and F on
Lm (Rn ; G). Thus Fm (Rn ; G) = Em (Rn ; G), and we may apply [22, Theorem 3.2].
5.4. The Support of a Flat Chain. For any flat chain T ∈ Fm (X; G), one may
now define, following T. Adams [1, Chapter 5], the closed set
spt T = {a ∈ X : T
B (a, r) 6= 0 for a.e. r > 0 } .
One readily checks that, in case T is a rectifiable chain, as in §3.3, spt T = Clos MT .
Also, for a chain of finite mass as defined below in §6, spt T coincides with the
support of an associated positive measure.
Lemma 5.4.1. For a general flat chain T ∈ Fm (X; G),
spt T = ∅
⇐⇒
T =0.
Proof. By Theorem 5.2.1(2) and Lemma 2.0.1(1), we may assume X is an `∞ space.
Then we may repeat the proof of [1, Lemma 6.7] with polyhedral chains replaced
by Lipschitz chains.
For a Lipschitz map u : X → R, one has, for a.e. r ∈ R, that
spt (T
{x : u(x) < r}) ⊂ (spt T ) ∩ {x : u(x) ≤ r} ,
and
T −T
{u < r} =
∞
X
(Pi − Pi
∞
X
{u < r}) =
i=1
Pi
{u > r} = T
{u > r}.
i=1
In particular, for a.e. r > 0,
T = T
{x : dist(x, spt T ) < r} =
∞
X
Pi
i=1
We find the following alternate characterization:
{x : dist(x, spt T ) < r} .
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
31
spt T is the smallest closed subset KPof X such that, for every δ > 0, there is a
∞
flat convergent representation T = i=1 Pi in terms of Lipschitz chains Pi with
supports in {x : dist(x, K) < δ}.
Lemma 5.4.2. Suppose T ∈ Fm (X; G).
(1) If f, g : X → Y are Lipschitz and f spt T = g spt T , then f# T = g# T .
(2) If p, q : X → Rk are Lipschitz and p spt T = q spt T , then
hT, p, yi = hT, q, yi for a.e. y ∈ Rk .
Proof. By Lemma 2.0.1(1), Y admits an isometric embedding Φ into a Banach
space. By Theorem 5.2.1(2) and Theorem 5.2.4(8), it is sufficient to prove that
(Φ ◦ f )# T = (Φ ◦ g)# T . Thus, we may now assume that Y itself is a Banach
space. For r > 0, we may thus define, exactly as in [10, 4.1.15], the map Gr =
f + Ψr ◦ (g − f ) : X → Y where
Ψr (y) = 0 for kyk ≤ r ,
Ψr (y) = (1 − rkyk)y for kyk ≥ r .
Then Lip Gr ≤ 2 Lip f + Lip g and kGr −Pgk − ∞ ≤ r.
∞
We can choose a representation T = i=1 Pi as in the above alternate characterization with δ = r/(1 + Lip f + Lip g). For every x ∈ spt Pi , we deduce that
kg(x) − f (x)k < r so that Gr (x) = f (x). Since §3.5 implies that f# Pi = Gr# Pi for
each i = 1, 2, · · · , we have from Theorem 5.2.3(1) that the relation
f# T = Gr# T
(43)
is true for all r > 0.
For any ε > 0, we may fix a Lipschitz chain P ∈ Lm (X; G) so that
F (T − P ) < ε/(1 + 2Lip f + Lip g)m+1
which, with Theorem 5.2.3(1), implies both
F (Gr# T − Gr# P ) < ε
and
F (g# P − g# T ) < ε .
(44)
Finally, we may now fix r = (1 + 2Lip f + Lip g)−m [M(P ) + M(∂P )]−1 ε. This will
guarantee that
MH# ([[0, 1]] × P ) < ε
and
MH# ([[0, 1]] × ∂P ) < ε
where H is the affine homotopy between Gr and g. So the homotopy formula turns
these mass bounds into the flat estimate
F (Gr# P − g# P ) < 2 ε .
(45)
Combining (43), (44), and (45) shows that
f# T − g# T = (f# T − Gr# T ) + (Gr# T − Gr# P ) + (Gr# P − g# P ) + (g# P − g# T )
has flat norm less than 0 + ε + 2ε + ε = 4ε, and we let ε ↓ 0 to prove conclusion(1).
For conclusion (2), we similarly define qr = p + Ψr ◦ (q − p) : X → Rk . Choosing
ε as above, we now find that
h T, p, y i = h T, qr , y i for a.e. y ∈ Rk
(46)
because, by §3.7, each hPi , p, yi = hPi , qr , yi for a.e. y. For ε > 0, we fix P
approximating T as above, and now verify that
Z
Z
F h T − P, qr , y i dy < ε and
F h P − T, q, y i dy < ε .
(47)
32
THIERRY DE PAUW AND ROBERT HARDT
The homotopy slice formula,
h P, qr , y i − h P, q, y i = ∂π# h P × [[0, 1]], hr , y i + π# h ∂P × [[0, 1]], hr , y i ,
where hr is the affine homotopy between q and qr and π(x, t) = x, and integration
using Theorem 3.8.1(9), allows us, as in [10, 4.3], to estimate
Z
F h P, qr − q, y i dy ≤ Ckqr − qkk∞ [M(P ) + M(∂P )] .
(48)
Choose r sufficiently small so that this term is less than ε. Then combine with (46)
and (47) and let ε ↓ 0 to conclude that
Z
F (h T, p, y i − h T, q, y i) dy = 0 .
Rk
5.5. Some Relations between Rm (X; G) and Fm (X; G). By the M density
of Lm (X; G) in Rm (X, G), we see that each T ∈ Rm (X, G) defines an element
of Fm (X; G). As in the proof of Theorem 5.2.2, we may use the integral slicing
inequality Theorem 5.2.3(3), Corollary 4.4.2, and Corollary 3.8.3 we deduce that
F (T ) = 0 only if T = 0 in Rm (X; G). Thus
(Rm (X; G), F ) ⊂ (Fm (X; G), F )
is a continuous and injective. We have another formula for the flat norm:
Theorem 5.5.1. For T ∈ Rm (X; G),
F (T ) = inf{M(R) + M(∂S) : Φ# T = R + ∂S, Φ is an isometric embedding of X
into a Banach space Y, R ∈ Rm (Y ; G), S ∈ Rm+1 (Y ; G) }
(49)
Proof. We may assume that T ∈ Lm (X; G). Letting G (T ) denote the righthand
side, we see from our definition in §5.2 that clearly F (T ) ≥ G (T ).
To verify that F (T ) ≤ G (T ), we first choose, for ε > 0, an isometric embedding
Φ of X into a Banach space Y and chains R ∈ Rm (Y ; G), S ∈ Rm+1 (Y ; G) so that
Φ# T = R + ∂S and G (T ) + ε > M(R) + M(∂S).
By (13), we obtain Lipschitz chains
and Qi ∈ Lm+1 (Y ; G) and
P∞Pi ∈ Lm (Y ; G)
P∞
M convergent representations R = i=1 Pi and S = i=1 Qi satisfying
∞
X
M(Pi ) < M(R) + ε and
i=1
∞
X
M(Qi ) < M(S) + ε .
i=1
Thus we may use Theorem 5.2.3(1) and Theorem 5.2.2 to conclude that
F (T ) = F (Φ# T ) = F (R + ∂S) = F
∞
X
Pi + ∂
i=1
= F
∞
X
i=1
≤
∞
X
(Pi + ∂Qi ) ≤
∞
X
∞
X
Qi
i=1
F (Pi + ∂Qi )
i=1
M(Pi ) + M(Qi ) < M(R) + M(S) + 2ε < G (T ) + 3ε .
i=1
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
33
Note that, in formula (49), each chain ∂S = Φ# T − R is rectifiable. The group
Fm (X; G) may alternately be developed as the completion of Rm (X; G) using the
formula (49) for the norm and using Corollary 3.8.3 and the boundary slice formula
of Theorem 5.2.4(6) to define what it means for a chain in Rm (X; G) to be the
boundary of a chain in Rm+1 (X; G).
5.6. A Slice-null Flat Chain is Zero. For m ≥ 1, a flat chain T ∈ Fm (X; G)
is called slice-null if, for every Lipschitz map p : X → Rm , hT, p, yi = 0 for a.e.
y ∈ Rm .
By Corollary 3.8.3, a slice-null rectifiable chain necessarily vanishes. Our goal in
this section is to establish this property for an arbitrary flat chain in an arbitrary
metric space.
Theorem 5.6.1. A flat chain T ∈ F (X; G) is slice-null if and only if T = 0.
Thus a flat chain will be uniquely determined by its zero dimensional slices. This
will be useful in later sections where we obtain rectifiability properties in terms of
such slices.
Before beginning the proof, we note an immediate property of slice-nullity.
Lemma 5.6.2. Suppose T ∈ Fm (X; G).
(1) If φ : X → X̃ is a Lipschitz map and T is slice-null, then φ# T is slice-null.
(2) If ψ : X → X̃ is an isometric embedding and ψ# T is slice-null, then T is
slice-null.
Proof. (1) Suppose T is slice-null, and p̃ : X̃ → Rm is Lipschitz. Then, by Theorem
5.2.4(7),
h φ# T, p̃, y i = φ# h T, p̃ ◦ φ, y i = 0
for a.e. y ∈ Rm . Thus φ# T is slice-null.
(2) Suppose ψ# T is slice-null, and p : X → Rm is Lipschitz. By Lemma 2.0.1(2),
m
−1
may choose a Lipschitz
(which is defined on
√ extension−1q : X̃ → R of p ◦ ψ
ψ(X)) with Lip q ≤ m Lip(p ◦ ψ ). Then, again by Theorem 5.2.4(7),
ψ# hT, p, yi = ψ# hT, q ◦ ψ, yi = hψ# T, q, yi = 0
for a.e. y ∈ Rm . For such y, hT, p, yi = 0 by Theorem 4.3.3(2). Hence T is
slice-null.
Proof of Theorem. The zero chain is clearly slice-null by definition of the slice §3.7.
We will now prove the nullity of a slice-null flat chain by considering cases in
increasing generality.
Case 1. X = Rn with the standard Euclidean norm. This case follow from
White’s result [22, Theorem 3.2] where he only required the vanishing of almost
n
all slices by the m
coordinate projections of Rn onto Rm . Here we use not only
the fact, proven in §5.3, that our flat chains are the same as the flat chains in [22],
but also that, the two notions of slicing also coincide. The formula Theorem 4.1.1
shows they coincide on polyhedral and Lipschitz chains. By flat completions of
either polyhedra or Lipschitz chains, we get the same notions of slicing for general
flat chains.
Case 2. X = Rn with an arbitrary norm. By finite-dimensionality, this new
norm on Rn is comparable to the standard Euclidean norm so that there is a λ > 1
so that
λ−1 |x − y| ≤ dist (x, y) ≤ λ|x − y|
34
THIERRY DE PAUW AND ROBERT HARDT
for all x, y ∈ Rm . Using the subscript Eucl to denote quantities computed with
respect to the standard Euclidean norm, we readily check that
m
m
λ−m HEucl
≤ H m ≤ λm HEucl
λ−m MEucl (P ) ≤ M(P )
λ−(m+1) FEucl (P ) ≤ F (P )
≤ λm MEucl (P ) ,
≤ λm+1 FEucl (P )
for P ∈ Lm (Rn ; G). Thus the group Fm (Rn ; G) of flat chains for the new norm
coincides with the group Fm (Rn ; G)Eucl for the Euclidean norm, and Case 2 reduces
to Case 1.
Case 3. X is a separable Banach space. Here X admits a linear isometric
embedding Ψ into the particular Banach space Z = C ([0, 1]) of continuous functions
on the unit interval equipped with the supremum norm.
For > 0 we may find a Lipschitz chain P ∈ Lm (Z; G) so that F (Ψ# T −P ) < ε.
Since Z = C ([0, 1]) enjoys the metric approximation property, we may choose a
finite rank linear map L : Z → Z so that kLk = 1 and
ε
kL(z) − zk ≤
M(P ) + M(∂P )
for all z in the compact set spt P . This inequality, along with the affine homotopy
H from the identity to L, allows us to estimate, exactly as in [HF,4.1.18],
F (L# P − P ) ≤ M H# ([[0, 1]] × P ) + M H# ([[0, 1]] × ∂P ) < ε .
(50)
With n = rank L, we may choose a linear isometry J between the image of L
and Rn equipped with a suitable norm. Starting again with an arbitrary slice-null
T ∈ Fm (X; G), we infer from Lemma 5.6.2(1) that, in Rn with this suitable norm,
the flat chain J# (L# Ψ# T ) = (J ◦L◦Ψ)# T must also be slice-null and hence vanish
by Case 2. From Theorem 5.2.1(2) we infer that
F (L# Ψ# T ) = F J# (L# Ψ# T ) = 0 .
We may now combine this with (50) and Theorem 5.2.1(4) to deduce
F (T ) = F (Ψ# T ) ≤ F (P ) + ε
≤ F (P − L# P ) + F (L# P − L# Ψ# T ) + F (L# Ψ# T ) + ε
≤ ε + ε + 0 + ε.
Letting ε ↓ 0, we conclude that the slice-null chain T must vanish.
Case 4. X is a separable metric space. Here, we may use an isometric embedding
Φ of X into `∞ (X) as in Lemma 2.0.1(1), observe that the closure X̃ of the linear
span of Φ(X) is a separable Banach space, and note that Φ itself thus defines an
isometry Φ̃ : X → X̃. As before, a slice-null T ∈ Fm (X; G) gives, by Lemma
5.6.2(1), the slice-null chain Φ̃# T ∈ Fm (X̃; G), which must, by Case 3, vanish. So
that F (T ) = F (Φ̃# T ) = 0 by Theorem 5.2.1(2).
Case 5. X is a general metric space. Here, for any T ∈ Fm (X; G), we first
construct an isometric embedding Φ0 of a complete separable metric space X0 into
X and a flat chain T0 ∈ Fm (X0 ; G) so that Φ0# T0 = T . To obtain Φ0 , X0 , T0 ,
choose a sequence of Lipschitz chains Qn ∈ Lm (X; G) that is F convergent to
T . Then the support Xn of Qn is a compact (separable) subset of X. Thus the
closure X0 of ∪∞
n=1 Xn is a closed separable subspace of X. Let Φ0 : X0 → X be the
canonical injection, and note that the formula defining Qn also defines a Lipschitz
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
35
chain Q0,n ∈ Lm (X0 ; G) such that Φ0# Q0,n = Qn . From Theorem 5.2.1(2), we
deduce that
F (Q0,j − Q0,k ) ≤ F (Φ0# (Q0,j − Q0,k )) = F (Qj − Qk ) → 0 as j, k → ∞ .
By completeness of Fm (X0 ; G), the F Cauchy sequence Q0,n has a limit, say
T0 ∈ Fm (X0 ; G). Moreover, by Theorem 5.2.1(4),
F (Φ0# T0 − T ) = lim F (φ0# Q0,n − T ) = lim F (Qn − T ) = 0 .
n→∞
n→∞
Assuming now that T = Φ0# T0 is slice-null, we deduce from Lemma 5.6.2(2)
that T0 ∈ Fm (X0 ; G) is slice-null and so vanishes by Case 4. We conclude that
T = 0 by Theorem 5.2.1(2).
5.7. The Slicing Mass of a Rectifiable Chain and its Comparability to M.
As in [13], [22], and [1], we may define the mass for any 0 dimensional flat chain
T ∈ F0 (X; G) by the formula
M(T ) = lim inf {M(R) : R ∈ R0 (X; G), F (R − T ) < δ} .
δ↓0
(51)
For a 0 dimensional rectifiable chain T ∈ R0 (X; G), this definition is, by the lower
semi-continuity Theorem 4.4.1, consistent with our definition using Hausdorff measure in §3.3
However, for general m, such lower semicontinuity, is unknown in general metric
spaces for the Hausdorff measure mass M of §3.3, even for polyhedral chains in a
finite dimensional Banach space. If lower semi-continuity fails, then the definition
(51) is actually inconsistent with §3.3.
As in [3] and [1], we find it useful to introduce another mass M̂ which is definitely
lower semicontinuous under flat convergence. Fortunately our new mass M̂ is, by
Theorem 5.7.4 below, comparable to M, with bounds depending only on m.
Suppose m ∈ {1, 2, · · · }, U is open in X, and p ∈ Lip1 (X, Rm ), that is p : X →
m
R and Lip p ≤ 1. For any T ∈ Rm (X; G) and µT measurable A ⊂ X, we observe
that the quantity
Z
Z
λT ,U,p (A) :=
µhT ,p,yi (U ∩ A) dy =
M hT, p, yi (U ∩ A) dy .
Rm
Rm
is countably subadditive in A, satisfies Carathéodory’s Criteria [10, 2.32(9)], and
defines a finite Borel regular measure on X. Clearly, λT ,U,p (A) ≤ M(T A), and
λT +T̃ ,U,p ≤ λT ,U,p + λT̃ ,U,p
(52)
by Theorem 3.8.1(1)(2)(3).
Lemma 5.7.1. λT ,U,p (X) ≤ lim inf k→∞ λTk ,U,p (X) whenever T, Tk ∈ Rm (X; G)
and limk→∞ F (Tk − T ) = 0.
Proof. Passing to a subsequence we may assume that
lim λTk ,U,p (X) = lim inf λTk ,U,p (X) and
k→∞
k→∞
∞
X
F (Tk − T ) < ∞ .
k=1
Then, by Theorem 5.2.3(3), limk→∞ F hTk , p, yi − hT, p, yi = 0 for a.e. y ∈ Rm .
For such y, we may, for ε > 0, choose a positive r0 so that µhT,p,yi {uU ≤ r0 } < ε
36
THIERRY DE PAUW AND ROBERT HARDT
where uU (x) = dist(x, X ∼ U ). Using Theorem 4.3.3(2), we have, for a.e. positive
r < r0 , that
lim F hTk , p, yi {uU > r} − hT, p, yi {uU > r} = 0 .
k→∞
so that, by Theorem 4.4.1,
M hT, p, yi
U ) ≤ ε + M hT, p, yi
≤ ε + lim inf M hTk , p, yi
k→∞
{uU > r}
{uU > r} ≤ ε + lim inf M hTk , p, yi
k→∞
U
.
Now let ε ↓ 0, integrate over y ∈ Rm , and apply Fatou’s Lemma to find
Z
λT ,U,p (X) =
M hT, p, yi U ) dy
Rm
Z
≤ lim inf
M hTk , p, yi U dy = lim inf λTk ,U,p (X) .
k→∞
k→∞
Rm
For T ∈ Rm (X; G), we now define the Slicing Mass M̂(T ) = µ̂T (X) using the
measure
µ̂T (A) = sup {
I
X
λT ,Ui ,pi (A) : I < ∞, pi ∈ Lip1 (X, Rm ), and
i=1
Ui are disjoint open sets in X } .
Note that, as above µ̂T is also a finite, Borel regular measure on X. One can check
that µ̂T is, as in [3], the ”supremum” measure
^
µ̂T =
{ λT ,U,p : U is open in X and p ∈ Lip1 (X, Rm ) } .
Since the finite sum of, or the supremum over an arbitrary family of, lower
semicontinuous functions is again lower semicontinuous, we now infer from Lemma
5.7.1 the:
Corollary 5.7.2. The slicing mass M̂ is F lower semicontinuous on Rm (X; G).
Next note that the above definitions give the integral slicing inequality
Z
Z
MhT A, p, yi dy =
M(hT, p, yi A) dy = λT,p,X (A) ≤ µ̂T (A)
Rm
(53)
Rm
for any p ∈ Lip1 (X, Rm ) and µ̂T measurable A ⊂ X.
Using (52), we also see that
µ̂T +T̃ ≤ µ̂T + µ̂T̃
and M̂(T + T̃ ) ≤ M̂(T ) + M̂(T̃ ) ,
(54)
and deduce from Corollary 3.8.3 that the slicing mass M̂ is a norm on the group
Rm (X; G).
First we have, from Theorem 3.8.1(9) and Fatou’s Lemma, the inequality
M̂ ≤ M
on Rm (X; G) ,
(55)
(where the mass M was originally defined in §3.1 using Hausdorff measure). To see
that these two norms are actually comparable, we start with the elementary:
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
37
Lemma 5.7.3. For any m dimensional normed vectorspace (V, k k ) there is a linear
map L : (V, k k ) → (Rm , | | ) such that
Lip L ≤ 1
Lip L−1 ≤ 2m .
and
Proof. We may assume that V = Rm equipped with some norm k k , and we wish to
compare the ordinary Euclidean unit ball B with linear images L(B) of the k k ball
B = {v ∈ Rm : kvk < 1}. We do this by finding an orthonormal basis w1 , · · · , wm
of Rm and positive numbers R1 ≤ · · · ≤ Rm so that
{±
Rm
R1
w1 · · · , ±
wm } ⊂ B ⊂
2
2
Then by choosing L so that L(wi ) =
m
\
L(B) ⊂ L
Ri
{x : |x · wi | ≤ Ri }
i=1
1
√
m
m
\
{x : |x · wi | ≤ Ri } .
i=1
wi for i = 1, · · · , m, we find that
=
m
\
1
{x : |x · wi | ≤ √ } ⊂ B
m
i=1
and
L(B) ⊃ L convex hull{
±wi
±Ri wi 1 } = convex hull{ √ } ⊃ B 0,
,
2
2m
2 m
which will complete the proof.
We choose wi and Ri by induction on m. The case m = 1 is easy. To obtain the
inductive step, we first choose vm ∈ ∂B to maximize Rm := max{ kvk : v ∈ ∂B}
and let wm = vm /|vm |. Then we choose w ∈ ∂B to be a maximizer of
Rm−1 := max{ |v||πm (v)| : v ∈ ∂B} ,
(56)
⊥
. Then 0 < Rm−1 ≤ Rm and w 6= ±wm .
where πm (v) = v − (wm · v)wm ∈ wm
Since B is a convex set, the quantity kvk|πm (v)| is a concave function of v · wm as
v varies on the convex loop
{v ∈ ∂B : πm (v) ∧ πm (wm ) = 0} .
Moreover, this function, having maximum value Rm−1 at v = w and vanishing when
v · wm = ±Rm , must have value in the range [ 21 Rm−1 , Rm−1 ] when v · wm = 0.
⊥
This means that any vector vm−1 maximizing | | in ∂B ∩ wm
must have
1
Rm−1 ≤ kvm−1 k ≤ Rm−1 .
2
⊥
, we let wm−1 = vm−1 /|vm−1 | and
Next working inside the hyperplane wm
⊥
Rm−2 := max{ |v||πm−1 (v)| : v ∈ wm
∩ ∂B} ,
⊥
⊥
where πm−1 is the orthogonal projection onto wm
∩ wm−1
. We find that a vector
⊥
⊥
vm−2 maximizing | | in ∂B ∩ wm ∩ wm−1 has kvm−2 k in [ 12 Rm−2 , Rm−2 ] and
let wm−2 = vm−2 /|vm−2 |. Continuing inductively we eventually obtain Rm ≥
Rm−1 ≥ Rm−2 ≥ · · · ≥ R1 > 0 and orthonormal wm , wm−1 , wm−2 , · · · , w1 with the
desired properties.
Theorem 5.7.4.
(2m)−m M(T ) ≤ M̂(T ) ≤ M(T )
for any T ∈ Rm (X; G).
38
THIERRY DE PAUW AND ROBERT HARDT
Proof. The second inequality was verified in (55). To prove the first one, we recall
from Lemma 4 of [17] that we may, in the parametric representation from §3.3 of
T , assume, for any λ > 1 and i ∈ {1, 2, · · · }, there is a norm k ki on Rm so that
1
kx − yki ≤ dist γi (x), γi (y) ≤ λ kx − yki
λ
for x, y ∈ Ki .
Then we choose, by Lemma 5.7.3, a linear map Li : (Rm , k ki ) → (Rm , | | ) such
that Lip Li ≤ 1 and Lip L−1
≤ 2m.
i
PI
For ε > 0, we find in §3.3 that M(T ) − i=1 M T γi (Ki ) < ε provided I is
large enough. We can then choose disjoint open sets Ui ⊃ γi (Ki ). Observing that
Lip (Li ◦ γi−1 ) ≤ 2mλ, we let pi : X → Rm be a Lipschitz extension of Li ◦ γi−1 .
From §3.7 we see that for Ui sufficiently close to γi (Ki ),
µT ,Ui ,pi >
1 m
M(T
2mλ2
Ui ) .
Thus,
−ε + M(T ) <
I
X
M(T
Ui ) < (2mλ2 )m
I
X
µT ,Ui ,pi ≤ (2mλ2 )m M̂(T ) ,
i=1
i=1
and we may let ε ↓ 0 and λ ↓ 1.
Remark 5.7.5. For T ∈ Rm (X; G),
M̂(T ) = M(T )
and µ̂T = µT
if either m = 0 or m = 1 or X is a Hilbert space.
In fact, for m = 0, the slice of a chain T ∈ R0 (x; G) by a constant map is T ,
and for m = 1, Lemma 5.7.3 is true with 2m replaced by 1. If X is a Hilbert
space,R and T has support in an m dimensional subspace W , then one readily checks
that Rm MhT, p, yi dy = M(T ) where p = q ◦ π with q : W → Rm being an
isometry and π being the nearest point projection of X onto W . It follows that
µ̂T = µT in case T is a polyhedral chain. The general case now follows using a finite
dimensional approximation of X and the strong polyhedral approximation theorem
of [10, 4.2.20].
6. The Slicing Mass of a Flat Chain
Now we may proceed as in equation (51) defining the slicing mass for any flat
chain T ∈ Fm (X; G)
M̂(T ) = lim inf{M̂(R) : R ∈ Rm (X; G), F (R − T ) < δ} ,
δ↓0
and note, by the lower semi-continuity of Corollary 5.7.2, that this is consistent
with the definition of M̂ for T ∈ Rm (X; G).
In case M̂(T ) < ∞, we say simply that T has finite mass. Also the situation
M̂(T ) = 0 implies that T = 0 because then M(Rk ) → 0 for some sequence Rk in
Rm (X; G) with F (Rk − T ) < k1 and hence
F (T ) = lim F (Rk ) ≤ lim M(Rk ) ≤ lim (2m)m M̂(Rk ) = 0 ,
k→∞
k→∞
k→∞
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
39
by (49) and Theorem 5.7.4. Putting these facts together with (54), we now conclude
that M̂ defines a complete norm on the group
Mm (X; G) = {T ∈ Fm (X; G) : M̂(T ) < ∞}
of flat chains of finite mass. In §8.2 below we show that Mm (X; G) = Rm (X; G)
in case the group G does not contain any nonconstant Lipschitz curves.)
Lemma 6.0.6. The slicing mass M̂ is F lower semicontinuous on Mm (X; G).
Proof. Assuming Ti , T ∈ Mm (X; G), limi→∞ F (Ti − T ) = 0 and ε > 0, we first
choose δ > 0 so that M̂(T ) < M̂(R)+ 3ε whenever R ∈ Rm (X; G) and F (T −R) < δ.
Second choose j so that M̂(Tj ) < lim inf i→∞ M̂(Ti ) + 3ε and F (Tj − T ) < 2δ . Third
choose R ∈ Rm (X; G) so that M̂(R) < M̂(Tj ) + 3ε and F (R − Tj ) < 2δ . Since
F (T − R) < δ, we conclude
M̂(T ) < M̂(R) +
2ε
ε
< M̂(Tj ) +
< lim inf M̂(Ti ) + ε .
i→∞
3
3
Lemma 6.0.7. For any flat chainPT ∈ Fm (X; G) of finite mass, we may, in the
∞
F convergent representation T = i=1 Pi of Theorem 5.2.3(3) insist that
M̂(T ) = lim M̂(P I )
I→∞
where P I =
I
X
Pi .
i=1
and that (see Theorem 5.2.3(3)), for any Lipschitz u : X → R,
{x : u(x) < r}) = lim M̂ P I
M̂ (T
I→∞
{x : u(x) < r}
for a.e. r > 0.
Proof. For each positive integer I, choose RI ∈ Rm (X; G) so that F (RI −T ) < 2−I
and
M̂(RI ) < 2−I + inf{M̂(R) : R ∈ Rm (X; G), F (R − T ) < 2−I } .
(57)
Then choose P I ∈ Lm (X; G) so that M̂(P I − RI ) ≤ M(P I − RI ) < 2−I , let
PI
P1 = P 1 , and let Pi+1 = P i+1 − P i so that P I = i=1 Pi . Then
∞
X
F (Pi ) ≤
i=2
≤
∞
X
(F (P i − Ri ) + F (Ri − T ) + F (T − Ri−1 ) + F (Ri−1 − P i−1 )
i=2
∞
X
2−i + 2−i + 2−i+1 + 2−i+1
< ∞.
i=2
Similarly, as I → ∞,
!
I
X
F T−
Pi = F (T − P I ) ≤ F (T − RI ) + F (RI − P I ) ≤ 2 · 2−I → 0 .
i=1
Moreover, by (57),
lim M̂(P I ) = lim M̂(RI ) = M̂(T ) .
I→∞
I→∞
(58)
40
THIERRY DE PAUW AND ROBERT HARDT
Next, for any Lipschitz u : X → R the set
D =
∞
[
r ∈ R : µ̂P I (u−1 {r}) > 0
I=1
is countable, and, for a.e. r ∈
/ D, we let where Ur = {u < r} and Vr = {u > r} and
infer from the absolute flat convergence of three series that
T =
∞
X
Pi =
i=1
∞
X
(Pi
Ur + Pi
Vr ) = T
Ur + T
Vr .
(59)
i=1
We claim that
M̂(T ) = M̂(T
Ur ) + M̂(T
Vr ) .
(60)
for a.e. positive r. Since, for each positive integer j, the number of points in the
set
1
Cj =
r ∈ R : lim lim inf µ̂P I u−1 ( [r − ε, r + ε] ) >
ε→0 I→∞
j
is finite, bounded by j supi M̂(P I ), the “concentration” set C = ∪∞
j=1 Cj is also
countable. Now for a.e. positive r ∈
/ C ∪ D and any δ > 0, we find that, forε < δ
sufficiently small there is a subsequence I 0 so that µ̂P I 0 u−1 ( [r − ε, r + ε] ) < δ.
For almost every such ε, the chain
Tε := T − T
Ur−ε − T
Vr+ε
is, by Theorem 5.2.3(3), the sum of three F convergent series which we may use
to estimate
!
∞
∞
∞
X
X
X
Pi Vr+ε
M̂(Tε ) = M̂
Pi −
Pi Ur−ε −
i=1
∞
X
= M̂
i=1
i=1
!
Pi − Pi
Ur−ε − Pi
Vr+ε
(61)
i=1
0
≤ lim
inf M̂ P I − P I
0
I →∞
≤ lim sup µ̂P I 0
0
0
Ur−ε − P I
Vr+ε
u−1 ( [r − ε, r + ε] ) < δ .
I 0 →∞
Similarly, we verify that
M̂(T
Ur − T
Ur−ε ) + M̂(T
Vr − T
Vr+ε ) < δ .
(62)
For a.e. such r, ε, we see that
M̂(T
Ur−ε ) + M̂(T
Vr+ε ) ≤ lim inf M̂(P I
I→∞
Ur−ε ) + lim inf M̂(P I
I→∞
Vr+ε )
≤ lim M̂(P I ) = M̂(T )
I→∞
= M̂ (T
≤ M̂(T
Ur−ε + T
Ur−ε ) + M̂(T
Vr+ε + Tε )
Vr+ε ) + M̂(Tε ) .
Letting δ, and hence ε, approach zero, and using (61) and (62), this inequality
becomes an equality and gives equation (60).
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
41
For a.e. positive r, we now deduce from (60) and (58) that
M̂(T ) = M̂(T
Ur ) + M̂(T
≤ lim inf M̂(P
I
I→∞
≤ lim sup M̂(P I
Vr )
Ur ) + lim inf M̂(P I Vr )
I→∞
Ur ) + lim inf M̂(P I ) − M̂(P I
I→∞
I→∞
= lim sup M̂(P
I
Ur )
Ur ) + lim M̂(P I ) − lim sup M̂(P I
I→∞
I→∞
Ur ) = M̂(T ) ,
I→∞
which implies the desired conclusion, M̂(T
Ur ) = limI→∞ M̂(P I
Ur ).
From Lemma 6.0.7 and the proof of Theorem 5.7.4 we readily deduce, for T ∈
Fm (X; G), the inequalities
M̂(T ) ≤ lim inf{M(R) : R ∈ Rm (X; G), F (R − T ) < δ} ≤ (2m)m M̂(T ) .
δ↓0
6.1. The Borel Regular Measure µ̂T . Here for T ∈ Mm (X; G) and any Borel
subset A of X, we will define the “restriction” T A ∈ Mm (X; G) so that the association A 7−→ M̂(T A) give a positive Borel regular measure µ̂T . The restriction
T A will then be defined for a general µ̂T measurable subset A of X.
For a.e. r <
we have, as in the proof of Theorem 5.2.3(2), the F convergence
Ps,
∞
of the series i=1 Pi {r < u < s} to a flat chain T {r < u < s} which also
equals
T {u < s} − T {u < r} = T {u > r} − T {u > s} .
Moreover, M̂(T
M̂(T
{r < u < s}) equals
{u < s}) − M̂(T
{u < r}) = M̂(T
{u > r}) − M̂(T
{u > s}) .
This readily gives, for a.e. r < s < t, the additivity
M̂(T
{r < u < t}) = M̂(T
{r < u < s}) + M̂(T
{s < u < t}) .
(63)
For any open subset U of X, we apply the above discussion to the Lipschitz
function uU (x) = dist(x, X ∼ U ). In particular, for a.e. s1 > s2 > · · · > si → 0,
one has, by (63) the convergence
∞
X
M̂(T
{sj+1 < u < sj }) = lim
J→∞
j=1
J
X
{sj+1 < u < sj }) ≤ M̂(T ) < ∞ .
M̂(T
j=1
Thus for a.e. 0 < r < s < sJ ,
M̂(T
{uU > r} − T
{uU > s}) = M̂(T
≤
∞
X
{s > uU > r})
M̂(T
{sj+1 < u < sj }) → 0
j=J
as J → ∞. The directed family T {uU > r}, corresponding to such r, is thus M
(and hence F ) Cauchy in Fm (X; G), and we let
T
U ∈ Fm (X; G)
denote the M̂ essential limit as r ↓ 0 of the chains T {uU > r}. Here the essential
limit (denoted ess − limr↓0 ) corresponds to an ordinary limit taken over some set
of almost all positive numbers approaching 0.
42
THIERRY DE PAUW AND ROBERT HARDT
The notation T U is consistent with that of Theorem 5.2.3(2). In fact, in the
special case U = {u < r} as before, the proof of Lemma 6.0.7 shows that, for
a.e. r, the chains T {uU > s} mass converge, as s essentially approaches 0, to
T {u < r} as defined before because
lim lim inf M̂(P I
{s ≥ uU ≥ 0}) = 0 .
s↓0 I→∞
We now define
µ̂T (U ) = M̂(T
U) ,
(64)
and observe that
µ̂T (U ) ≤
∞
X
∞
[
µ̂T (Ui ) whenever Ui are open in X and U ⊂
i=1
Ui
(65)
i=1
because
M̂(P I
{uU > s}) ≤ M̂(P I
{u∪∞
U
i=1 i
> s}) ≤
∞
X
M̂(P I
{uUi > s})
i=1
for almost all s and all I. Similarly we deduce from Lemma 6.0.7 that, for disjoint
open U, Ũ ⊂ X,
T
(U ∪ Ũ ) = T
Ũ , hence, µ̂T (U ∪ Ũ ) = µ̂T (U ) + µ̂T (Ũ ) .
U +T
(66)
because uU ∪Ũ = uU + uŨ = uU ∪Ũ charU + uU ∪Ũ charŨ so that
PI
{uU ∪Ũ > s} = P I
{uU > s} + P I
{uŨ > s}
for almost all s and all I. Another easy consequence of the definition (64) and
Lemma 6.0.6 is the lower semicontinuity
U ) ≤ ess − lim inf M̂(T
µ̂T (U ) = M̂(T
r↓0
≤ ess − lim inf lim inf M̂(Tk
r↓0
≤ lim inf M̂(Tk
k→∞
k→∞
{uU > r}
{uU > r}
(67)
U ) = lim inf µ̂Tk (U ) .
k→∞
whenever Tk ∈ Mm (X; g) and limk→∞ F (Tk − T ) = 0.
For any subset E of X we now define
µ̂T (E) = inf{ µ̂T (U ) : U ⊃ E and U is open in X } .
Clearly µ̂T (∅) = 0 and (65) readily implies that
µ̂T (E) ≤
∞
X
i=1
µ̂T (Ei ) whenever E ⊂
∞
[
Ei ⊂ X .
(68)
i=1
Moreover,
µ̂T (E ∪ Ẽ) = µ̂T (E) + µ̂T (Ẽ) for E, Ẽ ⊂ X with δ = dist(E, Ẽ) > 0 .
(69)
In fact, for ε > 0, we choose an open V ⊃ E ∪ Ẽ with µT (V ) ≤ µT (E ∪ Ẽ) + ε, let
U = V ∩ {x : dist(x, E) < 21 δ} , Ũ = V ∩ {x : dist(x, Ẽ) < 12 δ} ,
observe by (66) that
µ̂T (E) + µ̂T (Ẽ) ≤ µ̂T (U ) + µ̂T (Ũ ) = µ̂T (U ∪ Ũ ) ≤ µ̂T (V )
≤ µ̂T (E ∪ Ẽ) + ε ≤ µ̂T (E) + µ̂T (Ẽ) + ε ,
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
43
and let ε → 0.
Theorem 6.1.1. For any T ∈ Mm (X; G), µ̂T is a finite Borel regular measure on
X.
Proof. Combine (68), (69), and [10, 2.12,2.32(9),2.23].
6.2. The Restriction of a Finite Mass T to a µ̂T Measurable Set A.
Lemma 6.2.1. If T ∈ Mm (X; G) and U ⊂ V are open subsets of X, then
V −T
M̂(T
V ) − M̂(T
U ) = M̂(T
U ) = µT (V ∼ U ) .
I
Proof. Choosing P as in Lemma 6.0.7, we see that for a.e. s > 0,
{uU > s} = F − lim P I
{uV > s} − T
T
{uV > s} − P I
I→∞
{uU > s} .
Thus,
M̂(T
V ) − M̂(T
U ) ≤ M̂(T
{uV > s} − T
= ess- lim M̂ T
s↓0
≤ ess- lim lim inf M̂ P I
U)
{uU > s}
{uV > s} − P I
I→∞
s↓0
V −T
{uU > s}
= ess- lim lim M̂(P I
s↓0 I→∞
= ess- lim M̂ T
s↓0
= M̂(T
{uV > s}) − ess- lim lim M̂(P I
s↓0 I→∞
{uV > s} − ess- lim M̂ T {uU > s}
{uU > s}
s↓0
V ) − M̂(T
U ) = µ̂T (V ) − µ̂T (U ) = µ̂T (V ∼ U ) .
Theorem 6.2.2. For any T ∈ Mm (X; G) and µ̂T measurable A ⊂ X, there is a
unique chain
T A ∈ Mm (X; G) such that lim M̂ T Ui − T A = 0
i→∞
for any sequence of open sets Ui ⊃ A such that limi→∞ µT (Ui ∼ A) = 0. Also,
M̂(T
A) = lim M̂(T
i→∞
Ui ) = lim µ̂T (Ui ) = µ̂T (A) .
i→∞
Proof. From Lemma 6.2.1 we deduce that
Ui − T
M̂(T
≤ M̂ (T
Uj )
Ui − T
(Ui ∩ Uj )) + M̂ (T
Uj − T
(Ui ∩ Uj ))
(70)
≤ µ̂T (Ui ∼ Uj ) + µ̂T (Uj ∼ Ui ) ≤ µ̂T (Ui ∼ A) + µ̂T (Uj ∼ A) ,
and the theorem follows from the M̂ completeness of Mm (X; G).
Lemma 6.2.3. For any T ∈ Mm (X; G) and open set U ⊂ X,
T = T
U + T
(X ∼ U ) and M̂(T ) = M̂(T
U ) + M̂ (T
(X ∼ U )) .
Proof. Recall from (59) and (60) that, for a.e. positive s,
T = T
{uU < s} + T
{uU > s}
(71)
and
M̂(T ) = M̂(T
{uU < s}) + M̂(T
{uU > s}) .
(72)
44
THIERRY DE PAUW AND ROBERT HARDT
As s essentially converges to 0, the chains T {uU < s} mass converge to T U by
Lemma 6.2.1 because µ̂T ({0 < uU < s}) → 0, and the chains T {uU > s} mass
converge to T (X ∼ U ) by Theorem 6.2.2 because µ̂T ({uU > s} ∼ (X ∼ U )) → 0.
Taking this limit in (71) and (72) now gives the Lemma.
Corollary 6.2.4. For any T ∈ Mm (X; G) and µ̂T measurable set A ⊂ X,
lim M̂ T Ki − T A = 0
i→∞
for any sequence of closed sets Ki ⊂ A such that limi→∞ µT (A ∼ Ki ) = 0.
Proof. With the help of Lemma 6.2.3, we may apply Theorem 6.2.2 with A and Ui
replaced by X ∼ A and X ∼ Ki .
Theorem 6.2.5. For any T ∈ Mm (X; G) and disjoint µ̂T measurable sets Ai ⊂ X,
the series
∞
∞
[
X
Ai .
T Ai is M̂ convergent to T
i=1
i=1
P∞
P∞
Proof. Since i=1 M̂(T Ai ) = i=1 µT (Ai ) = µT ∪∞
i=1 Ai ≤ M̂(T ), the series is
M̂ convergent.
For ε > 0, choose, by Corollary 6.2.4, a closed Ki ⊂ Ai so that µT (Ai ∼ Ki ) <
2−i ε. Then since the Ki are disjoint closed sets, we may also choose, by Theorem
6.2.2, disjoint open sets Ui ⊃ Ki so that µT (Ui ∼ Ki ) < 2−i ε.
For these disjoint open sets, we infer from equation (66) that
!
∞
∞
X
[
Ui
M̂
T Ui − T
i=1
≤ M̂
i=1
I
X
!
T
∪Ii=1
Ui − T
Ui
+ M̂
i=1
≤ 0 + 2
∞
X
T
Ui
+ M̂ T
i=I+1
∞
X
µT (Ui ) ≤ 2
i=I+1
∞
X
∞
[
!
!
Ui
i=I+1
µT (Ai ) + 2−i → 0 as I → ∞ .
i=I+1
Letting j → ∞ in (70), we have
M̂(T
M̂ T
∞
[
Ui − T
i=1
Ui − T
∞
[
Ki ) ≤ µ̂T (Ui ∼ Ki ) < 2−i ε ,
!
≤ µ̂T
Ki
i=1
∞
[
!
Ui ∼ Ki
=
i=1
∞
X
µ̂T (Ui ∼ Ki ) < ε .
i=1
Similarly, using Lemma 6.2.3 as in the proof of Corollary 6.2.4,
M̂(T
M̂ T
∞
[
i=1
Ai − T
Ai − T
∞
[
i=1
Ki ) ≤ µ̂T (Ai ∼ Ki ) < 2−i ε ,
!
Ki
≤ µ̂T
∞
[
i=1
!
Ai ∼ Ki
=
∞
X
i=1
µ̂T (Ai ∼ Ki ) < ε .
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
45
Combining all of the above gives
M̂
≤
∞
X
i=1
∞
X
Ai − T
T
∞
[
!
Ai
i=1
Ai − T
M̂(T
∞
X
Ki ) +
+ M̂
M̂(T
Ui − T
Ki )
i=1
i=1
∞
X
Ui − T
T
i=1
+ M̂ T
∞
[
Ui − T
i=1
∞
[
i=1
∞
[
!
Ui
∞
[
!
Ki
+ M̂ T
i=1
Ki − T
i=1
∞
[
!
Ai
i=1
≤ ε + ε + 0 + ε + ε → 0 as ε → 0 .
7. The Group M0 (X; G) of 0 Dimensional Flat Chains of Finite Mass
Lemma 7.0.6. For any T ∈ M0 (X; G),
F (T ) ≤ kχ(T )k + diam(spt µT ) M̂(T ) .
Proof. Letting u(x) = dist(x, sptµT ), we may, by (59) and (60), find an an arbitrarily small positive ε so that M̂(T {u > ε}) = µT ({u > ε}) = 0 and
T = T {u < ε}. Replacing Pi by Pi {u < ε} in Lemma 6.0.7, we may write
PJ
P I = j=1 gj [[aj ]] where now dist(aj , sptµT ) < ε. For any isometric embedding Φ
of X into a Banach space Y and any point a ∈ spt µT , we may, as in [22, Theorem
2.1(4)], use the formula
Φ# P I = χ(P I )[[Φ(a)]] + ∂
J
X
gj [[Φ(aj ), Φ(a)]]
j=1
to estimate F (P I ) ≤ kχ(T )k + (ε + diam(spt µT )) M(T ), and then let I → ∞ and
ε → 0.
Lemma 7.0.7. For any T ∈ M0 (X; G) and a ∈ X, T
Proof. Since the chain T̃ = T {a} − χ T A) [[a]]
diam(spt µT̃ ) = 0, T̃ = 0 by Lemma 7.0.6.
{a} = χ(T
{a}) [[ a ]] .
has both χ(T̃ ) = 0 and
7.1. The G−valued Borel Measure ΨT . Let AT denote the σ algebra of µT
measurable subsets of X.
Theorem 7.1.1. For each T ∈ M0 (X; G), the set function
ΨT : AT → G ,
ΨT (A) = χ(T
A),
is a G−valued Borel measure on X whose total variation measure
( ∞
)
X
kΨT k(A) := sup
kΨT (Ei )k : {Ei } is a µT measurable partition of A
i=1
(73)
equals µT (A) for A ∈ AT . Conversely, any G−valued Borel measure on X of finite
total variation coincides with ΨT for some unique T ∈ M0 (X; G).
46
THIERRY DE PAUW AND ROBERT HARDT
Proof. Clearly χ(T ∅) = χ(0) = 0G . Suppose again that the Ai are disjoint µT
measurable subsets of X. Since, by 5.2.3(1),
!
!
∞
∞
∞
X
X
X
T Ai ≤ M̂
T Ai ≤
µT (Ai ) → 0 as I → ∞ ,
χ
i=I+1
i=I+1
i=I+1
Theorem 6.2.5 and the finite additivity of χ imply that
!
!
∞
∞
I
[
X
X
Ai = χ
T Ai =
χ(T Ai ) + χ
χ T
i=1
→
i=1
∞
X
i=1
Ai )
χ(T
as
∞
X
!
T
Ai
i=I+1
I→∞
i=1
∞
so that ΨT (∪∞
i=1 Ai ) = Σi=1 ΨT (Ai ). Also, for (73),
kΨT k ≤ µT
because
∞
X
kΨT (Ei )k ≤
i=1
∞
X
µT (Ei ) = µT (A) .
(74)
i=1
For the remainder of the proof, we modify the arguments of [22, Theorem 2.2] by
using an isometric embedding Φ of X into `∞ (X), as in Lemma 2.0.1(1).
For each positive integer k and function ` : X → Z, the `∞ (X) “half-open” cube
h `(x) `(x) + 1 ,
for x ∈ X }
Qk,` = { f : X → R : f (x) ∈
2k
2k
has diameter 2−k . Since these form a µT measurable (possibly uncountable) partition of `∞ (X) the family
Qk = { Φ−1 (Qk,` ) : µT Φ−1 (Qk,` ) > 0 }
is at most countable. For each Q ∈ Qk , weP
choose a point aQ ∈ Q ∩ spt µT and
observe, as in[22, Theorem 2.2] , that Tk = Q∈Qk χ(T Q)[[aQ ]] → T in F as
k → ∞. By lower semicontinuity,
X
µT (X) = M̂(T ) ≤ lim inf M̂(Tk ) =
kχ(T Q)k ≤ kΨT k(X) .
k→∞
Q∈Qk
Combining this with (74) gives, for any µT measurable A ⊂ X the opposite inequality,
µT (A) = µT (X) − µT (X ∼ A) ≤ kΨT k(X) − kΨT k(X ∼ A) = kΨT k(A) ,
and thus the conclusion that kΨT k = µT .
It follows that the correspondence T 7→ ΨT is injective because
M̂(T1 − T2 ) = |ΨT1 −T2 |(X) = |ΨT1 − ΨT2 |(X) for T1 , T2 ∈ M0 (X; G) .
Finally for the surjectivity, we assume that ψ̃ is an arbitrary G−valued Borel
measure on X of finite total variation. Then, as before, the family
Q˜k = { Φ−1 (Qk,` ) : |Ψ̃| Φ−1 (Qk,` ) i 0 }
is at most countable. For each Q ∈ Q˜k , we choose a point ãQ ∈ Q ∩ spt µT and
P
verify, as in [22, Theorem 2.2], that T̃k = Q∈Q˜k χ(T Q)[[ãQ ]] is F Cauchy
convergent, as k → ∞, to some T ∈ M0 (X; G) and that ΨT = Ψ̃.
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
47
7.2. Rectifiability. In [22], B.White proved that, in Rn , all flat G chains of finite mass are rectifiable if and only if G contains no nonconstant Lipschitz curve.
This condition fails for G = R and the rectifiability result is false; for example, the
P2n
weighted sums of point masses, i=0 2−n [[2−n i]], converge, as n → ∞, to a nonrectifiable, mass one, 0 dimensional flat chain. However, the “no-Lipschitz-curve”
condition is verified by any finite or discrete abelian group (e.g. Z or Z/jZ) as well
as the p−adic numbers or (R, | · |α ) with 0 < α < 1. In Theorem 7.2.2 and Theorem
8.2.3 below, we will generalize White’s result to hold in a general complete metric
space. First we need some properties of a Borel measure on X without “atoms”.
Theorem 7.2.1. Suppose that µ a finite positive Borel measure on X such that
µ({x}) = 0 for every x ∈ X.
(1) For each µ measurable A ⊂ X and each 0 < r < 1, there exists a µ
measurable subset Ar of A with µ(Ar ) = r µ(A).
(2) There exists a µ measurable function f : X → [0, 1] such that µ(f −1 ){t} = 0
for every t ∈ [0, 1].
Proof. Conclusion (1) follows from §8.20 of Mattila’s book [19]. Though the statement in [19] assumes µ to be a Radon measure, the proof, using Zorn’s lemma,
works for a general Borel measure without atoms.
To verify conclusion (2), we construct a sequence A0 , A1 , A2 , · · · of partitions
of X, as follows. Starting with A0 = {X}, we inductively define, for j = 0, 1, 2, · · · ,
Aj+1 = {A1/2 , A ∼ A1/2 : A ∈ Aj }
where A1/2 is, as in conclusion (1), a Borel measurable subset of A with µ measure
equal to 21 µ(A). Using these, we then construct a sequence f0 , f1 , f2 · · · of µ
measurable functions from X to [0, 1] as follows. Starting with f0 being the constant
function identically 1, we define inductively, for each j = 0, 1, 2, · · · ,
(
fj (x)
for x ∈ A1/2 and A ∈ Aj ,
fj+1 (x) =
j+1
fj (x) − 2
for x ∈ A ∼ A1/2 and A ∈ Aj .
It is plain that the sequence fj is nonincreasing in j, and we let f = inf j fj .
It remains to verify that each level set f −1 {t}, for t ∈ [0, 1], has µ measure zero.
For each positive integer k, there exists a unique integer `k such that
`k
1 + `k
≤ t <
.
2k
2k
Then, for each x ∈ f −1 {t} we may choose some integer j(x) ≥ k for which
t = f (x) ≤ fj(x) (x) < (1 + `k )2−k . Therefore
∞
h `
[
1 + `k k
f −1 {t} ⊂
fj−1
,
.
2k
2k
j=k
We also notice that our construction of the fj ’s implies that, for all j ≥ k, the sets
h `
1 + `k k
fj−1 k ,
2
2k
all coincide with one single set Bk ∈ Ak . Whence
µ f −1 {t} ≤ µ(Bk ) = 2−k µ(X) .
Letting k → ∞ completes the proof.
48
THIERRY DE PAUW AND ROBERT HARDT
Theorem 7.2.2. If G is a complete normed abelian group that does not contain
any nonconstant Lipschitz curve, then each T ∈ M0 (X; G) belongs to R0 (X; G).
Proof. With Theorem 7.2.1(2), we may now follow the idea of [22, Theorem 7.1]
which we repeat here for the reader’s convenience. Since µT (X) = M(T ) < ∞, the
set A = {a ∈ X : µT ({a}) > 0} of atoms of µT is at most countable, and
X
χ(T {a})[[ a ]] ∈ R0 (X; G) ,
T A =
a∈A
by Lemma 7.0.7 and Theorem 6.2.5. Thus it suffices to show that T̃ = T − T A
vanishes. If not, then we may, by Theorem 7.1.1, choose a Borel subset E with
kΨT̃ (E)k > 0 and hence ΨT̃ (E) 6= 0G . Since µT̃ has no atoms, we may apply Theorem 7.2.1(2) with X, µ replaced by E, µT̃ to choose a Borel measurable function
f : E → [0, 1] having µT̃ f −1 {t} = 0 for all t ∈ [0, 1].
It follows that the map
γ : [0, 1] → G , γ(t) = ΨT̃ f −1 [0, t]
for t ∈ [0, 1] ,
is continuous because, as 0 ≤ s ↑ t ≤ 1,
kγ(t) − γ(s)k = kΨT̃ f −1 (s, t] ≤ µT̃ f −1 (s, t] → µT̃ f −1 {t} = 0 ,
and similarly, lims↓t kγ(s) − γ(t)k = 0. We also see that γ is nonconstant because
γ(1) = ΨT̃ (E) 6= 0G while γ(0) = ΨT̃ f −1 {0} = 0G
because kΨT̃ k f −1 {0} = µT̃ f −1 {0} = 0. Finally, γ has finite length because,
for any 0 = t0 < t1 < · · · < tI = 1,
I
X
kγ(ti ) − γ(ti−1 )k ≤
i=1
I
X
µT̃ f −1 (ti−1 , ti ]
= µT̃ (E) ≤ M(T ) < ∞ .
i=1
Reparameterizing γ by arc-length thus gives a nonconstant Lipschitz curve in G,
contradicting the hypothesis and showing that T̃ = 0 and that T = T A is, in
fact, rectifiable.
8. The Group Mm (X; G) of m Dimensional Flat Chains of Finite Mass
Lemma 8.0.3. If T ∈ Mm (X; G) and p : X → Rk is Lipschitz with k ≤ m, then,
Z
M̂h T, p, y i dy ≤ (2m)m C(Lip p)k M̂(T ) .
(75)
Rk
For any µ̂T measurable A ⊂ X,
h T, p, y i
A = hT
A, p, y i for a.e. y ∈ Rk .
(76)
Proof. Choose a sequence Ri ∈ Rm (X; G) that is F convergent to T so that
M̂(Ri ) → M̂(T ). By passing to a subsequence, we may further assume that
P
∞
i=1 F (Ri − T ) < ∞. By Theorem 5.2.1(4) and Monotone Convergence
Z X
∞
∞ Z
X
F h Ri , p, y i − h T, p, y i dy =
F h Ri − T, p, y i dy
Rk i=1
i=1
Rk
≤ k k/2 (Lip p)k
∞
X
i=1
F (Ri − T ) < ∞ ,
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
49
In particular, for a.e. y ∈ Rk the sequence of slices h Ri , p, y i is F convergent to
h T, p, y i. By the lower semicontinuity of M̂, Fatou’s Lemma, Theorem 3.8.1(9),
and Theorem 5.7.4,
Z
Z
M̂h T, p, y i dy ≤
lim inf M̂h Ri , p, y i dy
i→∞
Z
≤ lim inf
Mh Ri , p, y i dy
Rk
Rk
i→∞
Rk
≤ lim inf C(Lip p)k M(Ri )
i→∞
= C(Lip p)k M(T ) ≤ (2m)m C(Lip p)k M̂(T ) .
Formula (76), for rectifiable T , follows from the definition in §3.7. For the general
flat chain T of finite mass, we will first treat special A as in the development
of the definition of P
T A given in Theorem 6.2.2. We may use a T convergent
∞
representation T = i=1 Pi with the Pi being Lipschitz chains. For any Lipschitz
u; X
→ R, we first deduce, as in Theorem 5.2.3(2), for a.e r ∈ R, the F convergence
P∞
of i=1 Pi Ur = T Ur where Ur = {u < r}. Then similarly, for a.e. y ∈ Rk ,
h T, p, y i
Ur = h T
Ur , p, y i for a.e. y ∈ Rk .
Next, as in Lemma 6.2.3, while approximating an arbitrary open set U by subsets
Ur and then a µ̂T measurable set A by open supersets, we have mass convergence,
and we may use (75) in verifying (76).
Lemma 8.0.4. (Absolute Continuity) If T ∈ Mm (X; G) and E is a µ̂T measurable
subset of X such that p(E) has Lebesgue measure zero for every Lipschitz map
p : X → Rm , then T E = 0.
Proof. In case m = 0, T E = T ∅ = 0. So we now assume m ≥ 1, and, for
contradiction, that T E 6= 0. Then, M̂(T K) = µ̂T (K) > 0 for some closed
K ⊂ E, and there is, by Theorem 5.6.1, a Lipschitz map p : X → Rm so that
hT K,
p, yi =
6 0 for y belonging to some set of positive measure in Rm . Let
P∞
T = i=1 Pi be a flat convergent representation in terms of Lipschitz chains Pi .
We may choose r1 > r2 > · · · approaching zero so that, with
Vj = {x ∈ X : dist(x, K) < rj } ,
P∞
we have flat convergence of the series i=1 Pi Vj = T VP
j for all j. By (76),
∞
we also have, for almost all y ∈ Rm , flat convergence of
i=1 hPi , p, yi Vj =
m
m
hT, p, yi Vj for every j. Note also that limj→∞ H p(Vj ) = H p(K) = 0 because
∩∞
/ p(Vj ), the lower semicontinuity of M on F0 (X; G)
j=1 p(Vj ) = p(K). For a.e. y ∈
(Lemma 6.0.6) and §3.7 imply that
M hT, p, yi
Vj
≤ lim inf M
I→∞
I
X
i=1
hPi , p, yi
Vj
= 0.
50
THIERRY DE PAUW AND ROBERT HARDT
Formula (76) and the integrability of MhT, p, · i now lead to the desired contradiction
Z
Z
0 <
MhT K, p, yi dy =
M(hT, p, yi K) dy
Rm
Rm
Z
≤ lim inf
M(hT, p, yi Vj ) dy
j→∞
m
ZR
≤ lim sup
MhT, p, yi dy = 0 .
j→∞
p(Vj )
8.1. Bounding the Total Variation of the Slice of a Flat Chain. Recall that
a real-valued integrable function f on Rk is of bounded variation , or simply BV, if
Z
kDf k(Rk ) := sup{
f div g dx : g ∈ C0∞ (Rk , Rk ), |g| ≤ 1 } < ∞ .
Rk
k
For
R a metric space X, a measurable function F : R → X is integrable if
dist
(F
(y),
x
)
dy
<
∞
for
some
point
x
∈
X.
Then, following [3], such an
0
0
Rk
k
integrable function F : R → X is BV if φ ◦ F ∈ BV (Rk , R) for every Lipschitz
function φ : X → R and
kDF k(Rk ) := sup{ kD(φ ◦ F )}(Rk ) : φ : X → R and Lip φ ≤ 1} < ∞ .
For the special case k = 1, we have, by [10, 4.5.9], that kDF k(R) coincides with
the classical essential variation
ess.var (F ) := sup{
I
X
dist F (ai ), F (ai−1 )
i=1
: a0 < a1 < · · · < aI are Lebesgue points of F } .
For higher dimensional k > 1, we also have from [10, 4.5.9] that
k Z
X
kDF k(Rk ) ≤
ess.var.(F ◦ χj,z ) dz ≤ kkDF k(Rk )
j=1
(77)
Rk−1
where χj,z (t) = (z1 , · · · , zj−1 , t, zj , · · · , zk−1 ) ∈ Rk . Thus bounded variation may
be checked using the essential variations of the restrictions to almost all coordinate
lines.
Theorem 8.1.1. For any flat chain T ∈ Fm (X; G) such that T and ∂T both have
finite mass and any Lipschitz p : X → Rk with k ≤ m, the slice function
F : Rk → Fm−k (X; G) , F (y) = h T, p, y i ,
is of bounded variation with
kDF k(Rk ) ≤ k(2m)m C(Lip p)k−1 [M̂(T ) + M̂(∂T )] .
Proof. The map F is integrable by Theorem 5.2.3(3).
In case k = 1, we may infer from Theorem 5.2.4(5) that, for a.e. numbers r < s,
h T, p, s i − h T, p, r i
= ∂(T
{x : r ≤ p(x) < s}) − (∂T )
{x : r ≤ p(x) < s} ,
(78)
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
51
so that
F h T, p, s i − h T, p, r i
≤ M̂ T
{x : r ≤ p(x) < s}
+ M̂ ∂T
(79)
{x : r ≤ p(x) < s} .
Thus, for each positive integer I and a.e. choice of numbers a0 < a1 < · · · < aI ,
I
X
F h T, p, ai i − h T, p, ai−1 i
i=1
≤
I
X
M̂ T
{x : r ≤ p(x) < s}
+ M̂ ∂T
{x : r ≤ p(x) < s}
i=1
≤ M̂(T ) + M̂(∂T ) ,
which gives the estimate
kDF k(R) = ess.var (F ) ≤ M̂(T ) + M̂(∂T ) .
(80)
In case k > 1, we write p = (p1 , p2 , · · · , pk ), y = (y1 , y2 , · · · , yk ), and define, for
each j ∈ {1, 2, · · · , k},
p̆i = (p1 , · · · , pi−1 , pi+1 , · · · , pk ) ,
y̆i = (y1 , · · · , yi−1 , yi+1 , · · · , yk ) .
We infer from Theorem 3.8.1(11)(12) and Theorem 5.2.3(3) that, for a.e. y ∈ Rk ,
F (y) = h T, p, y i = h· · · h h T, p1 , y1 i, pc , y2 i · · · , pk , yk i
= (−1)k−j h h T, p̆j , y̆j i, pj , yj i ,
which implies that
ess.var.(F ◦ χj,y̆j ) = ess.var. h h T, p̆j , y̆j i, pj , · i .
We insert this in formula (77), apply (80) to each F ◦χj,y̆j and use Theorem 5.2.4(6)
and estimate (75) to obtain the desired estimate
kDF k(Rk ) ≤
k Z
X
j=1
≤
Rk−1
k Z
X
j=1
ess.var. h h T, p̆j , y̆j i, pj , · i dy̆j
M̂h T, p̆j , y̆j i + M̂∂h T, p̆j , y̆j i dy̆j
Rk−1
≤ k(2m)m C(Lip p)k−1 [M̂(T ) + M̂(∂T )] .
We are now in a position to repeat exactly the beautiful proof of [3] (see also
[18]) to obtain the:
Corollary 8.1.2. For any Lipschitz map p of a separable metric space X0 into Rm
and any flat chain T ∈ Fm (X0 ; G) with T and ∂T having finite mass, the set
MT,p := {x ∈ X : h T, p, p(x) i ∈ M0 (X0 ; G) and µhT,p,p(x) i ({x}) > 0 }
is an H m rectifiable set.
52
THIERRY DE PAUW AND ROBERT HARDT
8.2. Rectifiability.
Lemma 8.2.1. Suppose that Ψ is a bilipschitz embedding of X into another metric
space Y and T ∈ Fm (X; G) has finite mass. Then T is rectifiable if and only if
Ψ# T is rectifiable.
Proof. The rectifiability of Ψ# T for a rectifiable T follows from §3.5. On the
other
P∞
hand, if Ψ# T is rectifiable, it has a mass convergent representation Ψ# T = i=1 Pi
as a sum of Lipschitz chains Pi in Y . Then this mass convergence gives the formula
Ψ# T = (Ψ# T )
Ψ(X) = (
∞
X
Pi )
Ψ(X) =
i=1
∞
X
Pi
Ψ(X) .
i=1
The H m rectifiable set and G orientation of the rectifiable chain Pi Ψ(X) in Y
defines a rectifiable chain Qi ∈ Rm (Ψ(X); G). Moreover, by mass convergence,
P∞
T̃ := i=1 (Ψ−1 )# Qi is, a rectifiable chain in X and
Ψ# T̃ =
∞
X
Ψ# (Ψ−1 )# Qi =
i=1
∞
X
Pi
Ψ(X) = Ψ# T
i=1
by §3.5. Following Theorem 5.2.3(1), T = T̃ is thus rectifiable.
Here we give the metric space generalization of [22] and m dimensional version
of Theorem 7.2.2. First we treat a special case:
Theorem 8.2.2. (Boundary Rectifiability) Suppose X is any complete metric space
and G is a complete normed abelian group that does not contain any nonconstant
Lipschitz curve. If S ∈ Rm+1 (X; G) and ∂S has finite mass, then ∂S ∈ Rm (X; G).
Proof. As in the proof of Corollary 8.1.2, we may write ∂S = Φ0# T for some
isometric embedding Φ0 of a separable metric space X0 into X and some finite mass
T ∈ Fm (X0 ; G). Then ∂T = 0 by Theorem 5.2.3(1) because Φ0# ∂T = ∂Φ0# T = 0.
It will be sufficient to show the rectifiability of T because then ∂S = Φ0# T would
also be rectifiable.
For any Lipschitz map p : X0 → Rm , we deduce from Theorem 5.2.3(3) and (75)
that, for a.e. y ∈ Rm , the slice hT, p, yi is a 0 dimensional flat G chain of finite
mass. Thus, by Theorem 7.2.2, hT, p, yi is rectifiable, that is, a countable, mass
convergent sum of G oriented point masses. Using (76) and Corollary 8.1.2, we
deduce that, for almost all y ∈ Rm ,
hT
MT,p , p, yi = hT, p, yi
MT,p = hT, p, yi .
(81)
It will now be convenient to define exactly as in §5.7, the finite Borel regular
measures λT,U,p and µ̂T .
For each positive integer i, there are a positive integer J(i) and, for, j =
1, · · · , J(i), disjoint open sets Ui,1 , · · · , Ui,J(i) ) and Lipschitz maps pi,1 , · · · , pi,J(i)
in Lip1 (X0 , Rm ) so that
µ̂T (X0 ) − i−1 <
J(i)
X
λT ,U (i,j),p(i,j) (X0 ) .
j=1
Let U = { U (i, j) }, P = { p(i, j) } corresponding to i = 1, 2, · · · , and j =
1, · · · , J(i). By the countability of P, we now obtain from Corollary 8.1.2 the
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
53
single H m rectifiable set
M =
[
MT,p .
p∈P
We claim that
T = T
M .
(82)
First note from (81)
λT,U,p (X0 ) = λT,U,p (M )
for any open U and any p ∈ P because MT,p ⊂ M ⊂ X0 . Thus
µ̂T (X0 ) = sup {
I
X
λT ,Ui ,pi (X0 ) : I < ∞, pi ∈ P, disjoint U1 , · · · , UI ∈ U }
i=1
= sup {
I
X
λT ,Ui ,pi (M ) : I < ∞, pi ∈ P, disjoint U1 , · · · , UI ∈ U }
i=1
≤ µ̂T (M ) ≤ µ̂T (X0 ) .
It then follows from (76) that, for an arbitrary p ∈ Lip1 (X0 , Rm ),
Z
Z
M h T, p, y i (X0 ∼ M ) dy
Mh T − T M, p, y i dy =
(83)
≤ λT,X0 ,p (X0 ∼ M ) ≤ µ̂T (X0 ∼ M ) = µ̂T (X0 ) − µ̂T (M ) = 0 .
Finally, for any Lipschitz map p : X0 → Rm ,
M, p, y i for a.e. y ∈ Rm
h T, p, y i = h T
because both are zero in case Lip p = 0, and we may apply (83) to p/Lip p in
case Lip p 6= 0. The slice-nullity Theorem 5.6.1 now implies the desired relation,
T = T M.
Next we consider the locally bilipschitz almost parameterization {γi , Ai } from
§3.1 and the corresponding partition
M =
∞
[
Mi
where
Mi = M ∩ γi (Ai ) for i = 1, 2, · · ·
i=0
m
and M0 = M ∼ ∪∞
(M0 ) = 0. By the H m measurability of
i=1 γi (Ai ) has H
∞
M and Lemma 8.0.4 this partition is µT measurable, and
T =
∞
X
T
Mi
i=1
P∞
is a M̂ convergent sum of flat chains with M̂(T ) = i=1 M̂(T Mi ). Thus it suffices
to show that each separate chain T Mi is rectifiable.
Letting Ψ : X0 → `∞ be an isometric embedding as in Lemma 2.0.1(1). The
Lipschitz map γi−1 ◦ Ψ−1 : Ψ ◦ γ(Ai ) → Ai ⊂ Rm admits a Lipschitz extension
√
β : `∞ → Rm (with Lip β ≤ m Lip γi−1 ). Similarly the Lipschitz map Ψ ◦ γi
admits a Lipschitz extension α : Rm → `∞ (with Lip α = Lip γi ). Moreover,
(α ◦ β)(x) = x for any x ∈ ClosΨ(Ai ) ,
in particular for x ∈ spt Ψ# (T
Ψ# (T
Mi ). From Lemma 5.4.2 and it follows that
Mi ) = (α ◦ β)# Ψ# (T
Mi ) = α# (β ◦ Ψ)# (T
Mi ) .
54
THIERRY DE PAUW AND ROBERT HARDT
But (β ◦ Ψ)# (T Mi ), being an m dimensional flat chain of finite mass in Rm is
necessarily rectifiable. So, being a Lipschitz push-forward of this chain, Ψ# (T Mi )
is also rectifiable. By Lemma 8.2.1,
P∞ T Mi itself is rectifiable.
We conclude that ∂S = Φ0# i=1 (T Mi ) is rectifiable.
Corollary 8.2.3. (Rectifiability of Flat Chains of Finite Mass) Again suppose X
is any complete metric space and G is any complete normed abelian group that does
not contain a nonconstant Lipschitz curve. Then every flat chain T ∈ Fm (X; G)
with finite mass is rectifiable.
Proof. By Lemma 2.0.1(1) and Lemma 8.2.1, we may assume that X itself is an
`∞ space so that we do not, by Theorem 5.2.1(6), have to bother with an isometric embedding Φ in our calculation of a flat norm. With R0 = T , we use §6 to
inductively choose Pi , Ri ∈ Lm (X; G) and Si ∈ Lm+1 (X; G) so that
M̂(Pi ) ≤ 2 M̂(Ri−1 ) , M̂(Ri ) + M̂(Si ) ≤ 2−i
P∞
P∞
for i = 1, 2, · · · . Inasmuch as i=1 M̂(Pi ) ≤ 2M̂(T ) + 2 and i=1 M̂(Si ) are both
finite, we have
Ri−1 = Pi + Ri + ∂Si ,
T = R + ∂S where R =
∞
X
Pi ∈ Rm (X; G) and S =
∞
X
Si ∈ Rm+1 (X; G) .
i=1
i=1
Since M̂(∂S) ≤ M̂(R) + M̂(T ) < ∞, the flat chain ∂S is, by Theorem 8.2.2,
rectifiable, and hence so is T = R + ∂S.
9. Virtual Flat Chains
Following [24], [13], [22], [1], we have defined flat chains as a completion of the
more elementary polyhedral or Lipschitz chains. However, many complete metric
spaces may, in various dimensions, contain no nonzero Lipschitz or even rectifiable
chains. To increase the chances of nevertheless having a nontrivial examples of
useful chains, we may alternately “add in” isometric copies of boundaries from
other spaces. We define a virtual flat chain as an equivalence class
[φ , T ]
of a pair (Φ, T ) of an isometric embedding Φ : X → `∞ (X) and a flat G chain
T ∈ Fm `∞ (X); G with spt T ⊂ Φ(X), where
(Φ1 , T1 ) ∼ (Φ2 , T2 )
if and only if Φ̃# T1 = T2
for some Lipschitz extension (see Lemma 2.0.1(2)) Φ̃ : `∞ (X) → `∞ (X) of Φ2 ◦Φ−1
1 .
By Lemma 5.4.2, this is well-defined independent of the choice of extension Φ̃. Using
the extension Lemma 2.0.1(2) as well as Theorem 5.4.2(1)(2) allows us to verify that
the following operations are well-defined:
spt [Φ, T ] = Φ−1 (spt T ) ,
F [Φ, T ] = F (T ) ,
∂[Φ, T ] = [φ, ∂T ] ,
M̂[Φ, T ] = M̂(T ) .
[Φ, T ] + [Φ0 , T0 ] = [Φ, T + Φ̃0# T0 ] ,
for any Lipschitz extension Φ̃0 of Φ ◦ Φ−1
0 ,
f# [Φ, T ] = [Ψ, f˜# T ]
for any Lipschitz f : spt T → Y ,
RECTIFIABLE AND FLAT G CHAINS IN A METRIC SPACE
55
any isometric embedding Ψ : Y → `∞ (Y ), and any Lipschitz extension f˜ of Ψ ◦ f ,
h [Φ, T ], p, y i = [Φ, h T, p̃, y i]
for any Lipschitz p : spt T → Rk
any Lipschitz extension p̃ of p ◦ Φ−1 , and a.e. y ∈ Rk . One may now carry over
the constructions and results of the present paper to the group F˜m (X; G) of m
dimensional virtual flat chains in X.
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