Contemporary Mathematics
Volume 665, 2016
http://dx.doi.org/10.1090/conm/665/13304
Measure theory and Lebesgue-like integration in two and
three dimensions over the Levi-Civita field
Khodr Shamseddine and Darren Flynn
Abstract. In this paper, we develop the foundations for a Lebesgue-like measure and integration theory over the spaces R2 and R3 , where R is the LeviCivita field. First we review the one-dimensional theory then we extend the
results to two and three dimensions. In particular, we introduce a measure
on R2 (resp. on R3 ) that has similar properties to those of the Lebesgue
measure of Real Analysis. Then we introduce a family of R-valued analytic
functions from which we obtain a larger family of measurable functions defined
on measurable subsets of R2 (resp. R3 ). We study the properties of measurable functions, we show how to integrate them over measurable subsets of R2
(resp. R3 ), and we show that the resulting integral satisfies similar properties
to those of the Lebesgue integral of Real Analysis.
1. Introduction
A Lebesgue-like measure and integration theory on the Levi-Civita spaces R2
and R3 will be presented. We recall that the elements of the Levi-Civita field R
and its complex counterpart C are functions from Q to R and C, respectively, with
left-finite support (denoted by supp). That is, below every rational number q, there
are only finitely many points where the given function does not vanish. For the
further discussion, it is convenient to introduce the following terminology.
Definition 1.1. (λ, ∼, ≈) For x = 0 in R or C, we let λ(x) = min(supp(x)),
which exists because of the left-finiteness of supp(x); and we let λ(0) = +∞.
Moreover, we denote the value of x at q ∈ Q with brackets like x[q].
Given x, y = 0 in R or C, we say x ∼ y if λ(x) = λ(y); and we say x ≈ y if
λ(x) = λ(y) and x[λ(x)] = y[λ(y)].
At this point, these definitions may feel somewhat arbitrary; but after having
introduced an order on R, we will see that λ describes orders of magnitude, the
relation ≈ corresponds to agreement up to infinitely small relative error, while ∼
corresponds to agreement of order of magnitude.
The sets R and C are endowed with formal power series multiplication and
componentwise addition, which make them into fields [2] in which we can isomorphically embed R and C (respectively) as subfields via the map Π : R, C → R, C
2010 Mathematics Subject Classification. 28E05, 26A42, 26E30, 12J25, 11D88, 46S10.
Key words and phrases. non-Archimedean analysis, Levi-Civita field, power series, measure
theory and integration, double and triple integrals.
c
2016
American Mathematical Society
289
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290
KHODR SHAMSEDDINE AND DARREN FLYNN
defined by
(1.1)
Π(x)[q] =
x
0
if q = 0
.
else
Definition 1.2. (Order in R) Let x, y ∈ R be given. Then we say that x > y
(or y < x) if x = y and (x − y)[λ(x − y)] > 0; and we say x ≥ y (or y ≤ x) if x = y
or x > y.
It follows that the relation ≥ (or ≤) defines a total order on R which makes
it into an ordered field. Note that, given a < b in R, we define the R-interval
[a, b] = {x ∈ R : a ≤ x ≤ b}, with the obvious adjustments in the definitions of the
intervals [a, b[, ]a, b], and ]a, b[. Moreover, the embedding Π in Equation (1.1) of R
into R is compatible with the order.
The order leads to the definition of an ordinary absolute value on R:
x
if x ≥ 0
|x| =
−x if x < 0;
which induces the same topology on R (called the order topology or valuation
topology) as that induced by the ultrametric absolute value:
|x|u = e−λ(x) ,
as was shown in [17]. Moreover, two corresponding absolute values are defined on
C in the natural way:
|x + iy| = x2 + y 2 ; and |x + iy|u = e−λ(x+iy) = max{|x|u , |y|u }.
Thus, C is topologically isomorphic to R2 provided with the product topology
induced by |·| (or |·|u ) in R.
We note in passing here that |·|u is a non-Archimedean valuation on R (resp.
C); that is, it satisfies the following properties
(1) |v|u ≥ 0 for all v ∈ R (resp. v ∈ C) and |v|u = 0 if and only if v = 0;
(2) |vw|u = |v|u |w|u for all v, w ∈ R (resp. v, w ∈ C); and
(3) |v + w|u ≤ max{|v|u , |w|u } for all v, w ∈ R (resp. v, w ∈ C): the strong
triangle inequality.
Thus, (R, | · |) and (C, | · |) are non-Archimedean valued fields.
Besides the usual order relations on R, some other notations are convenient.
Definition 1.3. (
, ) Let x, y ∈ R be non-negative. We say x is infinitely
smaller than y (and write x y) if nx < y for all n ∈ N; we say x is infinitely larger
than y (and write x y) if y x. If x 1, we say x is infinitely small; if x 1,
we say x is infinitely large. Infinitely small numbers are also called infinitesimals or
differentials. Infinitely large numbers are also called infinite. Non-negative numbers
that are neither infinitely small nor infinitely large are also called finite.
Definition 1.4. (The Number d) Let d be the element of R given by d[1] = 1
and d[t] = 0 for t = 1.
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MEASURE THEORY AND INTEGRATION OVER THE LEVI-CIVITA FIELD
291
Remark 1.5. Given m ∈ Z, then dm is the positive R-number given by
⎧
dd
· · · d if m > 0
⎪
⎪
⎪
⎪
⎪
⎪
⎨ m times
m
.
d =
1
if m = 0
⎪
⎪
⎪
⎪
⎪
⎪
⎩ 1
if m < 0
d−m
Moreover, given a rational number q = m/n (with n ∈ N and m ∈ Z), then dq is
the positive nth root of dm in R (that is, (dq )n = dm ) and it is given by
1 if t = q
.
dq [t] =
0 otherwise
It is easy to check that dq 1 if q > 0 and dq 1 if q < 0 in Q. Moreover,
for all x ∈ R (resp. C), the elements of supp(x) can be arranged in ascending
order, say supp(x) = {q1 , q2 , . . .} with qj < qj+1 for all j; and x can be written as
∞
x[qj ]dqj , where the series converges in the valuation topology [2].
x=
j=1
Altogether, it follows that R (resp. C) is a non-Archimedean field extension of
R (resp. C). For a detailed study of these fields, we refer the reader to the survey
paper [13] and the references therein. In particular, it is shown that R and C are
complete with respect to the natural (valuation) topology.
It follows therefore that the fields R and C are just special cases of the class of
fields discussed in [7]. For a general overview of the algebraic properties of formal
power series fields in general, we refer the reader to the comprehensive overview by
Ribenboim [6], and for an overview of the related valuation theory to the books
by Krull [4], Schikhof [7] and Alling [1]. A thorough and complete treatment of
ordered structures can also be found in [5].
Besides being the smallest ordered non-Archimedean field extension of the real
numbers that is both complete in the order topology and real closed, the LeviCivita field R is of particular interest because of its practical usefulness. Since
the supports of the elements of R are left-finite, it is possible to represent these
numbers on a computer [2]; and having infinitely small numbers in the field allows
for many computational applications. One such application is the computation
of derivatives of real functions representable on a computer [14], where both the
accuracy of formula manipulators and the speed of classical numerical methods are
achieved.
2. Measure Theory and Integration on R
Using the nice smoothness properties of power series (see [10] and the references
therein), we developed a Lebesgue-like measure and integration theory on R in
[11, 16] that uses the R-analytic functions (functions given locally by power seriesDefinition 2.4) as the building blocks for measurable functions instead of the step
functions used in the real case. This was possible in particular because the family
S(a, b) of R-analytic functions on a given interval I(a, b) ⊂ R (where I(a, b) denotes
any one of the intervals [a, b], ]a, b], [a, b[ or ]a, b[) satisfies the following crucial
properties.
(1) S(a, b) is an algebra that contains the identity function;
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KHODR SHAMSEDDINE AND DARREN FLYNN
(2) for all f ∈ S(a, b), f is Lipschitz on I(a, b) and there exists an antiderivative F of f in S(a, b), which is unique up to a constant;
(3) for all differentiable f ∈ S(a, b), if f = 0 on ]a, b[ then f is constant on
I(a, b); moreover, if f ≥ 0 on ]a, b[ then f is nondecreasing on I(a, b).
Notation 2.1. Let a < b in R be given. Then by l(I(a, b)) we will denote the
length of the interval I(a, b), that is
l(I(a, b)) = length of I(a, b) = b − a.
Definition 2.2. Let A ⊂ R be given. Then we say that A is measurable
if for every > 0 in R, there exist a sequence of mutually disjoint intervals (In )
∞
∞
In ⊂ A ⊂
Jn ,
and a sequence of mutually disjoint intervals (Jn ) such that
∞
l(In ) and
∞
n=1
n=1
l(Jn ) converge in R, and
∞
l(Jn ) −
n=1
∞
n=1
n=1
l(In ) ≤ .
n=1
Given a measurable set A, then for every k ∈ N, we can select a sequence
of
mutually disjoint intervals Ink and a sequence of mutually disjoint intervals Jnk
∞ ∞ l Ink and
l Jnk converge in R for all k,
such that
n=1
∞
Ink ⊂
n=1
n=1
∞
Ink+1 ⊂ A ⊂
n=1
∞
Jnk+1 ⊂
n=1
∞
Jnk and
n=1
∞
∞
l Jnk −
l Ink ≤ dk
n=1
n=1
for all k ∈ N. Since R is Cauchy-complete in the order topology, it follows that
∞ ∞ k l Jnk both exist and they are equal. We call the
lim
n=1 l In and lim
k→∞
k→∞ n=1
common value of the limits the measure of A and we denote it by m(A). Thus,
∞
∞
k
l In = lim
l Jnk .
m(A) = lim
k→∞
n=1
k→∞
n=1
Contrary to the real case,
∞
∞
sup
l(In ) : In ’s are mutually disjoint intervals and
In ⊂ A
n=1
and
inf
∞
n=1
l(Jn ) : Jn ’s are mutually disjoint intervals and A ⊂
n=1
∞
Jn
n=1
need not exist for a given set A ⊂ R. However, as shown in [16], if A is measurable
then both the supremum and infimum exist and they are equal to m(A). This shows
that the definition of measurable sets in Definition 2.2 is a natural generalization
of that of the Lebesgue measurable sets of real analysis that corrects for the lack
of suprema and infima in non-Archimedean ordered fields.
It follows directly from the definition that m(A) ≥ 0 for any measurable set
A ⊂ R and that any interval I(a, b) is measurable with measure m(I(a, b)) =
l(I(a, b)) = b − a. It also follows that if A is a countable union of mutually disjoint
∞
(bn − an ) converges then A is measurable with
intervals (In (an , bn )) such that
n=1
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MEASURE THEORY AND INTEGRATION OVER THE LEVI-CIVITA FIELD
m(A) =
∞
293
(bn − an ). Moreover, if B ⊂ A ⊂ R and if A and B are measurable,
n=1
then m(B) ≤ m(A).
In [16] we show that the measure defined on R above has similar properties to
those of the Lebesgue measure on R. For example, we show that any subset of a
measurable set of measure 0 is itself measurable and has measure 0. We also show
that any countable unions of measurable sets whose measures form a null sequence
is measurable and the measure of the union is less than or equal to the sum of the
measures of the original sets; moreover, the measure of the union is equal to the sum
of the measures of the original sets if the latter are mutually disjoint. Furthermore,
we show that any finite intersection of measurable sets is also measurable and that
the sum of the measures of two measurable sets is equal to the sum of the measures
of their union and intersection.
It is worth noting that the complement of a measurable set in a measurable
set need not be measurable. For example, [0, 1] and [0, 1] ∩ Q are both measurable
with measures 1 and 0, respectively. However, the complement of [0, 1] ∩ Q in [0, 1]
is not measurable. On the other hand, if B ⊂ A ⊂ R and if A, B and A \ B are all
measurable, then m(A) = m(B) + m(A \ B).
The example of [0, 1] \ [0, 1] ∩ Q above shows that the axiom of choice is not
needed here to construct a nonmeasurable set, as there are many simple examples
of nonmeasurable sets. Indeed, any uncountable real subset of R, like [0, 1] ∩ R for
example, is not measurable.
Then we define in [16] a measurable function on a measurable set A ⊂ R using
Definition 2.2 and R-analytic functions (Definition 2.4 below).
Definition 2.3. A sequence (an )∞
n=1 in R (or C) is said to be regular if the
union of the supports of all members of the sequence is a left-finite subset of Q.
Definition 2.4. Let a < b in R be given and let f : I(a, b) → R. Then we say
that f is R-analytic (or simply analytic) on I(a, b) if for all x ∈ I(a, b) there exists
a positive δ ∼ b − a in R, and there exists a regular sequence (an (x))∞
n=1 in R such
that, under weak convergence,
∞
f (y) =
an (x) (y − x)n for all y ∈ ]x − δ, x + δ[ ∩ I(a, b).
n=0
Definition 2.5. Let A ⊂ R be a measurable subset of R and let f : A → R
be bounded on A. Then we say that f is measurable on A if for all > 0 in R,
there exists a sequence of mutually disjoint intervals (In ) such that In ⊂ A for all
∞
∞
n,
l (In ) converges in R, m(A) −
l(In ) ≤ and f is R-analytic on In for all
n=1
n=1
n.
In [16], we derive a simple characterization of measurable functions and we
show that they form an algebra. Then we show that a measurable function is
differentiable almost everywhere and that a function measurable on two measurable
subsets of R is also measurable on their union and intersection.
We define the integral of an R-analytic function over an interval I(a, b) and we
use that to define the integral of a measurable function f over a measurable set A.
Definition 2.6. Let a < b in R, let f : I(a, b) → R be R-analytic on I(a, b),
and let F be an R-analytic anti-derivative of f on I(a, b). Then the integral of f
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294
KHODR SHAMSEDDINE AND DARREN FLYNN
over I(a, b) is the R number
f = lim F (x) − lim F (x).
x→a
x→b
I(a,b)
The limits in Definition 2.6 account for the case when the interval I(a, b) does
not include one or both of the end points; and these limits exist since F is Lipschitz
on I(a, b) [8].
Now let A ⊂ R be measurable, let f : A → R be measurable and let M
be a bound for |f | on A. Then for every k ∈ N, there exists a sequence of
∞
∞ ∞
Ink ⊂ A,
l Ink converges,
mutually disjoint intervals Ink n=1 such that
n=1
n=1
∞ m(A) −
l Ink ≤ dk , and f is R-analytic on Ink for all n ∈ N. Without loss
n=1
k
k+1
of generality, we may assume that
In ⊂ In for all n ∈ N and for all k ∈ N.
k
Since lim l In = 0, and since I k f ≤ M l Ink (proved in [16] for R-analytic
n→∞
n
functions), it follows that
f = 0 for all k ∈ N.
lim
n→∞
k
In
∞ f converges in R for all k ∈ N [15].
∞
∞
We show that the sequence
f
converges in R; and we define the
Ik
Thus,
n=1
k
In
n
n=1
unique limit as the integral of f over A.
k=1
Definition 2.7. Let A ⊂ R be measurable
and let f : A → R be measurable.
Then the integral of f over A, denoted by A f , is given by
∞ f=
lim
f.
∞
A
l(In ) → m(A)
n=1
∞
n=1
In
In ⊂ A
n=1
s are mutually disjoint
In
f is R-analytic on In ∀ n
It turns out that the integral in Definition 2.7 satisfies similar properties to those
of the Lebesgue integral on R [16]. In particular,
we
prove the linearity property
of the integral and that if |f | ≤ M on A then A f ≤ M m(A), where m(A) is the
measure of A. We also show that the sum of the integrals of a measurable function
over two measurable sets is equal to the sum of its integrals over the union and the
intersection of the two sets.
In [11], which is a continuation of the work done in [16] and complements it,
we show, among other results, that the uniform limit of a sequence of convergent
power series on an interval I(a, b) is again a power series that converges on I(a, b).
Then we use that to prove the uniform convergence theorem in R.
Theorem 2.8. Let A ⊂ R be measurable, let f : A → R, for each k ∈ N let
fk : A → R be measurable on A, and let the sequence (fk ) converge uniformly to f
on A. Then f is measurable on A, lim A fk exists, and
k→∞
lim
fk =
f.
k→∞
A
A
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MEASURE THEORY AND INTEGRATION OVER THE LEVI-CIVITA FIELD
295
In this paper, we generalize the results of [11, 16] to two and three dimensions.
In particular, we define a Lebesgue-like measure on R2 (resp. R3 ). Then we
define measurable functions on measurable sets using analytic functions in two
(resp. three) variables and show how to integrate those measurable functions using
iterated integration. The resulting double (resp. triple) integral satisfies similar
properties to those of the single integral in [11, 16] as well as those properties
satisfied by the double and triple integrals of real calculus.
3. Measure Theory and Integration on R2
3.1. Simple regions and Measurable Sets.
Definition 3.1 (Simple Region). Let G ⊂ R2 . Then we say that G is a
simple region if there exist constants a, b ∈ R with a ≤ b and R-analytic functions
g1 , g2 : I(a, b) → R with g1 < g2 on I(a, b) such that
G = {(x, y) ∈ R2 : y ∈ I(g1 (x), g2 (x)), x ∈ I(a, b)}
or
G = {(x, y) ∈ R2 : x ∈ I(g1 (y), g2 (y)), y ∈ I(a, b)}.
Definition 3.2 (Area of a Simple Region). Let G ⊂ R2 be a simple region. If
G is of the form G = {(x, y) ∈ R2 : y ∈ I(g1 (x), g2 (x)), x ∈ I(a, b)} then we define
the area of G, denoted by a(G), as
[g2 (x) − g1 (x)],
a(G) =
x∈I(a,b)
which is well-defined since g1 (x) and g2 (x) are both analytic on I(a, b) and hence
so is g2 (x) − g1 (x). On the other hand, if G is of the form G = {(x, y) ∈ R2 : x ∈
I(g1 (y), g2 (y)), y ∈ I(a, b)} then
[g2 (y) − g1 (y)].
a(G) =
y∈I(a,b)
It is a simple exercise to show that the intersection, union and difference of
two simple regions in R2 can each be written as a finite union of mutually disjoint
simple regions; we refer the interested reader to [3] for the proof of this statement.
Definition 3.3 (Measurable Set). Let A ⊂ R2 . Then we say that A is measurable if for every > 0 in R there exist a sequence of mutually disjoint sim∞
ple regions (Gn )∞
n=1 and a sequence of (mutually disjoint) simple regions (Hn )n=1
∞
∞
∞
∞
such that
Gn ⊂ A ⊂
Hn ,
a(Gn ) and
a(Hn ) both converge, and
∞
n=1
∞
a(Hn ) −
n=1
n=1
n=1
n=1
a(Gn ) < .
n=1
Remark 3.4. Let A ⊂ R be a measurable set. Then for every k ∈ N there
k ∞
are two sequences of mutually disjoint simple regions (Gkn )∞
n=1 and (Hn )n=1 such
∞
∞
∞
∞
∞
that
Gkn ⊂ A ⊂
Hnk ,
Gkn and
Hnk both converge, and
a(Hnk ) −
∞
n=1
n=1
a(Gkn )
n=1
n=1
n=1
n=1
k
<d .
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296
KHODR SHAMSEDDINE AND DARREN FLYNN
We show that
∞
∞
n=1
a(Gkn )
is a Cauchy sequence. Assume not. Then there
k=1
is a η ∈ R, η > 0, such that for every k ∈ N, there exists an l > k such that
∞
∞
a(Gln ) −
a(Gkn ) > η. Fix k0 ∈ N so that dk0 < η, then for every k k0 ,
n=1
∞
n=1
a(Hnk )
−
n=1
∞
n=1
a(Gkn ) < dk dk0 < η. Thus, for every k k0 in N,
∞
(3.1)
a(Hnk ) <
n=1
∞
a(Gkn ) + η.
n=1
However, from above we have that there exists an l0 > k0 such that
∞
a(Gkn0 )+η which implies by equation 3.1 that
∞
a(Gln0 ) >
∞
n=1
∞
a(Gln0 ) >
a(Hnk0 ) which is a
n=1
n=1
n=1 ∞
∞
∞
∞
l0
k0
k
contradiction because by definition
Gn ⊂ A ⊂
Hn . Thus,
a(Gn )
n=1
n=1
∞n=1
∞ k=1
k
is a Cauchy sequence. Using a similar argument, we show that
a(Hn )
is
n=1
k=1
∞
also a Cauchy sequence. Since R is Cauchy complete it follows that lim
a(Gkn )
k→∞ n=1
∞
∞
∞
and lim
a(Hnk ) both exist; and hence lim
a Hnk −
a Gkn
exists.
k→∞ n=1
For every k ∈ N, we have that
∞
n=1
k→∞ n=1
∞
Gkn ⊂
Hnk ;
n=1
n=1
∞
n=1
it follows that
a(Hnk ); combining this with the fact that for every k ∈ N,
∞
n=1
∞
a(Gkn ) n=1
∞
a(Hnk )−
n=1
a(Gkn )
< dk , we infer that
0 lim
k→∞
∞
∞
k k
lim dk = 0
a Hn −
a Gn
n=1
k→∞
n=1
and hence
lim
k→∞
It follows that lim
∞
k→∞ n=1
∞
a(Hnk ) −
n=1
∞
a(Gkn )
= 0.
n=1
a(Gkn ) = lim
∞
k→∞ n=1
a(Hnk ).
Definition 3.5 (The Measure of a Measurable Set). We define the common
limit in Remark 3.4 to be the measure of A and we denote it by m(A). Thus,
m(A) = lim
k→∞
∞
n=1
a(Gkn ) = lim
k→∞
∞
a(Hnk ).
n=1
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MEASURE THEORY AND INTEGRATION OVER THE LEVI-CIVITA FIELD
297
Proposition 3.6. Let A ⊂ R2 be a measurable set. Then
∞
∞
a(Hn ) : Hn ’s are mutually disjoint simple regions, A ⊂
Hn ,
m(A) = inf
n=1
∞
and
= sup
n=1
a(Hn ) converges
n=1
∞
a(Gn ) : Gn ’s are mutually disjoint simple regions,
n=1
and
∞
∞
Gn ⊂ A,
n=1
a(Gn ) converges .
n=1
Proof. First we show that the infimum exists and is equal to m(A). Since
A is a measurable set we know that for every k ∈ N, there exist two sequences of
k ∞
mutually disjoint simple regions (Gkn )∞
n=1 and (Hn )n=1 such that
∞
Gkn ⊂
n=1
∞
n=1
a(Gkn ) and
∞
n=1
∞
∞
Gk+1
⊂A⊂
n
n=1
n=1
Hnk ,
n=1
∞
a(Hnk ) both converge, and
∞
Hnk+1 ⊂
n=1
a(Hnk ) −
∞
n=1
a(Gkn ) < dk . By
definition,
m(A) = lim
k→∞
∞
∞
a Gkn = lim
a(Hnk ).
k→∞
n=1
n=1
Moreover, for every k ∈ N, we have that
∞
∞
a(Gkn ) m(A) n=1
a(Hnk ).
n=1
It remains to be shown that if (Hn )∞
n=1 is a sequence of mutually disjoint
∞
∞
∞
Hn and
a(Hn ) converges, then
a(Hn ) simple regions such that A ⊂
lim
∞
k→∞ n=1
n=1
a(Hnk )
n=1
= m(A). Suppose not. Then there is a sequence of mutually
disjoint simple regions (Hn )∞
n=1 such that
m(A) >
∞
n=1
∞
a(Hn ) converges, A ⊂
n=1
∞
Hn , and
n=1
a(Hn ). Let k0 ∈ N be such that
n=1
m(A) −
k0
d
We have from above that
∞
n=1
<
Gkn0 ⊂ A ⊂
∞
n=1
2
∞
n=1
a(Hn )
.
∞
Hn , and since (Gkn0 )∞
n=1 and (Hn )n=1
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298
KHODR SHAMSEDDINE AND DARREN FLYNN
are both sequences of mutually disjoint simple regions it follows that
∞
a(Hn ). But
n=1
∞
n=1
∞
a(Hnk0 ) m(A), and hence m(A) −
n=1
∞
n=1
a(Gkn0 ) ∞
n=1
∞
n=1
a(Gkn0 ) a(Hnk0 ) −
a(Gkn0 ) < dk0 . Thus,
∞
a(Hn ) −
n=1
∞
a(Gkn0 ) =
n=1
∞
n=1
∞
a(Hn ) − m(A)
+
m(A) −
∞
a(Gkn0 )
n=1
a(Hn ) − m(A)
+ dk0
n=1
<
=
1
2
∞
a(Hn ) − m(A)
n=1
∞
a(Hn )
n=1
+
2
a(Hn ) − m(A)
∞
<0
n=1
But this contradicts the fact that
∞
n=1
(Hn )∞
n=1
m(A) −
a(Gkn0 ) ∞
a(Hn ), so a sequence such as
n=1
cannot exist.
A similar argument shows that
∞
∞
a(Gn ) : Gn ’s are mutually disjoint simple regions,
Gn ⊂ A,
sup
n=1
n=1
∞
a(Gn ) converges
n=1
exists and is equal to m(A).
Proposition 3.7. Let A, B ⊂ R2 be measurable sets with A ⊂ B. Then
m(A) ≤ m(B).
Proof. Suppose not. Then m(A) > m(B); let η = m(A) − m(B), then
η > 0. Since A is measurable there is a sequence of mutually disjoint simple regions
∞
∞
∞
(Gn )∞
Gn ⊂ A,
a(Gn ) converges, and m(A) −
a(Gn ) <
n=1 such that
n=1
η
4.
n=1
n=1
Since B is measurable there is a sequence of mutually disjoint simple regions
∞
∞
∞
(Hn )∞
Hn ,
a(Hn ) converges, and
a(Hn ) − m(B) < η4 .
n=1 such that B ⊂
n=1
n=1
n=1
It follows that
∞
∞
η
η
a(Hn ) < m(B) + < m(A) − <
a(Gn ).
4
4 n=1
n=1
However,
∞
n=1
so
∞
n=1
a(Gn ) ≤
∞
Gn ⊂ A ⊂ B ⊂
∞
Hn ,
n=1
a(Hn ) and thus we have reached a contradiction.
n=1
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MEASURE THEORY AND INTEGRATION OVER THE LEVI-CIVITA FIELD
299
Proposition 3.8. Let A ⊂ R2 be a measurable set with m(A) = 0 and let
B ⊂ A. Then B is measurable and m(B) = 0.
Proof. Let > 0 in R be given. Since A is measurable and since m(A) = 0,
then for every k ∈ N there exists a sequence of mutually disjoint simple regions
∞
∞
∞
(Hnk )∞
Hnk ,
a(Hnk ) converges, and
a(Hnk ) − m(A) =
n=1 such that A ⊂
n=1
∞
a(Hnk )
n=1
n=1
n=1
< d . For every n ∈ N, let Gn = ∅ which is a simple region.
k
∞
Let k0 ∈ N be such that dk0 < . Then
∞
a(Gn ) =
n=1
∞
B⊂
n=1
∞
n=1
Gn ⊂ B ⊂
n=1
∞
n=1
Hnk0 and
∞
n=1
a(Hnk0 )−
a(Hnk0 ) < dk0 < . Hence B is measurable. Since for every k ∈ N,
∞
Hnk it follows that 0 ≤ m(B) n=1
a(Hnk ) < dk . Letting k → ∞ we obtain
that m(B) = 0.
Proposition 3.9. Let A ⊂ R2 be countable. Then A is measurable and
m(A) = 0.
Proof. Since A is a countable set there is a sequence of points ((xn , yn ))∞
n=1
∞
such that A =
{(xn , yn )}. Let > 0 in R be given. For every n ∈ N let
n=1
Hn =
1
1
1
1
(x, y) ∈ R2 : x ∈ xn − (dn ) 2 , xn + (dn ) 2 , y ∈ yn − (dn ) 2 , yn + (dn ) 2 .
Then for every n ∈ N, Hn is a simple region with a(Hn ) = 4dn . Thus, lim a(Hn ) =
n→∞
∞
n
a(Hn ) converges. For every j ∈ N, let Gj = ∅. Then
lim 4d = 0, and hence
n→∞
∞
Gj ⊂ A ⊂
j=1
∞
i=1
∞
i=1
Hi ,
n=1
∞
j=1
a(Hi ) −
∞
a(Gj ) and
a(Hi ) converge, and
i=1
∞
a(Gj ) =
j=1
∞
a(Hi ) =
i=1
∞
4di =
i=1
4d
< ,
1−d
which proves that A is measurable. Furthermore, since A ⊂
∞
∞
Hi , m(A) i=1
a(Hi ) < . Taking the limit as → 0 shows that m(A) = 0.
i=1
∞
Proposition 3.10. Let (Hk )∞
k=1 and (Gn )n=1 be sequences of mutually disjoint
∞
∞
a(Hk ) and
a(Gn ) both converge. Then there exists
simple regions such that
k=1
n=1
a sequence of mutually disjoint simple regions (Tm )∞
m=1 such that
∞
∞
∞
Hk
Gn =
Tm
k=1
and
∞
n=1
m=1
a(Tm ) converges.
m=1
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300
KHODR SHAMSEDDINE AND DARREN FLYNN
Proof. First note that for every k, n ∈ N, there is a finite collection of mutulk,n
k,n
k,n lk,n
)m=1 such that Hk ∩Gn =
Tm . We show that
ally disjoint simple regions (Tm
m=1
l
k,n k,n ∞
)m=1 )k=1 )∞
the collection (((Tm
n=1 is mutually disjoint; so consider k1 , n1 , m1 ∈ N
and k2 , n2 , m2 ∈ N such that either k1 = k2 , n1 = n2 or m1 = m2 . Of course
k1 ,n1
k2 ,n2
if k1 = k2!= k and n1 = n2" = n then Tm
and Tm
are both elements
1
2
k,n
of the set Tm : 1 ≤ m ≤ lk,n and hence they are disjoint. If n1 = n2 then
k1 ,n1
k2 ,n2
k1 ,n1
⊂ Hk1 ∩ Gn1 ⊂ Gn1 and Tm
⊂ Hk2 ∩ Gn2 ⊂ Gn2 ; and hence Tm
and
Tm
1
2
1
k2 ,n2
Tm2 are disjoint since Gn1 and Gn2 are. The same argument holds if k1 = k2 .
k,n lk,n ∞
∞
Since (((Tm
)m=1 )k=1 )∞
n=1 is a countable collection it may be rewritten as (Tm )m=1 ,
∞
a(Tm ) converges. Thus, (Tm )∞
and rearrange terms if necessary so that
m=1 is a
m=1
collection of mutually disjoint simple regions such that
∞
∞
∞
Hk
Gn =
Tm
n=1
k=1
and
∞
m=1
a(Tm ) converges.
m=1
Proposition 3.11. For every k ∈ N, let (Gkn )∞
n=1 be a countable sequence
∞
∞ a(Gkn ) converges. Then there
of mutually disjoint simple regions such that
k=1 n=1
exists a collection of mutually disjoint simple regions (Hm )∞
m=1 such that
∞
Hm =
m=1
∞
∞ Gkn and
k=1 n=1
∞
a(Hm ) converges.
m=1
∞
Proof. First we note that ((Gkn )∞
n=1 )k=1 is a countable collection of simple re0 ∞
gions and so may be rewritten as (Hn )n=1 . To obtain the desired sequence (Hm )∞
m=1
we begin by defining H1 = H10 . Next we observe that, for every n, j ∈ N, Hn0 \Hj0
tn,j
is given by a finite number of mutually disjoint simple regions (Fin,j )i=1
. Thus, for
every n ∈ N, we have that
⎞
⎛
tn,j
n−1
n−1
0 0 n−1 Hn \Hj =
Hj0 ⎠ =
Fin,j .
Hn0 \ ⎝
j=1
j=1
j=1 i=1
However, using the same argument as in the proof of Proposition 3.10, we infer
n,j
n−1
' t
that for every n ∈ N,
Fin,j can be expressed as the union of a finite number
j=1 i=1
n
of mutually disjoint simple regions (Fin )li=1
.
We define
H2 = F12 , . . . , Hl2 +1 = Fl22
Hl2 +2 = F13 , . . . , Hl2 +l3 +1 = Fl33
Hl2 +l3 +2 = F14 , . . . , Hl2 +l3 +l4 +1 = Fl44
..
.
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MEASURE THEORY AND INTEGRATION OVER THE LEVI-CIVITA FIELD
301
Thus, by construction, the Hm ’s are mutually disjoint and lim a(Hm ) = 0,
∞
so
m→∞
a(Hm ) converges. Moreover,
m=0
∞
∞
Hm =
m=1
Hn0 =
n=1
∞
∞ Gkn .
k=1 n=1
2
Proposition 3.12. For each k ∈ N, let A
k ⊂ R be measurable, with
∞
∞
∞
Ak is measurable and m
Ak m(Ak ). Morelim m(Ak ) = 0. Then
k→∞
k=1
k=1
∞ k=1
∞
over, if the Ak ’s are mutually disjoint then m
Ak =
m(Ak ).
k=1
k=1
∞
Proof. First note that, since lim m(Ak ) = 0, we have that
k→∞
m(Ak ) con-
k=1
verges. Now let > 0 in R be given. Since each Ak is measurable, it follows
that, for every k ∈ N, there are two sequences of mutually disjoint simple regions
∞
∞
∞
∞
k ∞
(Gkn )∞
Gkn ⊂ Ak ⊂
Hnk ,
a(Gkn ) and
a(Hnk )
n=1 and (Hn )n=1 such that
both converge, and
∞
n=1
a(Hnk )
n=1
∞
−
n=1
n=1
a(Gkn )
k→∞
we infer that lim
∞
∞ k=1 n=1
∞
k→∞ n=1
∞
a(Gkn ) = lim
n=1
k→∞ n=1
n=1
< d .
∞
Since lim m(Ak ) = 0 and since 0 ≤
n=1
k
a(Gkn ) ∞
n=1
a(Hnk ) < m(Ak ) + dk ,
a(Hnk ) = 0. Thus,
∞
∞ k=1 n=1
a(Gkn ) and
a(Hnk ) both converge.
Using the proof of Proposition 3.11, there exist two sequences of mutually
∞
∞
∞ ∞
disjoint simple regions (Gn )∞
Gn =
Gkn and
n=1 and (Hn )n=1 such that
∞
n=1
Hn =
∞
∞ k=1 n=1
n=1
Hnk .
∞
Therefore,
\
Hn
n=1
∞
Gn
=
n=1
∞ ∞
Hnk
\
k=1 n=1
∞ ∞
\
Hnk
k=1 n=1
∞
For every k ∈ N we have that
k=1 n=1
n=1
∞ ∞
Gkn ⊂
Gkn
∞
n=1
=
k=1 n=1
∞ ∞
Gkn
.
k=1 n=1
Hnk , and hence
∞
k=1
∞
Hnk \
n=1
k ∞
Gn n=1
∞
Gkn
.
n=1
∞
Moreover, since for every k ∈ N, the sequences
and Hnk n=1 are both
mutually disjoint, we can arrange them in such a way that for every n ∈ N, Gkn ⊂
∞
∞
∞
Hnk . Thus, for every k ∈ N,
Hnk \
Gkn =
(Hnk \Gkn ). It follows that
n=1
∞
n=1
Hn
\
n=1
∞
n=1
Gn
n=1
=
∞ ∞
(Hnk \Gkn ).
k=1 n=1
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302
KHODR SHAMSEDDINE AND DARREN FLYNN
Therefore,
∞
a(Hn ) −
n=1
∞
(
a(Gn ) = m
n=1
=
∞
∞
m
∞
∞
Since
∞
Ak ⊂
k=1 n=1
m
=m
k=1 n=1
(Hnk \Gkn )
−
1−d
∞ ∞
k k
Hn \Gn
=
a(Gkn ))
∞ ∞
m(Hnk \Gkn )
k=1 n=1
∞
∞
=
a(Hnk )
−
n=1
k=1
∞
a(Gkn )
n=1
<
Ak is measurable.
k=1
∞
k=1
Gn
(a(Hnk )
k=1 n=1
∞
d
k
d =
)
n=1
n=1
k=1
∞
∞
k=1
which proves that
\
Hn
n=1
∞
∞
∞
Hnk we have that
Ak
m
∞ ∞
∞ ∞
Hnk
k=1 n=1
k=1
a(Hnk )
k=1 n=1
∞
∞
m(Ak ) + dk <
m(Ak ) + .
k=1
k=1
The above holds for any > 0 in R; and hence m
∞
∞
Ak
k=1
m(Ak ).
k=1
Now, assume that the Ak ’s are mutually disjoint, and let > 0 in R be given.
∞
There exists a K ∈ N such that
m(Ak ) < 2 . Since
Ak is measurable there
k>K
k=1
exists a sequence of mutually disjoint simple regions (Hn )∞
n=1 such that
∞
∞
∞
∞
Hn ,
a(Hn ) converges, and
a(Hn ) − m
Ak < 2 .
n=1
n=1
n=1
∞
k=1
∞
Because the Ak ’s and the Hn ’s are mutually disjoint, and because
∞
Hn , we can find for every k ∈ {1, . . . , K} a sequence of mutually disjoint sim-
ple regions (Hnk )∞
n=1 such that
n=1
Ak ⊂
k=1
n=1
∞
Ak ⊂
k=1
Hn1 ,
∞
n=1
Hn2 , . . . ,
∞
n=1
K
∞
n=1
a(Hnk ) converges, Ak ⊂
∞
n=1
Hnk ⊂
∞
Hn , and
n=1
HnK are mutually disjoint. Thus,
m(Ak ) k=1
K
k=1
∞
n=1
m
∞
Hnk
n=1
a(Hn ) < m
=
K ∞
a(Hnk )
k=1 n=1
∞
Ak
k=1
+ .
2
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MEASURE THEORY AND INTEGRATION OVER THE LEVI-CIVITA FIELD
303
Therefore,
∞
m(Ak ) =
k=1
K
m(Ak ) +
k=1
m(Ak ) < m
k>K
∞
Ak
k=1
∞
Taking the limit as → 0 yields that
+ + =m
2 2
m(Ak ) m(
k=1
∞
∞
Ak
+ .
k=1
Ak ).
k=1
Proposition 3.13. Let K ∈ N be given and for each k ∈ {1, . . . , K}, let
K
'
Ak ⊂ R2 be measurable. Then
Ak is measurable and
k=1
K
min {m(Ak ) : k ∈ {1, . . . , K}} .
Ak
m
k=1
Proof. Using induction on K, it suffices to show that if A and B are measurable sets in R2 then so is A ∩ B, and m(A ∩ B) ≤ min{m(A), m(B)}. So let A, B ⊂
R2 be measurable and let > 0 in R be given. Since A and B are measurable,
∞
B ∞
there exist four sequences of mutually disjoint simple regions (GA
n )n=1 , (Gn )n=1 ,
∞
∞
∞
∞
B ∞
GA
HnA ,
GB
HnB ;
(HnA )∞
n=1 , and (Hn )n=1 such that
n ⊂ A ⊂
n ⊂ B ⊂
n=1
∞
n=1
n=1
n=1
n=1
∞
∞
∞
a GA
a GB
a HnA , and
a HnB all converge; and
n ,
n ,
n=1
n=1
n=1
∞
∞
∞
∞
and
a HnA −
a GA
a HnB −
a GB
n n .
2
2
n=1
n=1
n=1
n=1
By Proposition 3.10, there exist two sequences of mutually disjoint simple re∞
gions (Hn )∞
n=1 and (Gn )n=1 such that
∞
Hn =
n=1
∞
and
∞
HnA
n=1
a(Hn ) and
n=1
∞
HnB
∞
and
n=1
∞
∞
Gn =
n=1
GA
n
n=1
∞
a(Gn ) both converge. Obviously
n=1
∞
GB
n
;
n=1
Gn ⊂ A ∩ B ⊂
n=1
∞
Hn .
n=1
Since
∞
n=1
Hn \
∞
Gn
=
n=1
(
∞
Hn \
n=1
(
⊂
⊂
∞
n=1
∞
n=1
( ∞
n=1
\
Hn
HnA
\
GA
n
∞
)
GA
n
n=1
∞
n=1
GA
n
)
∞
GB
n
n=1
(
∞
Hn
n=1
) ( ∞
n=1
\
HnB
\
∞
)
GB
n
n=1
∞
)
GB
n
,
n=1
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304
KHODR SHAMSEDDINE AND DARREN FLYNN
we have that
∞
a(Hn ) −
n=1
∞
a(Gn )
n=1
+
∞
∞
A A
a Hn −
a Gn
n=1
∞
n=1
∞
a HnB −
n=1
B
a Gn
n=1
+ =
2 2
which proves that A ∩ B is measurable. Using the result of Proposition 3.7,
we have that m(A ∩ B) m(A) and m(A ∩ B) m(B). Thus, m(A ∩ B) min{m(A), m(B)}.
Proposition 3.14. Let A, B ⊂ R2 be measurable. Then
m(A ∪ B) = m(A) + m(B) − m(A ∩ B).
Proof. First we note that, by Proposition 3.10 and Proposition 3.11, A∪B and
A∩B are measurable. Now let > 0 in R be given. Since A∪B is measurable there
exists a sequence of mutually disjoint simple regions (Hn )∞
n=1 such that A ∪ B ⊂
∞
∞
∞
Hn ,
a(Hn ) converges and
a(Hn ) − m(A ∪ B) < 2 . Since A\(A ∩ B),
n=1
n=1
n=1
B\(A ∩ B), and A ∩ B are mutually disjoint subsets of A ∪ B, there exist three
1 ∞
2 ∞
3 ∞
subsequences of (Hn )∞
n=1 denoted by (Hn )n=1 , (Hn )n=1 , and (Hn )n=1 such that
∞
∞
∞
Hn1 , B\(A ∩ B) ⊂
Hn2 , and (A ∩ B) ⊂
Hn3 . Note that
A\(A ∩ B) ⊂
n=1
∞
Hn1
n=1
n=1
∞
Hn2
n=1
∞
n=1
Hn3
n=1
∞
=
Hn .
n=1
Since A = [A\(A ∩ B)] (A ∩ B), we have that
∞
∞
∞
∞
A⊂
Hn1
Hn3 , and hence m(A) a(Hn1 ) +
a(Hn3 ).
n=1
Similarly,
B⊂
n=1
∞
Hn2
n=1
∞
n=1
Hn3
, and hence m(B) n=1
∞
n=1
a(Hn2 ) +
n=1
∞
a(Hn3 ).
n=1
Thus,
m(A) + m(B) =
∞
n=1
∞
n=1
Since
∞
n=1
a(Hn1 ) +
a(Hn ) +
∞
n=1
∞
a(Hn3 ) +
∞
n=1
a(Hn2 ) +
∞
a(Hn3 )
n=1
a(Hn3 )
n=1
a(Hn ) m(A ∪ B) + 2 , it follows that
∞
n=1
a(Hn3 ) − m(A ∩ B) ≤
∞
n=1
a(Hn ) − m(A ∪ B) ≤
.
2
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MEASURE THEORY AND INTEGRATION OVER THE LEVI-CIVITA FIELD
305
Therefore,
m(A) + m(B) m(A ∪ B) + m(A ∩ B) + .
Since this holds for any > 0 in R, we infer that
m(A) + m(B) m(A ∪ B) + m(A ∩ B).
Next we prove the other inequality. Since A and B are measurable, there
B ∞
exist two sequences of mutually disjoint simple regions (HnA )∞
n=1 and (Hn )n=1 such
∞
∞
∞
∞
that A ⊂
HnA , B ⊂
HnB ;
a(HnA ) and
a(HnB ) both converge; and
∞
n=1
n=1
a(HnA )
n=1
< m(A) +
2
and
n=1
∞
n=1
a(HnB )
n=1
< m(B) +
2.
B can be written as the union of the two mutually disjoint subsets B\(A ∩ B)
B,1 ∞
and A ∩ B. It follows that (HnB )∞
n=1 can be split into two subsequences (Hn )n=1
∞
∞
and (HnB,2 )∞
HnB,1 and A ∩ B ⊂
HnB,2 . Thus,
n=1 where B\(A ∩ B) ⊂
n=1
A ∪ B = A ∪ [B\(A ∩ B)] ⊂
∞
HnA
n=1
n=1
∞
HnB,1
,
n=1
and hence
m(A ∪ B) Since A ∩ B ⊂
∞
a(HnA )
+
n=1
∞
n=1
∞
n=1
∞
a(HnB,1 )
m(A) + +
a(HnB,1 ).
2 n=1
HnB,2 , we have that m(A ∩ B) ∞
n=1
a(HnB,2 ). Thus,
∞
∞
m(A ∪ B) + m(A ∩ B) m(A) + +
a(HnB,1 ) +
a(HnB,2 )
2 n=1
n=1
B
a(Hn )
≤ m(A) + +
2 n=1
≤ m(A) + m(B) + .
Since this holds for any > 0 in R, we infer that
m(A ∪ B) + m(A ∩ B) m(A) + m(B).
3.2. Analytic Functions.
Definition 3.15 (Finite Simple Region and Order of Magnitude). Let A ⊂ R2
be a simple region. First assume that A is of the form
A = {(x, y) ∈ R2 : y ∈ I(h1 (x), h2 (x)), x ∈ I(a, b)}
where a b, h1 , h2 : I(a, b) → R are analytic functions, and h1 < h2 . In this case,
we define
λx (A) = λ(b − a) and λy (A) = i(h2 (x) − h1 (x))
where i(h2 (x) − h1 (x)) is the index of the analytic function h2 (x) − h1 (x) on I(a, b),
given by
i(h2 (x) − h1 (x)) = min{λ(h2 (x) − h1 (x)) : x ∈ I(a, b)},
which exists as shown in [17]. On the other hand, if A is of the form
A = {(x, y) ∈ R2 : x ∈ I(h1 (y), h2 (y)), y ∈ I(a, b)},
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306
KHODR SHAMSEDDINE AND DARREN FLYNN
we define
λy (A) = λ(b − a) and λx (A) = i(h2 (y) − h1 (y)).
We call λx (A) and λy (A) the orders of magnitude of A in x and y, respectively. If
λx (A) = λy (A) = 0 then we say that A is a finite simple region.
Definition 3.16 (Analytic Functions on R2 ). Let A ⊂ R2 be a simple region.
Then we say that f : A → R2 is an analytic function on A if, for every (x0 , y0 ) ∈ A,
there exist a simple region A0 containing (x0 , y0 ) that satisfies λx (A0 ) = λx (A) and
λy (A0 ) = λy (A), and a regular sequence (aij )∞
i,j=0 such that for every s, t ∈ R, if
(x0 + s, y0 + t) ∈ A ∩ A0 then
∞
∞
f (x0 + s, y0 + t) =
aij si tj = f (x0 , y0 ) +
aij si tj ,
i,j=0
i,j=0
i+j=0
where the power series converges in the weak topology [9, 15].
The following proposition follows directly from Definition 3.16.
Proposition 3.17. Let A ⊂ R2 be a simple region and let f : A → R2 be an
analytic function on A. Then for a fixed x, the function g(y) := f (x, y) is analytic
on Ix = {y ∈ R : (x, y) ∈ A}; and for a fixed y, h(x) := f (x, y) is analytic on
Iy = {x ∈ R : (x, y) ∈ A}.
Proposition 3.18. Let A ⊂ R2 be a simple region and let f : A → R be
analytic on A. Then f is bounded on A.
Proof. Without loss of generality, we may assume that A is the form
A = {(x, y) ∈ R2 : y ∈ [h1 (x), h2 (x)], x ∈ [a, b]},
with h1 , h2 : [a, b] → R analytic on [a, b]. Let F (x, y) : [0, 1]2 → R be given by
F (x, y) = f ((b − a)x + a, (h2 (x) − h1 (x))y + h1 (x)).
Then F is analytic on [0, 1]2 and f is bounded on A if and only if F is bounded on
[0, 1]2 .
For every (v, w) ∈ R2 let N ((v, w), η) = {(x, y) ∈ R2 : (x − v)2 + (y − w)2 <
η}. Since F is analytic on [0, 1]2 , it follows that, for every (x0 , y0 ) ∈ [0, 1]2 ∩ R2 ,
there exists a real η(x0 , y0 ) > 0 and a regular sequence (aij (x0 , y0 ))∞
i,j=0 such that
for every (x, y) ∈ N ((x0 , y0 ), η(x0 , y0 )) ∩ [0, 1]2 , we have that
∞
F (x, y) =
aij (x0 , y0 )(x − x0 )i (y − y0 )j .
i,j=0
+
*
2
2
2
∩R
is an open cover of [0, 1]2 ∩
The set N (x, y), η(x,y)
:
(x,
y)
∈
[0,
1]
∩R
2
R2 which is a compact subset of the Euclidean space R2 ; hence we can select a
finite subcover. Thus, there exists a finite set of points {(xk , yk )}m
k=1 contained in
+
*
m
η(xk ,yk )
2
2
2
2
2
∩ R . It follows that
N (xk , yk ),
[0, 1] ∩ R such that [0, 1] ∩ R ⊂
2
[0, 1] ⊂
2
m
k=1
N ((xk , yk ), η(xk , yk )). Let
k=1
⎧
⎨
l = min
1km ⎩
min
⎧
∞
⎨ ⎩
i,j=0
⎫⎫
⎬⎬
supp(aij (xk , yk ))
⎭⎭
,
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MEASURE THEORY AND INTEGRATION OVER THE LEVI-CIVITA FIELD
307
which exists by the regularity of the sequence (aij (xk , yk )) for each k. It follows
from the above that |F (x, y)| < dl−1 for every (x, y) ∈ [0, 1]2 . Thus F is bounded
on [0, 1]2 and hence f is bounded on A.
Remark 3.19. Let A, f , F , {(xk , yk )}m
k=1 , and l be as in Proposition 3.18 and
the proof thereof. Then
"
!
l = min {λ(f (x, y)) : (x, y) ∈ A} = min λ(F (x, y)) : (x, y) ∈ [0, 1]2 .
2
2
Thus, l is independent of our choice of the finite set {(xk , yk )}m
k=1 in [0, 1] ∩ R in
the proof of Proposition 3.18 above.
Definition 3.20. Let A, f and l be as in Proposition 3.18 and Remark 3.19.
Then we call l the index of f on A and we denote it by i(f ). Thus,
i(f ) := l = min {λ(f (x, y)) : (x, y) ∈ A} .
Proposition 3.21. Let A ⊂ R2 be a finite simple region, let f, g : A → R2 be
analytic functions on A, and let α ∈ R be given. Then f + αg and f · g are analytic
functions on A.
Proof. Let (x0 , y0 ) ∈ A be given. Since f and g are analytic on A, there exist
finite η1 , η2 > 0 such that for every s, t ∈ R, if s2 + t2 < η12 and (x0 + s, y0 + t) ∈ A
∞
∞ then f (x0 + s, y0 + t) =
aij si tj , and if s2 + t2 < η22 and (x0 + s, y0 + t) ∈ A
i=0 j=0
then g(x0 + s, y0 + t) =
∞
∞ bkl sk tl . Let η = min{η1 , η2 }. Then for every s, t ∈ R
k=0 l=0
satisfying s2 + t2 < η 2 and (x0 + s, y0 + t) ∈ A, we have that
(f + αg)(x0 + s, y0 + t) = f (x0 + s, y0 + t) + αg(x0 + s, y0 + t)
=
∞ ∞
aij si tj + α
i=0 j=0
=
=
∞ ∞
i=0 j=0
∞ ∞
∞ ∞
bkl sk tl
k=0 l=0
(aij + αbij )si tj
cij si tj ,
i=0 j=0
where cij = aij + αbij . Hence f + αg is analytic on A. Moreover,
(f · g)(x0 + s, y0 + t) = f (x0 + s, y0 + t) · g(x0 + s, y0 + t)
⎞
⎛
∞ ∞ ∞
∞
aij si tj ⎠
bkl sk tl
=⎝
i=0 j=0
=
∞ ∞
k=0 l=0
aij bkl sn tm
n=0 m=0 i+k=n j+l=m
Note that the Cauchy product of two power series converging weakly in R also
converges [15]; we infer that the same is true for the Cauchy product of two (weakly)
converging power series in two variables. Defining
aij bkl
enm =
i+k=n j+l=m
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308
KHODR SHAMSEDDINE AND DARREN FLYNN
yields
(f · g)(x0 + s, y0 + t) =
∞ ∞
enm sn tm .
n=0 m=0
Hence f · g is analytic on A.
Corollary 3.22. Let A ⊂ R2 be a simple region, let f, g : A → R2 be analytic
functions on A, and let α ∈ R be given. Then f +αg and f ·g are analytic functions
on A.
Proof. Without loss of generality, we may assume that A is of the form A =
{(x, y) ∈ R2 : y ∈ I(h1 (x), h2 (x)), x ∈ I(a, b)} with h1 , h2 : I(a, b) → R analytic on
I(a, b). Let A0 = {(x, y) ∈ R2 : y ∈ I(d−λy (A) h1 (dλx (A) x), d−λy (A) h2 (dλx (A) x)), x ∈
I(d−λx (A) a, d−λx (A) b)}. Then A0 is a finite simple region. Moreover, the functions F, G : A0 → R, given by F (x, y) = f (dλx (A) x, dλy (A) y) and G(x, y) =
g(dλx (A) x, dλy (A) y) are both analytic on A0 so by Proposition 3.21 F + αG and
F · G are analytic on A0 . It follows that f + αg and f · g are analytic functions on
A.
Proposition 3.23. Let A ⊂ R2 be a finite simple region and let f : A → R be
an analytic function on A. Let a, b ∈ R be such that a < b and b − a is finite, let
g : I(a, b) → R be an R-analytic function on I(a, b) such that for every x ∈ I(a, b),
(x, g(x)) ∈ A. Then the function F : I(a, b) → R, given by F (x) = f (x, g(x)) is an
R-analytic function on I(a, b).
Proof. Without loss of generality, we may assume that i(f ) = 0 on A. Now let
x0 ∈ A be given. By the definition of analytic functions there exist finite η1 , η2 > 0
∞
such that if |h| < η1 and x0 + h ∈ I(a, b) then g(x0 + h) = g(x0 ) +
an hn and if
n=1
∞
∞ s2 + t2 < η22 and (x0 + s, g(x0 ) + t) ∈ A then f (x0 + s, g(x0 ) + t) =
Define F : R → R by F (X) = (X + (
2
∞
bij si tj .
i=0 j=0
n 2
an X ) )[0]. Then F is continuous
n=1
on R and hence we can choose a real number η ∈ (0, η21 ] such that if |h| < η and
∞
η2
x0 + h ∈ I(a, b) then F (h[0]) < 22 , and hence, since h2 + (
an hn )2 is different
n=1
from F (h[0]) by at most an infinitely small amount, it follows that if |h| < η and
∞
an hn )2 < η22 . Thus, for any |h| < η such that x0 +h ∈
x0 +h ∈ I(a, b) then h2 +(
2
n=1
∞
I(a, b), we have that h + (
n=1
an hn )2 < η22 and (x0 + h, g(x0 ) +
N, Vij (h) can be rewritten as Vij (h) =
∞
bij hi (
∞
an hn ) ∈ A, and
n=1
an hn ) j .
n=1
i,j=0
i+j=0
∞
bij hi (
an hn )j . Then for every
n=1
∞
cijm hm . Since for every i, j
m=1
hence f (x0 + h, g(x0 + h)) = f (x0 , g(x0 )) +
Now, for every i, j ∈ N, let Vij (h) =
∞
∞
i, j ∈
∈ N,
cijm hm converges, it follows that for every i, j ∈ N and for every q ∈ Q,
m=1
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MEASURE THEORY AND INTEGRATION OVER THE LEVI-CIVITA FIELD
∞
309
(cijm hm )[q] converges absolutely in R [15]. This, in addition to the fact that
m=1
∞
for every q ∈ Q,
(bij hi (
∞
n=1
i,j=0
i+j=0
an hn )j )[q] =
∞
∞
(cijm hm )[q] converges in R
i,j=0 m=1
i+j=0
allows us to change the order of summation in the triple sum. Thus, for every
q ∈ Q, we have that
∞
∞
∞
n
an h
(cijm hm )[q]
[q] = f (x0 , g(x0 ))[q] +
f x0 + h, g(x0 ) +
n=1
= f (x0 , g(x0 ))[q] +
i,j=0 m=1
i+j=01
∞ ∞
(cijm hm )[q];
m=1 i,j=0
i+j=0
and hence
f (x0 + h, g(x0 + h)) = f (x0 , g(x0 )) +
∞ ∞
cijm hm
m=1 i,j=0
i+j=0
⎛
= f (x0 , g(x0 )) +
∞
∞
⎜ ⎟ m
⎜
cijm ⎟
⎝
⎠h
m=1
Letting em =
∞
∞
⎞
i,j=0
i+j=0
cijm , we obtain that f (x0 + h, g(x0 + h)) = f (x0 , g(x0 )) +
i,j=0
i+j=0
em hm . Thus, for every x ∈ I(a, b) there exists a finite η > 0 such that for every
m=1
|h| < η, f (x0 + h, g(x0 + h)) is given by a power series about x0 , which means that
f (x, g(x)) is an R-analytic function on I(a, b).
Corollary 3.24 follows directly from Proposition 3.23 using a scaling argument
similar to that for Corollary 3.22.
Corollary 3.24. Let A ⊂ R2 be a simple region and let f : A → R be an
analytic function on A. Let a, b ∈ R be such that a < b, let g : I(a, b) → R be an
R-analytic function on I(a, b) such that for every x ∈ I(a, b), (x, g(x)) ∈ A. Then
the function F : I(a, b) → R, given by F (x) = f (x, g(x)) is an R-analytic function
on I(a, b).
The proof of Proposition 3.25 below is similar to that of Proposition 3.23 and
Corollary 3.24 above. We skip the details here and refer the interested reader to
[3].
Proposition 3.25. Let S ⊂ R2 be a simple region and let f : S → R be an
analytic function on S. Let H ⊂ R2 be a simple region and let g : H → R be an
analytic function on H such that for every (x, y) ∈ H, (x, g(x, y)) ∈ S. Then the
function F : H → R, given by F (x, y) = f (x, g(x, y)) is analytic on H.
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310
KHODR SHAMSEDDINE AND DARREN FLYNN
3.3. Measurable Functions on R2 .
Definition 3.26 (Measurable Function). Let A ⊂ R2 be a measurable set and
let f : A → R be bounded on A. Then we say that f is measurable on A if for every
> 0 in R there exists a sequence of mutually disjoint simple regions (Gn )∞
n=1 such
∞
∞
∞
Gn ⊂ A ,
a(Gn ) converges, m(A) −
a(Gn ) < , and for all n ∈ N f
that
n=0
n=1
n=1
is analytic on Gn .
Proposition 3.27. Let A ⊂ R2 be a measurable set and let f : A → R be
a measurable function on A. Then f is given locally by a power series almost
everywhere on A.
Proof. Let A0 = {(x, y) ∈ A : f is not given locally by a power series about
(x, y)} We show that A0 is measurable and m(A0 ) = 0. Let > 0 be given in R.
Since f is measurable on A, there exists a sequence of mutually disjoint open simple
∞
∞
∞
Gn ⊂ A,
a(Gn ) converges, m(A)−
a(Gn ) 2 ,
regions (Gn )∞
n=1 such that
n=1
n=1
n=1
and f is analytic on Gn for all n.
Also, since A is measurable, there exists a sequence of mutually disjoint simple
∞
∞
∞
Hn ,
a(Hn ) converges,
a(Hn ) − m(A) regions (Hn )∞
n=1 such that A ⊂
n=1
2.
n=1
n=1
∞
Since f is given by a power series around every point in
Gn and since
n=1
A0 ⊂ A, we have that
A0 ⊂ A\
∞
Gn
⊂
n=1
∞
Hn
n=1
\
∞
Gn
.
n=1
∞
Since both (Gn )∞
n=1 and (Hn )n=1 consist of mutually disjoint sets, we can
rearrange the members of the sequences so that for every n ∈ N, Gn ⊂ Hn . Thus,
∞
n=1
Hn
\
∞
Gn
n=1
=
∞
(Hn \Gn ) .
n=1
The Hn \Gn ’s are mutually disjoint, and hence for every n ∈ N, Hn \Gn may be
expressed as the union of a finite number of mutually disjoint simple regions. It
∞
∞
Hn \
Gn may be rewritten as the union of countably many
follows that
n=1
n=1
mutually disjoint simple regions (Hn0 )∞
n=1 .
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MEASURE THEORY AND INTEGRATION OVER THE LEVI-CIVITA FIELD
For every n ∈ N, let G0n = ∅. Then we have
∞
a(Hn0 ) −
n=1
∞
a(G0n ) =
n=1
=
=
∞
n=1
∞
∞
n=1
G0n ⊂ A0 ⊂
∞
n=1
311
Hn0 . Moreover,
a(Hn0 )
a(Hn ) −
n=1
∞
∞
a(Gn )
n=1
a(Hn ) − m(A)
n=1
+
m(A) −
∞
a(Gn )
n=1
+
2 2
=
∞
So A0 is measurable. Since A0 ⊂
n=1
Hn0 we have that m(A0 ) This is true for all > 0 in R; hence m(A0 ) = 0.
∞
n=1
a(Hn0 ) .
Proposition 3.28. Let A ⊂ R2 be a simple region and let f : A → R be a
∂
∂
∂
f (x, y) and ∂y
f (x, y) both exist and ∂x
f (x, y) =
measurable function such that ∂x
∂
f
(x,
y)
=
0
everywhere
in
A
(see
[12]
for
the
definition
of
partial
derivatives).
∂y
Then f (x, y) is constant on A
Proof. Fix two arbitrary points (x0 , y0 ), (x1 , y1 ) ∈ A. By Proposition 3.27,
we have that f (x, y) is analytic almost everywhere on A, and hence by Corollary
3.17, we have that for a fixed y, the function gy (x) := f (x, y) is R-analytic almost
everywhere on {x ∈ R : (x, y) ∈ A} and for a fixed x, the function hx (y) := f (x, y) is
R-analytic almost everywhere on {y ∈ R : (x, y) ∈ A}. Therefore, gy0 (x) = f (x, y0 )
∂
is R-analytic almost everywhere on {x ∈ R : (x, y0 ) ∈ A} and since ∂x
f (x, y0 ) =
gy0 (x) = 0 everywhere on {x ∈ R : (x, y0 ) ∈ A} it follows that gy0 (x) is constant
[8]. Thus f (x0 , y0 ) = f (x1 , y0 ). Using a similar argument, making use of the fact
∂
that ∂y
f (x, y) = 0 everywhere in A, we show that f (x1 , y0 ) = f (x1 , y1 ), and hence
f (x0 , y0 ) = f (x1 , y1 ).
Proposition 3.29. Let A, B ⊂ R2 be measurable sets and let f : A, B → R be
a measurable function on both A and B. Then f is a measurable function on A ∩ B
and A ∪ B.
Proof. Let > 0 in R be given. Since f is measurable on A there exists a
∞
sequence of mutually disjoint simple regions (GA
n )n=1 such that for every n ∈ N, f
∞
∞
∞
GA
a(GA
a(GA
is analytic on GA
n,
n ⊂ A,
n ) converges, and m(A) −
n ) < 2.
n=1
n=1
n=1
Also, since f is measurable on B there exists a sequence of mutually disjoint simple
∞
∞
B
regions (GB
GB
n )n=1 such that for every n ∈ N, f is analytic on Gn ,
n ⊂ B,
∞
n=1
∞
∞
n=1
n=1
converges, and m(B) −
<
n=1
∞
B
GA
Gn can be rewritten as the union of mutually disjoint simn
a(GB
n)
ple regions
∞
n=1
n=1
G0n
⊂ A ∪ B such that
a(GB
n)
∞
n=1
2.
a(G0n ) converges and for every n ∈ N, f
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312
KHODR SHAMSEDDINE AND DARREN FLYNN
∞
is analytic on G0n . Since
n=1
G0n =
∞
n=1
GA
n
∞
n=1
⊂ A ∪ B and since the
GB
n
G0n ’s are mutually disjoint, it follows that
∞
∞
∞
m(A ∪ B) −
+ .
a(G0n ) m(A) −
a(GA
)
+
m(B)
−
a(GB
n
n) 2
2
n=1
n=1
n=1
Thus, f ismeasurable
A ∪ B.
on
∞
∞
'
Also,
GA
GB
can be rewritten as a countable union of mutually
n
n
n=1
n=1
disjoint simple regions
∞
∞
G1n ⊂ A∩B, such that
n=1
on G1n .
n=1
a(G1n ) converges and for every
n ∈ N, f is analytic
Using Proposition 3.14 we have that m(A ∩ B) = m(A) + m(B) − m(A ∪ B),
and
∞
∞
∞
∞
1
A
B
0
Gn = m
Gn + m
Gn − m
Gn .
m
n=1
n=1
n=1
It follows that
m(A ∩ B) −
∞
a(G1n )
= m(A ∩ B) − m
n=1
= m(A) + m(B) − m(A ∪ B) + m
n=1
∞
G1n
n=1
∞
G0n
−m
n=1
= m(A) −
m(A) −
∞
n=1
∞
a(GA
n ) + m(B) −
a(GA
n ) + m(B) −
n=1
∞
n=1
∞
∞
−m
GA
n
n=1
a(GB
n ) − m(A ∪ B) +
∞
GB
n
n=1
∞
a(G0n )
n=1
a(GB
n)
n=1
+ = .
2 2
Thus, f is measurable on A ∩ B.
Proposition 3.30. Let A ⊂ R2 be measurable, let f, g : A → R be both
measurable on A, and let α ∈ R be given. Then f + αg and f · g are measurable on
A.
Proof. Fix > 0 in R. Since f and g are both measurable on A, there exist
g ∞
sequences of mutually disjoint simple regions (Gfn )∞
n=1 and (Gn )n=1 such that, for
∞
∞
f
every n ∈ N, f is analytic on Gfn and g is analytic on Ggn ,
Gn ⊂ A,
Ggn ⊂ A,
∞
n=1
∞
n=1
a(Gfn )
and
∞
n=1
a(Ggn )
both converge, m(A) −
∞
n=1
n=1
a(Gfn )
n=1
< /2, and m(A) −
a(Ggn ) < /2.
∞
∞
f ' g
Gn
Gn can be rewritten as the union of mutually disjoint simn=1
ple regions
∞
n=1
Ti ⊂ A such that
i=1
∞
a(Ti ) converges, and for every i ∈ N, f and g
i=1
are analytic on Ti .
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MEASURE THEORY AND INTEGRATION OVER THE LEVI-CIVITA FIELD
Since
∞
Ti =
i=1
∞
n=1
Gfn
'
∞
n=1
313
Ggn
and since the Ti ’s are mutually disjoint,
the Gfn ’s are mutually disjoint and the Ggn ’s are mutually disjoint, we obtain that
∞ ∞
a(Ti ) = m(A) − m
Ti
m(A) −
i=1
(
i=1
∞
m(A) − m
n=1
m(A) −
=
)
∞
(
+ m(A) − m
Gfn
a(Gfn )
+
m(A) −
n=1
∞
)
Ggn
n=1
∞
a(Ggn )
n=1
< .
We know from Corollary 3.22 that for each i ∈ N, f + αg and f · g are analytic
on Ti . Thus, there exists a sequence of mutually disjoint simple regions (Ti )∞
i=1
∞
∞
∞
such that
Ti ⊂ A,
a(Ti ) converges, m(A) −
a(Ti ) < , and for every i ∈ N,
i=1
i=1
i=1
f + αg and f · g are R-analytic on Ti . Therefore, f + αg and f · g are measurable
on A.
3.4. Integration on R2 .
Definition 3.31 (Integration of Analytic Functions on Simple Regions). Let
H ⊂ R2 be a simple region and let f : H → R be an analytic function. First
assume that H = {(x, y) ∈ R2 : y ∈ I(h1 (x), h2 (x)), x ∈ I(a, b)}, where a, b ∈ R,
a b, and h1 , h2 : I(a, b) → R are analytic on I(a, b) with h1 < h2 . We define the
integral of f over H as the iterated integral
⎡
⎤
⎢
⎥
f (x, y) =
f (x, y)⎦ .
⎣
(x,y)∈H
x∈I(a,b)
y∈I(h1 (x),h2 (x))
We note
that, for each x ∈ I(a, b), f (x, y) is R-analytic on I(h1 (x), h2 (x)); hence
f (x, y) is well-defined and it yields an R-analytic function F (x) on
y∈I(h1 (x),h2 (x))
I(a, b). Then
f (x, y) =
(x,y)∈H
F (x)
x∈I(a,b)
is well-defined.
On the other hand, if H is given by H = {(x, y) ∈ R2 : x ∈ I(h1 (y), h2 (y)), y ∈
I(a, b)} then we define the integral of f over H as the iterated integral
⎡
⎤
⎢
⎥
f (x, y) =
f (x, y)⎦ .
⎣
(x,y)∈H
y∈I(a,b)
x∈I(h1 (y),h2 (y))
Lemma 3.32. Let G ⊂ R be a simple region and let α ∈ R be given. Then
α = αa(G).
2
(x,y)∈G
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KHODR SHAMSEDDINE AND DARREN FLYNN
Proof. Without loss of generality, we may assume that G = {(x, y) ∈ R2 :
y ∈ I(g1 (x), g2 (x)), x ∈ I(a, b)}, where a, b ∈ R, a b, and g1 , g2 : I(a, b) → R
are R-analytic on I(a,b) with g1 < g2 . Thus, by the linearity of the single integral
[11, 16], we have that
⎡
⎤
⎡
⎤
⎢
⎥
⎢
⎥
α=
α⎦ =
α⎣
1⎦
⎣
(x,y)∈G
x∈I(a,b)
y∈I(g1 (x),g2 (x))
⎡
⎢
⎣
=α
x∈I(a,b)
⎤
x∈I(a,b)
⎥
1⎦ = α
y∈I(g1 (x),g2 (x))
y∈I(g1 (x),g2 (x))
[g2 (x) − g1 (x)] = αa(G).
x∈I(a,b)
Using Definition 3.31 and the linearity of the single integral, we readily obtain
the following result which will be used later to prove the linearity property for the
integral of a measurable function over a measurable subset of R2 .
Lemma 3.33. Let H ⊂ R2 be a simple region, let f, g : H → R be analytic
functions on H, and let α ∈ R be given. Then
(f + αg)(x, y) =
f (x, y) + α
g(x, y).
(x,y)∈H
(x,y)∈H
(x,y)∈H
Lemma 3.34. Let G ⊂ R be a simple region and let f : G → R be analytic
such that f ≤ 0 on G. Then
f (x, y) 0.
2
(x,y)∈G
Proof. Without loss of generality we may assume that G = {(x, y) ∈ R2 :
y ∈ I(g1 (x), g2 (x)), x ∈ I(a, b)} where a, b ∈ R, a b, and g1 , g2 : I(a, b) →
R are analytic on I(a, b) with g1 < g2 on I(a, b). Since f ≤ 0 on G, then for
every x ∈ I(a, b), we have that f (x, y) is a non-positive R-analytic function on
I(g1 (x),
g2 (x)). It follows from Proposition 4.4 in [16] that for every x ∈ I(a, b),
f (x, y) 0. Thus, using Proposition 4.4 of [16] again, we obtain that
y∈I(g1 (x),g2 (x))
f (x, y) =
(x,y)∈G
x∈I(a,b)
⎡
⎢
⎣
⎤
⎥
f (x, y)⎦ 0.
y∈I(g1 (x),g2 (x))
region and let f, h : G → R be
Corollary 3.35. Let G ⊂ R2 be a simple f (x, y) h(x, y).
analytic functions such that f g on G. Then
(x,y)∈G
(x,y)∈G
Corollary 3.36. Let G be a simple region, let f : G → R be an analytic
function, and let M be a bound for |f | on G. Then
f (x, y) ≤
|f (x, y)| M a(G).
(x,y)∈G
(x,y)∈G
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MEASURE THEORY AND INTEGRATION OVER THE LEVI-CIVITA FIELD
315
Definition 3.37. [The Integral of a Measurable Function over a Measurable
Set] Let A ⊂ R2 be a measurable set, let f : A → R be a measurable function,
and let M be a bound for |f | on A. Since f is measurable then for every k ∈
N there exists a sequence of mutually disjoint simple regions (Gkn )∞
n=1 such that
∞
∞
for every n ∈ N, f is analytic on Gkn ,
Gkn ⊂ A,
a(Gkn ) converges, and
∞
m(A) −
n=1
n=1
a(Gkn )
n=1
d . Without loss of generality we may assume that for every
k
⊂ Gk+1
k and n in N,
n .
∞
Since
a(Gkn ) converges we have that lim a(Gkn ) = 0. From Corollary 3.36
n→∞
n=1
f (x, y)| M a(Gkn ), and hence lim
f (x, y) = 0.
we have that |
Gkn
(x,y)∈Gk
n
Therefore, for every k ∈ N,
n→∞
(x,y)∈Gk
n
∞
f (x, y) converges. Now, fix > 0 in R
n=1(x,y)∈Gk
n
k0
and select a k0 ∈ N such that M d . Let i > j k0 be given in N. We can
∞
∞
∞
write
Gin \
Gjn as the union of mutually disjoint simple regions (Gi,j
n )n=1
n=1
such that
∞
n=1
n=1
a(Gi,j
n ) converges and
∞
a(Gi,j
n ) =
n=1
∞
a(Gin ) −
n=1
∞
a(Gjn )
n=1
m(A) −
∞
a(Gjn ) dj dk0 .
n=1
It follows that
∞ ∞ ∞
f (x, y) −
f (x, y) = f (x, y)
n=1
n=1(x,y)∈Gi
n=1(x,y)∈Gi,j
(x,y)∈Gjn
n
n
∞ ∞
∞
k0
f (x, y) M a(Gi,j
)
=
M
a(Gi,j
,
n
n ) Md
n=1 n=1
n=1
(x,y)∈Gi,j
n
Thus,
∞
∞
f (x, y)
is a Cauchy sequence and hence it converges. We
k=1
f (x, y).
call the limit of this sequence the integral of f over A and we denote it by
n=1(x,y)∈Gk
n
(x,y)∈A
Thus,
f (x, y) =
∞
(x,y)∈A
lim
a(Gn )→m(A),
n=1
∞
Gn ⊂A,Gn ’s
∞
f (x, y)
n=1
(x,y)∈Gn
mutually disjoint,
n=1
f
is analytic on Gn for every n
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316
KHODR SHAMSEDDINE AND DARREN FLYNN
Proposition 3.38. Let A ⊂ R2 be measurable and let α ∈ R be given. Then
the function f : A → {α} is measurable on A and
f=
α = αm(A).
(x,y)∈A
(x,y)∈A
Proof. Since A is measurable, for every k ∈ N there exists a sequence of
∞
∞
mutually disjoint simple regions (Gkn )∞
Gkn ⊂ A,
a(Gkn ) conn=1 such that
∞
n=1
verges, and m(A) −
n=1
moreover,
a(Gkn )
k
< d . f is analytic on
Gkn
n=1
for each n and for each k;
f=
(x,y)∈Gk
n
(x,y)∈Gk
n
α = αa(Gkn ).
From Definition 3.37, it follows that
∞
∞
∞
f = lim
f = lim
αa(Gkn ) = α lim
a(Gkn ) = αm(A).
k→∞
(x,y)∈A
k→∞
n=1
(x,y)∈Gk
n
k→∞
n=1
n=1
Proposition 3.39. Let A ⊂ R2 be measurable and let f : A → R be a measurable function such that f ≤ 0 on A. Then
f (x, y) 0.
(x,y)∈A
Proof. Let > 0 in R be given. There exists a sequence of mutually disjoint
∞
simple regions (Gn )∞
Gn ⊂ A,
n=1 such that f is analytic on Gn for each n,
∞
∞
a(Gn ) converges, and m(A) −
n=1
n=1
a(Gn ) . We have from Lemma 3.34 that
n=1
for every n ∈ N,
f (x, y) 0.
(x,y)∈Gn
Thus,
∞ n=1
It follows that
f (x, y) 0.
(x,y)∈Gn
f (x, y) = lim
→0
(x,y)∈A
∞
f (x, y) 0.
n=1
(x,y)∈Gn
Proposition 3.40. Let A ⊂ R2 be measurable, let f, g : A → R be measurable
functions on A, and let α ∈ R be given. Then
(f + αg)(x, y) =
f (x, y) + α
g(x, y).
(x,y)∈A
(x,y)∈A
(x,y)∈A
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MEASURE THEORY AND INTEGRATION OVER THE LEVI-CIVITA FIELD
317
Proof. Let > 0 in R be given. Since f and g are measurable on A there exist
g ∞
two sequences of mutually disjoint simple regions (Gfn )∞
n=1 and (Gn )n=1 such that
∞
∞
f
g
Gn and
Gn are both subsets of A, for every n ∈ N f is analytic on Gfn and
n=1
n=1
g is analytic on Ggn ,
and m(A) −
∞
n=1
∞
n=1
a(Gfn ) and
∞
n=1
a(Ggn ) 2 .
a(Ggn ) converge, and m(A) −
By Proposition 3.10 we can write
∞
Gfn
n=1
ally disjoint simple regions (Tn )∞
n=1 such that
'
∞
∞
n=1
a(Gfn ) 2
∞
n=1
Ggn
as the union of mutu∞
a(Tn ) converges, m(A)−
n=1
a(Tn )
n=1
, and for every n ∈ N f and g are both analytic on Tn . It follows from Lemma
3.33 that, for every n ∈ N,
(f + αg)(x, y) =
f (x, y) + α
g(x, y).
(x,y)∈Tn
(x,y)∈Tn
(x,y)∈Tn
Therefore,
∞
(f + αg)(x, y) =
n=1
(x,y)∈Tn
∞ f (x, y) + α
n=1 T
n
∞
g(x, y).
n=1
(x,y)∈Tn
Thus,
(f + αg)(x, y) = lim
→0
(x,y)∈A
= lim
→0
∞ (f + αg)(x, y)
n=1
(x,t)∈Tn
∞
f (x, y) + α lim
→0
n=1
(x,y)∈Tn
=
(x,y)∈A
f (x, y) + α
∞
g(x, y)
n=1
(x,y)∈Tn
g(x, y).
(x,y)∈A
Corollary 3.41. Let A ⊂ R2 be measurable and let f, g : A → R be measurable functions such that f g on A. Then
f (x, y) g(x, y).
(x,y)∈A
(x,y)∈A
Corollary 3.42. Let A ⊂ R2 be measurable, let f : A → R be a measurable
function on A, and let M be a bound for |f | on A. Then
f (x, y) |f (x, y)| M m(A).
(x,y)∈A
(x,y)∈A
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318
KHODR SHAMSEDDINE AND DARREN FLYNN
Proposition 3.43. Let A, B ⊂ R2 be measurable sets and let f be a measurable
function on A and B. Then
f (x, y) =
f (x, y) +
f (x, y) −
f (x, y).
(x,y)∈A∪B
(x,y)∈A
(x,y)∈B
(x,y)∈A∩B
Proof. Let > 0 in R be given. Since f is a measurable function on A ∪
B there exists a sequence of mutually disjoint simple regions (Gn )∞
n=1 such that
∞
∞
∞
a(Gn ) converges,
Gn ⊂ A ∪ B, m(A ∪ B) −
a(Gn ) 2 , and for every
n=1
n=1
n=1
n ∈ N, f is analytic on Gn .
We can arrange (Gn )∞
n=1 into three sequences of mutually disjoint simple regions
∞
∞
∞
2 ∞
3 ∞
a(G1n ),
a(G2n ), and
a(G3n )
(G1n )∞
n=1 , (Gn )n=1 , and (Gn )n=1 such that
converge;
∞
G1n
n=1
∞
m(A ∪ B) −
n=1
⊂ A\(A ∩ B),
a(G1n ) −
∞
n=1
n=1
∞
n=1
a(G2n ) −
f (x, y) = lim
→0
(x,y)∈A∪B
= lim
→0
+ lim
→0
= lim
→0
+ lim
→0
− lim
→0
G2n
∞
n=1
∞
→0
− lim
→0
f (x, y) + lim
∞
→0
∞
∞
f (x, y) + lim
→0
n=1
(x,y)∈G1n
∞
f (x, y) + lim
→0
n=1
(x,y)∈G2n
∞
∞
n=1
f (x, y)
n=1
(x,y)∈G3n
∞
f (x, y)
n=1
(x,y)∈G3n
f (x, y)
n=1
(x,y)∈G3n
f (x, y) +
(x,y)∈G1n
(x,y)∈G3n
⎜
⎝
f (x, y)
n=1
(x,y)∈G2n
f (x, y)
n=1
(x,y)∈G3n
∞
⊂ (A ∩ B);
a(G3n ) . Thus,
n=1
(x,y)∈G1n
⎛
+ lim
n=1
n=1
G3n
f (x, y)
∞
n=1
∞
⊂ B\(A ∩ B), and
n=1
(x,y)∈Gn
⎛
∞
⎜
= lim
⎝
→0
n=1
f (x, y) +
(x,y)∈G2n
∞
⎞
⎟
f (x, y)⎠
⎞
⎟
f (x, y)⎠
(x,y)∈G3n
f (x, y).
n=1
(x,y)∈G3n
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MEASURE THEORY AND INTEGRATION OVER THE LEVI-CIVITA FIELD
319
2 ∞
3 ∞
By our choice of (G1n )∞
n=1 , (Gn )n=1 , and (Gn )n=1 ,
⎛
⎞
∞
⎜
⎟
lim
f (x, y) +
f (x, y)⎠ =
f (x, y),
⎝
→0
n=1
(x,y)∈G1n
(x,y)∈G3n
⎛
∞
⎜
lim
⎝
→0
n=1
⎞
⎟
f (x, y)⎠ =
f (x, y) +
(x,y)∈G2n
and
lim
→0
(x,y)∈A
(x,y)∈G3n
∞
f (x, y) =
n=1
(x,y)∈G3n
(x,y)∈A
f (x, y),
(x,y)∈B
f (x, y).
(x,y)∈A∩B
Thus, we finally get that
f (x, y) =
f (x, y) +
f (x, y) −
(x,y)∈A∪B
(x,y)∈B
f (x, y).
(x,y)∈A∩B
Proposition 3.44. Let A ⊂ R2 be measurable, let f : A → R, and for each
k ∈ N let fk : A → R be a measurable functionon A such that the sequence (fk )∞
k=1
converges uniformly to f on A. Then lim
fk (x, y) exists; moreover, if f is
k→∞(x,y)∈A
measurable on A then
lim
k→∞
(x,y)∈A
fk (x, y) =
f (x, y).
(x,y)∈A
Proof. Let > 0 in R be given and let
if m(A) = 0
.
0 = m(A)
if m(A) = 0
Then there exists a k0 ∈ N such that for every i, j k0 , |fi (x, y) − fj (x, y)| 0
for every (x, y) ∈ A. Thus,
fi (x, y) −
fj (x, y) = (fi (x, y) − fj (x, y))
(x,y)∈A
(x,y)∈A
(x,y)∈A
|fi (x, y) − fj (x, y)| 0 m(A) .
(x,y)∈A
∞
fk (x, y)
is a Cauchy sequence; since R is Cauchy complete, it
k=1
follows that lim
fk (x, y) exists.
Thus,
(x,y)∈A
k→∞(x,y)∈A
Now assume that f is measurable on A. Let > 0 be given in R and let 0 be
defined as above. Since (fk )∞
k=1 converges uniformly to f , there exists a k ∈ N such
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320
KHODR SHAMSEDDINE AND DARREN FLYNN
that, for every i k, we have that |fi (x, y) − f (x, y)| 0 for every (x, y) ∈ A. It
follows that
fi (x, y) −
f (x, y) |fi (x, y) − f (x, y)| 0 m(A) ,
(x,y)∈A
(x,y)∈A
(x,y)∈A
and hence lim
fk (x, y) =
f (x, y).
k→∞(x,y)∈A
(x,y)∈A
Just as we used the measure theory and integration in one dimension to extend
the results to two dimensions, we can use the measure theory and integration in
two dimensions developed above to extend the results to three dimensions (and, by
induction, to higher dimensions). In the following section, we only present the key
steps needed for going from two to three dimensions; the details of the theory in
three dimensions are similar to those in two dimensions and hence we will leave out
those details but we refer the interested reader to [3].
4. Measure Theory and Integration on R3
Definition 4.1 (Simple Region). Let S ⊂ R3 . Then we say S is a simple
region in R3 if there exists a simple region A ⊂ R2 and two analytic functions
h1 , h2 : A → R such that h1 < h2 everywhere on A and
S = {(x, y, z) ∈ R3 : z ∈ I(h1 (x, y), h2 (x, y)), (x, y) ∈ A}
or
S = {(x, y, z) ∈ R3 : y ∈ I(h1 (x, z), h2 (x, z)), (x, z) ∈ A}
or
S = {(x, y, z) ∈ R3 : x ∈ I(h1 (y, z), h2 (y, z)), (y, z) ∈ A}.
Definition 4.2 (Volume of a Simple Region). Let S ⊂ R3 be a simple region
with A, h1 and h2 as in Definition 4.1. If S is of the form
S = {(x, y, z) ∈ R3 : z ∈ I(h1 (x, y), h2 (x, y)), (x, y) ∈ A}
then we define the volume of S, denoted by v(S), as follows:
[h2 (x, y) − h1 (x, y)].
v(S) =
(x,y)∈A
We define v(S) similarly if S = {(x, y, z) ∈ R3 : y ∈ I(h1 (x, z), h2 (x, z)), (x, z) ∈ A}
or S = {(x, y, z) ∈ R3 : x ∈ I(h1 (y, z), h2 (y, z)), (y, z) ∈ A}.
Definition 4.3 (Finite Simple Region and Order of Magnitude). Let S be a
simple region given by
S = {(x, y, z) ∈ R3 : z ∈ I(h1 (x, y), h2 (x, y)), (x, y) ∈ A}.
We define λx (S) = λx (A), λy (S) = λy (A), and λz (S) = i(h2 (x, y) − h1 (x, y)), the
index of the analytic function h2 − h1 on A; we call these the orders of magnitude
of S in x, y and z, respectively. We say that S is a finite region in R3 if λx (S) =
λy (S) = λz (S) = 0.
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MEASURE THEORY AND INTEGRATION OVER THE LEVI-CIVITA FIELD
321
Definition 4.4 (Analytic Function in R3 ). Let S ⊂ R3 be a simple region
and let f : S → R. Then we say that f is an analytic function on S if for every
(x0 , y0 , z0 ) ∈ S, there exists a simple region A ⊂ R3 containing (x0 , y0 , z0 ) and
a regular sequence (aijk )∞
i,j,k=0 in R such that λx (A) = λx (S), λy (A) = λy (S),
λz (A) = λz (S), and if (x0 + r, y0 + s, z0 + t) ∈ S ∩ A then
f (x0 + r, y0 + s, z0 + t) =
∞ ∞ ∞
aijk r i sj tk ,
i=0 j=0 k=0
where the power series converges in the weak topology.
Exactly as in the two-dimensional case, we can show that if A ⊂ R3 is a simple
region, f, g : A → R are two analytic functions on A, and α ∈ R then f + αg and
f ·g are analytic functions on A. Moreover, we have the following result whose proof
is similar to that of the corresponding result in two dimensions above and which is
is useful for defining the iterated integral of an analytic function on a simple region
in R3 .
Proposition 4.5. Let A ⊂ R3 be a simple region, let f : A → R ba an analytic
function on A, let B ⊂ R2 be a simple region, and let g : B → R be an analytic
function on B such that for every (x, y) ∈ B, (x, y, g(x, y)) ∈ A. Then F : B → R,
given by F (x, y) = f (x, y, g(x, y)), is an analytic function on B.
Definition 4.6 (Measurable Set). Let S ⊂ R3 . Then we say that S is a
measurable set if for every > 0 there exists two sequences of mutually disjoint
∞
∞
∞
∞
simple regions, (Gn )∞
Gn ⊂ S ⊂
Hn ,
v(Gn )
n=1 and (Hn )n=1 , such that
and
∞
v(Hn ) converge, and
n=1
∞
v(Hn ) −
n=1
∞
n=1
n=1
n=1
v(Gn ) < .
n=1
Definition 4.7 (Measure of a Measurable Set). Let S ⊂ R3 be a measurable
set. Then for every k ∈ N, there exist two sequences of mutually disjoint simple
∞
∞
∞
k ∞
regions, (Gkn )∞
Gkn ⊂ S ⊂
Hnk ,
v(Gkn ) and
n=1 and (Hn )n=1 , such that
∞
n=1
v(Hnk )
converge, and
∞
v(Hnk ) −
∞
n=1
v(Gkn )
n=1
n=1
k
< d . We note that since for every
n=1
n=1
k ∞
k ∈ N, (Gkn )∞
n=1 and (Hn )n=1 are mutually disjoint we can arrange them so that
∞
∞
∞
∞
Gkn ⊂
Gk+1
⊂S⊂
Hnk+1 ⊂
Hnk .
n
n=1
n=1
n=1
n=1
∞
∞
We show that
v(Gkn )
is a Cauchy sequence. Let > 0 in R be given.
n=1
k=1
Let k0 ∈ N be large enough so that dk0 < . Then for l > k > k0 , we have that
∞
∞
∞
∞
Gln ⊂ S ⊂
Hnk and hence
v(Gln ) v(Hnk ). Thus,
n=1
n=1
n=1
n=1
0
∞
n=1
v(Gln ) −
∞
n=1
v(Gkn ) ∞
v(Hnk ) −
n=1
A similar argument shows that the sequence (
∞
v(Gkn ) < dk < dk0 < .
n=1
∞
n=1
v(Hnk ))∞
k=1 is Cauchy.
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322
KHODR SHAMSEDDINE AND DARREN FLYNN
∞
∞
Since R is Cauchy complete lim
v(Gkn ) and lim
v(Hnk ) both exist, and
k→∞ n=1
k→∞ n=1
∞
∞
hence lim
v(Hnk ) −
v(Gkn ) exists; moreover,
k→∞
n=1
lim
k→∞
n=1
∞
v(Hnk )
n=1
−
∞
v(Gkn )
= lim
k→∞
n=1
∞
v(Hnk ) − lim
k→∞
n=1
∞
v(Gkn ).
n=1
Furthermore, for every k ∈ N, we have that
0
∞
v(Hnk )
n=1
and hence
0 lim
k→∞
−
∞
v(Gkn ) < dk ,
n=1
∞
n=1
v(Hnk )
−
∞
v(Gkn )
0.
n=1
It follows that
∞
∞
∞
∞
k
k
lim
v(Hn ) −
v(Gn ) = 0; and hence lim
v(Gkn ) = lim
v(Hnk ).
k→∞
n=1
k→∞
n=1
n=1
k→∞
n=1
We call the common limit the measure of S and we denote it by m(S).
Proposition 4.8. Let S ⊂ R3 be a measurable set. Then
∞
m(S) = inf
v(Hn ) : Hn ’s are mutually disjoint simple regions,
n=1
S⊂
∞
Hn , and
n=1
= sup
∞
v(Hn ) converges
n=1
∞
v(Gn ) : Gn ’s are mutually disjoint simple regions,
n=1
∞
n=1
Gn ⊂ S, and
∞
v(Gn ) converges .
n=1
Definition 4.9 (Measurable Function). Let S ⊂ R3 be a measurable set and
let f : S → R be bounded on S. Then we say that f is measurable on S if for every
> 0, there exists a sequence of mutually disjoint simple regions (Gn )∞
n=1 such that
∞
∞
∞
Gn ⊂ S,
v(Gn ) converges, m(S) −
v(Gn ) < , and f is analytic on Gn
n=1
n=1
for every n ∈ N.
n=1
Definition 4.10 (Integral of an analytic function over a simple region in R3 ).
Let S ⊂ R3 be a simple region, and let f : S → R be an analytic function on S.
First assume that S is of the form
S = {(x, y, z) ∈ R3 : z ∈ I(h1 (x, y), h2 (x, y)), (x, y) ∈ A}
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MEASURE THEORY AND INTEGRATION OVER THE LEVI-CIVITA FIELD
323
where A is a simple region in R2 and where h1 , h2 : A → R are analytic on A.
Then we define
⎡
⎤
⎢
⎥
f (x, y, z) =
f (x, y, z)⎦
⎣
(x,y,z)∈S
(x,y)∈A
z∈I(h1 (x,y),h2 (x,y))
and we call this the integral of f over S. Note that for fixed x and y, f (x, y, z) is
an R-analytic function on the interval I(h1 (x, y), h2 (x, y)), and hence
f (x, y, z)
z∈I(h1 (x,y),h2 (x,y))
is well defined. Moreover, F (x, y) :=
f (x, y, z) is an analytic
z∈I(h1 (x,y),h2 (x,y))
function on A; thus, the integral is well-defined.
If S is of the form
S = {(x, y, z) ∈ R3 : y ∈ I(h1 (x, z), h2 (x, z)), (x, z) ∈ A}
then we define the integral of f over S by
⎡
⎢
f (x, y, z) =
⎣
(x,y,z)∈S
(x,z)∈A
⎤
⎥
f (x, y, z)⎦ .
y∈I(h1 (x,z),h2 (x,z))
Finally, if S is of the form
S = {(x, y, z) ∈ R3 : x ∈ I(h1 (y, z), h2 (y, z)), (y, z) ∈ A}
then we define the integral of f over S by
⎡
⎢
f (x, y, z) =
⎣
(x,y,z)∈S
(y,z)∈A
⎤
⎥
f (x, y, z)⎦ .
x∈I(h1 (y,z),h2 (y,z))
It follows readily from
Definition 4.10 that if S ⊂ R3 is a simple region and
M = M v(S).
M ∈ R a constant then
(x,y,z)∈S
Definition 4.11 (Integral of a Measurable Function over a Measurable Set).
measurable on S. Then the
Let S ⊂ R3 be measurable and let
f : S → R be
integral of f over S, denoted by
f (x, y, z) or
f , is given by
(x,y,z)∈S
f=
S
∞
lim
→ m(S)
⊂S
(Gn ) are mutually disjoint
f is analytic on Gn ∀ n
v(Gn )
n=1∞
∪n=1 Gn
S
∞ f.
n=1 G
n
That the limit exists follows the same arguments as in the one-dimensional and
two-dimensional cases.
With the definition above, the triple integral has similar properties to those
of the single and double integrals discussed in Section 2 and Section 3 above. In
particular,
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324
KHODR SHAMSEDDINE AND DARREN FLYNN
• If S ⊂ R3 is measurable, f, g : S → R are measurable on S, and α ∈ R
then f + αg is measurable on S, with
[f (x, y, z) + αg(x, y, z)] =
f (x, y, z) + α
g(x, y, z).
(x,y,z)∈S
(x,y,z)∈S
(x,y,z)∈S
• If S ⊂ R3 is measurable and M ∈ R a given constant then
M = M m(S).
(x,y,z)∈S
• If S ⊂ R is measurable and f, g : S → R are measurable with f g
everywhere on S then
f (x, y, z) g(x, y, z).
3
(x,y,z)∈S
(x,y,z)∈S
• If S ⊂ R3 is measurable and f : S → R is a measurable function satisfying
|f | ≤ M on S then
f (x, y, z) |f (x, y, z)| M m(S).
(x,y,z)∈S
(x,y,z)∈S
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Department of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada
E-mail address: [email protected]
Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T
2N2, Canada
E-mail address: [email protected]
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