Banach-Stone type theorems on spaces of
probability measures
György Pál Gehér
University of Reading
Reading, UK
Oxford Functional Analysis Seminar, 7 February 2017
Joint work with T. Titkos (Alfréd Rényi Institute of
Mathematics, Budapest, Hungary)
Some history
Theorem (Banach-Stone)
Let K be a compact Hausdorff space and C (K ) be the Banach
space of all real-valued continuous functions endowed with the
usual supremum norm kf k∞ . Assume that T : C (K ) → C (K ) is a
linear operator which is onto, and it is an isometry, i.e.
kTf k∞ = kf k∞
(f ∈ C (K )).
Then T is a product of
a composition operator Ch ∈ B(C (K )) with the symbol
h : K → K being a homeomorphism and
a multiplication operator Mτ ∈ B(C (K )) with continuous
symbol τ : K → {−1, 1};
that is we have
(Tf )(t) = (Mτ Ch f )(t) = τ (t) · f (h(t))
(t ∈ K ).
(1)
Note that obviously all operators of the form (1) is an isometry of
C (K ).
Let us denote by D(R) the set of all probability distribution
functions f : R → R. The Kolmogorov-Smirnov distance
between f , g ∈ D(R) is defined by
ρ(f , g ) := kf − g k∞ = sup |f (t) − g (t)|.
t∈R
We call a map φ : D(R) → D(R) a Kolmogorov-Smirnov
isometry if the following is satisfied:
ρ(φ(f ), φ(g )) = ρ(f , g )
(f , g ∈ D(R)).
Note that equivalently, we can consider the set of all Borel
probability measures PR , and define the notions of
Kolmogorov-Smirnov distance and Kolmogorov-Smirnov isometries.
For the Kolmogorov-Smirnov distance between µ and ν ∈ PR we
have
ρ(µ, ν) = sup µ (−∞, t] − ν (−∞, t] .
t∈R
Theorem (Dolniar-Molnár, 2008, JMAA, original phrasing)
Let φ : D(R) → D(R) be an arbitrary onto Kolmogorov-Smirnov
isometry. Then there exists a strictly increasing bijection ψ : R → R
such that φ has one of the following two forms:
(1) φ(f )(t) = f (ψ(t))
(t ∈ R), or
(2) φ(f )(t) = 1 − f (ψ(−t)−)
(t ∈ R).
Theorem (Dolniar-Molnár, re-phrasing)
Let ϕ : PR → PR be an arbitrary surjective Kolmogorov-Smirnov
isometry. Then there exists a homeomorphism h : R → R such that
ϕ is of the following form:
ϕ(µ)(A) = µ(h−1 [A])
(µ ∈ PR , A ∈ BR ),
i.e. ϕ(µ) is the push-forward measure of µ by h.
Note that the relation between h and ψ is that h = ψ −1 .
The Lévy distance between f , g ∈ D(R) is defined by
L(f , g ) := inf {t ∈ R : f (t − ε) − ε ≤ g (t) ≤ f (t + ε) + ε} .
ε>0
We call a map φ : D(R) → D(R) a Lévy isometry if the following
is satisfied:
L(φ(f ), φ(g )) = L(f , g )
(f , g ∈ D(R)).
Equivalently, the Lévy distance between µ, ν ∈ PR is
L(µ, ν) := inf µ (−∞,t − ε] − ε ≤ ν (−∞, t]
ε>0
≤ µ (−∞, t + ε] + ε ∀t ∈ R
and we call a map ϕ : PR → PR a Lévy isometry if we have
L(ϕ(µ), ϕ(ν)) = L(µ, ν)
(µ, ν ∈ PR ).
Note that the Lévy distance metrises the weak convergence.
Theorem (Molnár, 2011, JMAA, original phrasing)
Let φ : D(R) → D(R) be an arbitrary onto Lévy isometry. Then
there exists a number c ∈ R such that φ has one of the following
two forms:
(1) φ(f )(t) = f (t + c)
(t ∈ R), or
(2) φ(f )(t) = 1 − f ((−t + c)−)
(t ∈ R).
Theorem (Molnár, re-phrasing)
Let ϕ : PR → PR be an arbitrary surjective Lévy isometry. Then
there exists an isometry ψ : R → R such that ϕ is of the following
form:
ϕ(µ)(A) = µ(ψ −1 [A])
(µ ∈ PR , A ∈ BR ),
i.e. ϕ(µ) is the push-forward measure of µ by the isometry ψ.
Note that in case of (1) ψ is a translation by −c, and in case of (2)
ψ is a reflection in the point c2 .
Let I denote the set of all non-degenerate intervals of R. The
Kuiper distance between µ and ν ∈ PR
dKu (µ, ν) : = sup(fµ (t) − fν (t)) + sup(fν (t) − fµ (t))
t∈R
t∈R
= sup {|µ(I ) − ν(I )| : I ∈ I} .
Molnár posed me the problem of characterising all surjective Kuiper
isometries on the space of all continuous probability measures
PRc := µ ∈ PR ∀x ∈ R : µ({x}) = 0 .
For any x ∈ R let us define the function
rx : R \ {x} → R,
rx (t) =
1
t −x
and let
r∞ : R → R,
r∞ (t) = t
The answer for Molnár’s question:
Theorem (G., ≥2017, HJM)
Let φ : PRc → PRc be a surjective Kuiper isometry. Then there exists
a homeomorphism g : R → R and an x ∈ R ∪ {∞} such that φ has
the following form:
φ(µ)(A) = µ(g ◦ rx [A])
(µ ∈ PRc , A ∈ BR ).
The answer for Molnár’s question:
Theorem (G., ≥2017, HJM)
Let φ : PRc → PRc be a surjective Kuiper isometry. Then there exists
a homeomorphism g : R → R and an x ∈ R ∪ {∞} such that φ has
the following form:
φ(µ)(A) = µ(g ◦ rx [A])
(µ ∈ PRc , A ∈ BR ).
The problem on PR was more difficult, the answer reads as follows:
Theorem (G., ≥2017, HJM)
Let ϕ : PR → PR be a surjective Kuiper isometry. Then there exists
a homeomorphism g : R → R such that
ϕ(µ)(A) = µ(g [A])
(µ ∈ PR , A ∈ BR ).
The answer for Molnár’s question:
Theorem (G., ≥2017, HJM)
Let φ : PRc → PRc be a surjective Kuiper isometry. Then there exists
a homeomorphism g : R → R and an x ∈ R ∪ {∞} such that φ has
the following form:
φ(µ)(A) = µ(g ◦ rx [A])
(µ ∈ PRc , A ∈ BR ).
The problem on PR was more difficult, the answer reads as follows:
Theorem (G., ≥2017, HJM)
Let ϕ : PR → PR be a surjective Kuiper isometry. Then there exists
a homeomorphism g : R → R such that
ϕ(µ)(A) = µ(g [A])
(µ ∈ PR , A ∈ BR ).
We emphasise again that the reverse directions are easy to verify in
the previous theorems.
Our idea with T. Titkos was to generalize Molnár’s theorem for the
Lévy-Prokhorov metric. Let X be a real and separable Banach
space and BX be the set of Borel sets on X . The symbol PX
stands for the set of all Borel probability measures on X . The
so-called Lévy-Prokhorov distance metrises the topology of weak
convergence, and it is given by
π(µ, ν) := inf ε > 0 ∀ A ∈ BX : µ(A) ≤ ν(Aε ) + ε ,
where
Aε :=
[
Bε (x)
and
Bε (x) := z ∈ X kx − zk < ε .
x∈A
Note that in case when X = R, the Lévy and the Lévy-Prokhorov
metrics are different.
Theorem (G. & Titkos, ≥2017)
Let (X , k · k) be a separable real Banach space and ϕ : PX → PX be
a surjective Lévy–Prokhorov isometry. Then there exists a surjective
affine isometry ψ : X → X which implements ϕ, i.e. we have
(ϕ(µ)) (A) = µ(ψ −1 [A])
(∀ A ∈ BX ),
where ψ −1 [A] denotes the inverse-image set {ψ −1 (a) | a ∈ A}.
Sketch of the proof
Let
FX :=
(
X
)
X
λi δxi #I < ℵ0 ,
λi = 1, λi > 0, xi ∈ X (∀ i ∈ I )
i∈I
i∈I
be the set of all finitely supported measures, and ∆X stand for
the set of all Dirac measures.
The closed support of µ ∈ PX will be denoted by Sµ .
There will be four major steps:
1
The action on ∆X ;
2
Isolated atoms on the vertices of the convex hull of the
support;
3
The story beyond vertices;
4
The action on FX and PX .
1. The action on ∆X
Proposition
Assume that µ, ν ∈ PX . Then the following are equivalent:
(i) π(µ, ν) = 1,
(ii) d(Sµ , Sν ) := inf {d(x, y ) | x ∈ Sµ , y ∈ Sν } ≥ 1,
(iii) Sν ∩ Sµ1 = ∅.
1. The action on ∆X
Proposition
Assume that µ, ν ∈ PX . Then the following are equivalent:
(i) π(µ, ν) = 1,
(ii) d(Sµ , Sν ) := inf {d(x, y ) | x ∈ Sµ , y ∈ Sν } ≥ 1,
(iii) Sν ∩ Sµ1 = ∅.
We define the unit distance set of a set of measures A ⊆ PX by
A u = ν ∈ PX ∀ µ ∈ A : π(µ, ν) = 1 .
Observe that A u depends only on the shape of Sµ ’s (µ ∈ A ).
Proposition
Suppose that µ ∈ PX . Then the following are equivalent:
(i) ({µ}u )u = {µ},
(ii) there exists an x ∈ X such that µ = δx .
1. The action on ∆X
Lemma
There exists a surjective affine isometry ψ : X → X such that
ϕ(δx ) = δψ(x)
(∀ x ∈ X ).
Proof. Since
u
ϕ ({µ}u )u = {ϕ(µ)}u
(∀ µ ∈ PX ),
there exists a bijective map ψ : X → X such that
ϕ(δx ) := δψ(x)
(∀ x ∈ X ).
1. The action on ∆X
Lemma
There exists a surjective affine isometry ψ : X → X such that
ϕ(δx ) = δψ(x)
(∀ x ∈ X ).
Proof. Since
u
ϕ ({µ}u )u = {ϕ(µ)}u
(∀ µ ∈ PX ),
there exists a bijective map ψ : X → X such that
ϕ(δx ) := δψ(x)
(∀ x ∈ X ).
Observe that
π(δx1 , δx2 ) = min{1, kx1 − x2 k}
This is possible only if ψ is an isometry. (∀ x1 , x2 ∈ X ).
1. The action on ∆X
From now on we may and do assume without loss of generality that
ϕ(δx ) = δx
(∀ x ∈ X ).
Our aim will be to show that ϕ acts identically on the whole of PX .
We define the following continuous function for each µ ∈ PX :
Wµ : X → [0, 1],
Wµ (x) : = π(δx , µ)
(if µ ∈
/ ∆X )
= inf {ε > 0 | 1 ≤ µ(Bε (x)) + ε}
o
n
= min ε > 0 | 1 ≤ µ(Bε (x)) + ε
which will be called the witness function of µ. Clearly, we have
Wµ (x) = Wϕ(µ) (x) (∀ x ∈ X ).
It is natural to expect that the shape of the witness function carries
some information about the measure.
2. Isolated atoms on the vertices of the convex hull
Our aim: Let µ ∈ FX and x̂ be a vertex of conv(Sµ ). Is x̂ an
isolated atom of ϕ(µ)? If so, do we have µ({x̂}) = (ϕ(µ))({x̂})?
2. Isolated atoms on the vertices of the convex hull
Our aim: Let µ ∈ FX and x̂ be a vertex of conv(Sµ ). Is x̂ an
isolated atom of ϕ(µ)? If so, do we have µ({x̂}) = (ϕ(µ))({x̂})?
Proposition
Let µ ∈ FX \ ∆X , K = conv(Sµ ), and assume that x̂ is a vertex of
K . Set λ̂ := µ({x̂}) ∈ (0, 1). Then for every ϑ ∈ PX with Sϑ ⊆ K
the following two conditions are equivalent:
(i) ϑ = λ̂ · δx̂ + (1 − λ̂) · ϑe where ϑe ∈ PX with S e ⊆ K \ Br (x̂) for
some r > 0,
ϑ
(ii) there exist a number 0 < ρ ≤ 1 − λ̂ and a half-line e starting
from x̂ such that the restriction Wϑ |e is of the following form:

if kx − x̂k ≥ 1,
 1
kx − x̂k
if 1 − λ̂ < kx − x̂k < 1,
Wϑ |e (x) =

1 − λ̂
if 1 − λ̂ − ρ ≤ kx − x̂k ≤ 1 − λ̂.
In particular, we have that if µ ∈ FX , #Sµ ≤ 2, ν ∈ PX and
Wµ ≡ Wν , then µ = ν.
2. Isolated atoms on the vertices of the convex hull
Illustrating the proof of (i)=⇒(ii):
We concentrate on the finite dimensional subspace spanned by K .
The support Sµ consists of the red points in K .
The support Sϑ is the union of {x̂} and Sϑe.
x̂
r
Sϑe
K
2. Isolated atoms on the vertices of the convex hull
{f = ĉ}
{f = c}
x̂
Sϑe
ρ
K
2.
Recall that Wϑ (x) = min{ε > 0 | 1 ≤ ϑ(Bε (x)) + ε}
{f = ĉ}
{f = c}
e
1
x̂
Sϑe
ρ
K
2.
Recall that Wϑ (x) = min{ε > 0 | 1 ≤ ϑ(Bε (x)) + ε}
{f = ĉ}
{f = c}
e
1
1 − λ̂
x̂
Sϑe
ρ
K
2.
Recall that Wϑ (x) = min{ε > 0 | 1 ≤ ϑ(Bε (x)) + ε}
{f = ĉ}
{f = c}
e
1
1 − λ̂
1 − λ̂
x̂
Sϑe
ρ
K
2. Isolated atoms on the vertices of the convex hull
The proof of the reverse direction is similar. 2. Isolated atoms on the vertices of the convex hull
The proof of the reverse direction is similar. Corollary (Our aim)
Let µ ∈ FX be arbitrary. Then we have Sϕ(µ) ⊆ K , moreover, x̂ is
an isolated atom of ϕ(µ) with (ϕ(µ))({x̂}) = λ̂.
Proof: Since
π(δx , µ) = 1
(x ∈ X \ K 1 ),
the ϕ-invariance of the witness function gives
Wϕ(µ) (x) = π(δx , ϕ(µ)) = 1
(x ∈ X \ K 1 ),
and hence we conclude
Sϕ(µ) ∩ B1 (x) = ∅
(x ∈ X \ K 1 ).
Consequently,
Sϕ(µ) ⊆ X \ (X \ K 1 )1 ⊆ K
and an application of (i) ⇐⇒ (ii) completes the proof. 3. The story beyond vertices
Our aim: Suppose that m ∈ N pieces of atoms of a measure
ϑ ∈ PX have been already detected. We would like to get
information of the remaining part of ϑ in terms of the
Lévy–Prokhorov distances between ϑ and some measures which are
supported on at most m + 1 points.
3. The story beyond vertices
Our aim: Suppose that m ∈ N pieces of atoms of a measure
ϑ ∈ PX have been already detected. We would like to get
information of the remaining part of ϑ in terms of the
Lévy–Prokhorov distances between ϑ and some measures which are
supported on at most m + 1 points.
For an s > 0 we define the s-Lévy–Prokhorov distance by
πs = πs,k·k : PX × PX → [0, 1]
πs (µ, ν) := inf {ε > 0 | ∀ A ∈ BX : s · µ(A) ≤ s · ν(Aε ) + ε}
and the s-witness function of µ ∈ PX by
Ws,µ : X → R,
Ws,µ (x) := πs (δx , µ).
3. The story beyond vertices
Our aim: Suppose that m ∈ N pieces of atoms of a measure
ϑ ∈ PX have been already detected. We would like to get
information of the remaining part of ϑ in terms of the
Lévy–Prokhorov distances between ϑ and some measures which are
supported on at most m + 1 points.
For an s > 0 we define the s-Lévy–Prokhorov distance by
πs = πs,k·k : PX × PX → [0, 1]
πs (µ, ν) := inf {ε > 0 | ∀ A ∈ BX : s · µ(A) ≤ s · ν(Aε ) + ε}
and the s-witness function of µ ∈ PX by
Ws,µ : X → R,
Ws,µ (x) := πs (δx , µ).
It is not too hard to see that for every µ, ν ∈ PX we have
πs,k·k (µ, ν) = s · π1, 1 k·k (µ, ν),
s
in particular πs is a distance.
3. The story beyond vertices
Lemma
Let s > 0, µ ∈ FX \ ∆X . Set K = convSµ . Assume that x̂ is a
vertex of K and set λ̂ := µ({x̂}) ∈ (0, 1). Then for every ϑ ∈ PX
with Sϑ ⊆ K the following two conditions are equivalent:
(i) ϑ = λ̂ · δx̂ + (1 − λ̂) · ϑe where ϑe ∈ PX with S e ⊆ K \ Br (x̂) for
some r > 0,
ϑ
(ii) there exist a number 0 < ρ ≤ s(1 − λ̂) and a half-line e
starting from x̂ such that Ws,ϑ |e has the following form:

s


 kx − x̂k
Ws,ϑ |e (x) =
 s(1 − λ̂)


if kx − x̂k ≥ s,
if s(1 − λ̂) < kx − x̂k < s,
if s(1 − λ̂) − ρ ≤ kx − x̂k ≤
≤ s(1 − λ̂).
As a consequence we have that if µ ∈ FX , #Sµ ≤ 2, ν ∈ PX and
Ws,µ ≡ Ws,ν , then µ = ν.
3. The story beyond vertices
Assume that m ∈ N pieces of atoms of ϑ ∈ PX have been already
detected. Let us fix a point x∈ X . We are interested in Wwe ,ϑe(x).
y1,1
ρ1
y1,2
y2,4
y2,1
ρ2
ρ3
x
y3,1
y3,3
y3,2
y2,2
y2,3
Figure: An example when X = R2 with the `∞ -norm.
The main lemma (Part 1)
Let x ∈ X and {yj,l | 1 ≤ j ≤ k, 1 ≤ l ≤ dj } ⊂ X be some pairwise
different points such that for every 1 ≤ j ≤ k we have
ρj := kx − yj,1 k = kx − yj,l k (∀ 1 ≤ l ≤ dj ).
Assume that ρj > ρj+1 > 0 (∀ 1 ≤ j ≤ k − 1). We set
wj,l := ϑ({yj,l }) > 0 (∀ 1 ≤ j ≤ k, 1 ≤ l ≤ dj ),
wj :=
dj
X
wj,l = ϑ({yj,1 , . . . , yj,dj }) (∀ 1 ≤ j ≤ k),
l=1
e := 1 −
w
k
X
wj ,
j=1
ηr :=
dj
r X
X
j=1 l=1
wj,l · δyj,l
r
X
+ 1−
wj · δx ∈ FX
j=1
(∀ 0 ≤ r ≤ k).
The main lemma (Part 2)
Furthermore, denote by ϑe ∈ PX the measure which satisfies
ϑ=
dj
k X
X
e
e · ϑ.
wj,l · δyj,l + w
j=1 l=1
e -witness function of ϑe can be expressed in terms of the
Then the w
Lévy–Prokhorov distances of ϑ and ηr ’s in the following way:

π(δx , ϑ) if x is not (P1 )



π(ηr , ϑ) if x is (Pr ) but not (Pr +1 )
Wwe ,ϑe(x) =
with some 1 ≤ r < k



π(ηk , ϑ) if x is (Pk )
where for every 1 ≤ r ≤ k the property (Pr ) means
π(ηr −1 , ϑ) ≤ ρr .
(Pr )
4. The action on FX and PX
Let us assume that we showed that φ(µ) = µ was satisfied for
every µ ∈ FX , i.e. that φ|FX was the identity map. Since FX is
dense in PX with respect to the Lévy-Prokhorov metric and φ is
continuous with respect to the Lévy-Prokhorov metric, we could
conclude that φ is the identity map.
4. The action on FX and PX
Let us assume that we showed that φ(µ) = µ was satisfied for
every µ ∈ FX , i.e. that φ|FX was the identity map. Since FX is
dense in PX with respect to the Lévy-Prokhorov metric and φ is
continuous with respect to the Lévy-Prokhorov metric, we could
conclude that φ is the identity map.
Our aim: We will consider a µ ∈ FX and show that ϕ(µ) = µ.
We will use the notation ϑ = ϕ(µ).
4. The action on FX and PX
x1
x1
x2
conv(Sϑ )
x3
We know what is conv(Sϑ ) and that Wµ ≡ Wϑ .
4. The action on FX and PX
x1
x1
x2
conv(Sϑ )
x3
We know what is conv(Sϑ ) and that Wµ ≡ Wϑ .
Therefore we can detect an isolated atom of ϑ at x1 which has the
same weight as µ has at x1 .
...
...
4. The action on FX and PX
x1
x1
x2
Sϑ(1) ⊆
x3
ϑ = ϑ({x1 }) · δx−1 + (1 − ϑ({x1 })) · ϑ(1) .
We compute the (1 − ϑ({x1 }))–witness function of ϑ(1) .
4. The action on FX and PX
x1
x1
x2
Sϑ(1) ⊆
x3
ϑ = ϑ({x1 }) · δx−1 + (1 − ϑ({x1 })) · ϑ(1) .
We compute the (1 − ϑ({x1 }))–witness function of ϑ(1) .
If #Sµ was 3, then we are done. Moreover, we infer
ϕ(µ) = µ
(µ ∈ FX , #Sµ ≤ 3).
4. The action on FX and PX
x1
x1
x2
Sϑ(1) ⊆
x3
ϑ = ϑ({x1 }) · δx−1 + (1 − ϑ({x1 })) · ϑ(1) .
We compute the (1 − ϑ({x1 }))–witness function of ϑ(1) .
If #Sµ was 3, then we are done. Moreover, we infer
ϕ(µ) = µ
(µ ∈ FX , #Sµ ≤ 3).
Otherwise, x2 must be an isolated atom of ϑ(1) , and hence of ϑ.
...
4. The action on FX and PX
x1
x1
x2
x2
Sϑ(2) ⊆
x3
ϑ = ϑ({x1 }) · δx1 + ϑ({x2 }) · δx2 + (1 − ϑ({x1 }) − ϑ({x2 })) · ϑ(2) .
We compute the (1 − ϑ({x1 }) − ϑ({x2 }))–witness function of ϑ(2) .
4. The action on FX and PX
x1
x1
x2
x2
Sϑ(2) ⊆
x3
ϑ = ϑ({x1 }) · δx1 + ϑ({x2 }) · δx2 + (1 − ϑ({x1 }) − ϑ({x2 })) · ϑ(2) .
We compute the (1 − ϑ({x1 }) − ϑ({x2 }))–witness function of ϑ(2) .
If #Sµ was 4, then we are done. Moreover, we infer
ϕ(µ) = µ
(µ ∈ FX , #Sµ ≤ 4).
4. The action on FX and PX
x1
x1
x2
x2
Sϑ(2) ⊆
x3
ϑ = ϑ({x1 }) · δx1 + ϑ({x2 }) · δx2 + (1 − ϑ({x1 }) − ϑ({x2 })) · ϑ(2) .
We compute the (1 − ϑ({x1 }) − ϑ({x2 }))–witness function of ϑ(2) .
If #Sµ was 4, then we are done. Moreover, we infer
ϕ(µ) = µ
(µ ∈ FX , #Sµ ≤ 4).
Otherwise, x3 must be an isolated atom of ϑ(2) , and hence of ϑ.
...
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Thank you for your
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