Topological Spaces with Skorokhod Representation Property
We give a survey of recent results that generalize and develop a classical theorem of Skorokhod on representation of weakly convergent sequences of probability measures by almost everywhere convergent sequences of mappings.
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nasplib_isofts_kiev_ua-123456789-1658272025-02-09T21:43:56Z Topological Spaces with Skorokhod Representation Property Топологічні простори із властивістю зображення Скорохода Banakh, T.O. Bogachev, V.I. Kolesnikov, A.V. Статті We give a survey of recent results that generalize and develop a classical theorem of Skorokhod on representation of weakly convergent sequences of probability measures by almost everywhere convergent sequences of mappings. Наведено огляд отриманих останнім часом результатів, що узагальнюють класичну теорему Скорохода про зображення слабко збіжної послідовності ймовірнісних мір майже напевно збіжними послідовностями відображень. This work has been supported by the projects RFBR 04-01-00748, the Scientific Schools Grant 1758.2003.1, DFG 436 RUS 113/343/0(R), INTAS 03-51-5018, DFG-Forschergruppe “Spectral Analysis, Asymptotic Distributions, and Stochastic Dynamics”, and Centro di Ricerca Matematica Ennio De Giorgi at Scuola Normale Superiore di Pisa. 2005 Article Topological Spaces with Skorokhod Representation Property / T.O. Banakh, V.I. Bogachev, A.V. Kolesnikov // Український математичний журнал. — 2005. — Т. 57, № 9. — С. 1171–1186. — Бібліогр.: 41 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165827 519.21 en Український математичний журнал application/pdf Інститут математики НАН України |
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Статті Статті Banakh, T.O. Bogachev, V.I. Kolesnikov, A.V. Topological Spaces with Skorokhod Representation Property Український математичний журнал |
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We give a survey of recent results that generalize and develop a classical theorem of Skorokhod on representation of weakly convergent sequences of probability measures by almost everywhere convergent sequences of mappings. |
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Article |
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Banakh, T.O. Bogachev, V.I. Kolesnikov, A.V. |
| author_facet |
Banakh, T.O. Bogachev, V.I. Kolesnikov, A.V. |
| author_sort |
Banakh, T.O. |
| title |
Topological Spaces with Skorokhod Representation Property |
| title_short |
Topological Spaces with Skorokhod Representation Property |
| title_full |
Topological Spaces with Skorokhod Representation Property |
| title_fullStr |
Topological Spaces with Skorokhod Representation Property |
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Topological Spaces with Skorokhod Representation Property |
| title_sort |
topological spaces with skorokhod representation property |
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Інститут математики НАН України |
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2005 |
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Статті |
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| citation_txt |
Topological Spaces with Skorokhod Representation Property / T.O. Banakh, V.I. Bogachev, A.V. Kolesnikov // Український математичний журнал. — 2005. — Т. 57, № 9. — С. 1171–1186. — Бібліогр.: 41 назв. — англ. |
| series |
Український математичний журнал |
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| fulltext |
UDC 519.21
T. O. Banakh (Lviv Nat. Univ., Ukraine),
V. I. Bogachev, A. V. Kolesnikov (Moscow State Univ., Russia)
TOPOLOGICAL SPACES
WITH THE SKOROKHOD REPRESENTATION PROPERTY*
TOPOLOHIÇNI PROSTORY
IZ VLASTYVISTG ZOBRAÛENNQ SKOROXODA
We give a survey of recent results that generalize and develop a classical theorem of Skorokhod on represen-
tation of weakly convergent sequences of probability measures by almost everywhere convergent sequences of
mappings.
Navedeno ohlqd otrymanyx ostannim çasom rezul\tativ, wo uzahal\nggt\ klasyçnu teoremu Sko-
roxoda pro zobraΩennq slabko zbiΩno] poslidovnosti jmovirnisnyx mir majΩe napevno zbiΩnymy
poslidovnostqmy vidobraΩen\.
1. Introduction. A classical result of A. V. Skorokhod [1, 2], obtained half a century
ago and now-a-days included in textbooks on probability and measure theory, states that
for every sequence of Borel probability measures µn on a complete separable metric
space X that converges weakly to a Borel probability measure µ0, one can find Borel
mappings ξn : [0, 1] → X, n = 0, 1, . . . , such that the image of Lebesgue measure
λ under ξn is µn for every n ≥ 0 and lim
n→∞
ξn(t) = ξ0(t) for almost all t ∈ [0, 1].
Numerous extensions of this useful result have been found since then, see, e.g., Blackwell
and Dubbins [3], Dudley [4], Fernique [5], Jakubowski [6], Letta and Pratelli [7], Schief
[8], Szczotka [9], Wichura [10], and a series of our papers [11 – 15]; a section on this
topic is presented in volume 2 of the book [16]; for related problems and applications see
also Choban [17], Cuesta-Albertos and Matrán-Bea [18], Tuero [19]. The purpose of this
work is to survey recent achievements in this direction.
One of the most important extensions of Skorokhod’s theorem was discovered inde-
pendently by Blackwell and Dubbins [3] and Fernique [5], who showed that all Borel
probability measures on a Polish space X can be parameterized simultaneously by map-
pings from [0, 1] with the preservation of the above correspondence. More precisely, to
every Borel probability measure µ on X one can associate a Borel mapping ξµ : [0, 1] →
→ X such that the image of Lebesgue measure under ξµ equals µ and whenever mea-
sures µn converge weakly to µ, one has lim
n→∞
ξµn
(t) = ξµ(t) for almost all t ∈ [0, 1].
It has been recently shown in [11] that this result can be derived from its simple one
dimensional case and general topological and functional-analytic results. In addition, it
has been shown in [11] that there are interesting links between the Skorokhod param-
eterization of probability measures on topological spaces and topological properties of
those spaces. A study of nonmetrizable topological spaces to which the Skorokhod and
Blackwell – Dubbins – Fernique theorems can be extended was initiated in the works [6]
and [11, 12], respectively. Most of the results obtained fall into the following two groups:
given a set M of Borel probability measures on a topological space X, on parameter-
izes the measures in M by Borel mappings from [0, 1] to X in the above sense such
∗ This work has been supported by the projects RFBR 04-01-00748, the Scientific Schools Grant
1758.2003.1, DFG 436 RUS 113/343/0(R), INTAS 03-51-5018, DFG-Forschergruppe “Spectral Analysis,
Asymptotic Distributions, and Stochastic Dynamics”, and Centro di Ricerca Matematica Ennio De Giorgi at
Scuola Normale Superiore di Pisa.
c© T. O. BANAKH, V. I. BOGACHEV, A. V. KOLESNIKOV, 2005
1171 ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1172 T. O. BANAKH, V. I. BOGACHEV, A. V. KOLESNIKOV
that this correspondence is continuous either (a) for all convergent sequences in M or
(b) only for certain subsequences. Certainly, by varying the initial set M one obtains
a lot of particular cases. For example, one can take for M a convergent sequence, as
Skorokhod did, or the space of all measures, as in the Blackwell – Dubbins – Fernique
theorem, or a uniformly tight sequence, as in Jakubowski’s work, or some compact set.
The principal concept in this area is a space with the strong Skorokhod property
for Radon measures defined as a space on which all Radon probability measures ad-
mit a simultaneous parameterization µ �→ ξµ by Borel mappings from [0, 1] endowed
with Lebesgue measure such that one obtains the aforementioned correspondence be-
tween weak convergence of measures and almost everywhere convergence of mappings.
There are very interesting links between the Skorokhod representation property (and its
versions) and other probabilistic properties of topological nature such as the Prokhorov
property, open mappings and others. These matters are discussed in Sections 2 – 4. A very
important for applications version of Skorokhod’s result is due to Jakubowski. Section 5
is devoted to developments of his idea. Jakubowski’s theorem applies to a large class
of spaces, but does not provide a simultaneous parameterization of all Radon probability
measures. Section 6 contains a list of open problems. A common feature of many positive
results on Skorokhod parameterization (strong or weak) for the most diverse topological
spaces is that such spaces are obtained by applying certain topological operations on some
“building blocks” that are constructed from metric spaces by means of special surjections
or injections.
2. Notation and terminology. We assume throughout that X is a Tykhonoff (i.e.,
completely regular) topological space. Let Cb(X) be the space of all bounded contin-
uous functions on X and let B(X) be the Borel σ -field of X. The symbol P(X)
denotes the space of all Borel probability measures on X. Let P0(X) and Pr(X) de-
note, respectively, the spaces of all Baire and Radon (i.e., inner compact regular) prob-
ability measures on X. A probability measure µ on a space X is called discrete if
µ(X \ C) = 0 for some countable subset C ⊂ X. Dirac’s measure at x is denoted
by δx.
The weak topology on P(X), Pr(X) or P0(X) is the restriction of the weak topol-
ogy on the linear space of all bounded Borel (or Baire) measures that is generated by the
seminorms
pf (µ) =
∣∣∣∣∣∣
∫
X
f(x)µ(dx)
∣∣∣∣∣∣ , f ∈ Cb(X).
Thus, a sequence of measures µn converges weakly to a measure µ precisely when
lim
n→∞
∫
X
f(x)µn(dx) =
∫
X
f(x)µ(dx) ∀ f ∈ Cb(X).
It is well known that the weak topology is generated by the base of sets
W (µ,U, a) := {ν ∈ Pr(X) : ν(U) > µ(U)− a},
where U is open in X and a > 0. Weak convergence is denoted by µn ⇒ µ. Recall
that the weak topology is Hausdorff on P0(X) and Pr(X) and that P(X) = P0(X) =
= Pr(X) for any completely regular Souslin space X. See [16] for additional informa-
tion about weak convergence of probability measures.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
TOPOLOGICAL SPACES WITH THE SKOROKHOD REPRESENTATION PROPERTY 1173
If (X,A) and (Y,B) are measurable spaces and f : X → Y is a measurable map-
ping, then the image of a measure µ on X under the mapping f is denoted by µ ◦ f−1
and defined by the formula
µ ◦ f−1(B) = µ
(
f−1(B)
)
, B ∈ B.
We recall that a family M of nonnegative Borel measures on a topological space X
is called uniformly tight if, for every ε > 0, there exists a compact set Kε ⊂ X such
that µ(X\Kε) < ε for all µ ∈M.
We shall call a topological space X sequentially Prokhorov if every sequence of
Radon probability measures on X that converges weakly to a Radon measure is uni-
formly tight.
Definition 2.1. (i) We shall say that a family M of Borel probability measures on
a topological space X has the strong Skorokhod property if, to every measure µ ∈ M,
one can associate a Borel mapping ξµ : [0, 1] → X with λ ◦ ξ−1
µ = µ, where λ is
Lebesgue measure, such that if a sequence of measures µn ∈ M converges weakly to a
measure µ ∈M, then
lim
n→∞
ξµn
(t) = ξµ(t) for almost all t ∈ [0, 1]. (2.1)
If (2.1) holds under the additional assumption that {µn} is uniformly tight, then M is
said to have the uniformly tight strong Skorokhod property.
(ii) We shall say that a topological space X has the strong Skorokhod property for
Radon measures if the family Pr(X) of all Radon measures has that property. If the
family of all discrete probability measures on X has the strong Skorokhod property, then
X is said to have that property for discrete measures.
We shall abbreviate the strong Skorokhod property as SSP. The strong Skorokhod
property for Radon measures will be abbreviated as SSPr. Obviously, the SSP and SSPr
coincide for any Radon space, i.e., a space on which every Borel measure is Radon (we
recall that every Souslin space is Radon).
The uniformly tight Skorokhod property for X is defined analogously. In a similar
manner we define also the strong and uniformly tight strong Skorokhod properties for
probability measures with finite supports and two-point supports.
It may be useful to consider a more general property when in Definition 2.1 one can
take any probability space (Ω,F , P ) in place of the unit interval [0, 1] with Lebesgue
measure. Such a property can be called the extended SSP. However, in this survey we
consider the SSP with Lebesgue measure.
We shall use the terms Skorokhod parameterization and Skorokhod representation for
mappings µ �→ ξµ of the type described in this definition.
With this definition, the two principle results mentioned above can be restated as fol-
lows.
Theorem 2.1. (i) (Skorokhod). Any weakly convergent sequence of Borel probability
measures on a Polish space has the strong Skorokhod property.
(ii) (Blackwell – Dubbins – Fernique). Every Polish space has the strong Skorokhod
property.
It is clear that a sequentially Prokhorov space has the strong Skorokhod property for
Radon measures if and only if it has the uniformly tight strong Skorokhod property for
Radon measures.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1174 T. O. BANAKH, V. I. BOGACHEV, A. V. KOLESNIKOV
An advantage of dealing with Radon measures is that the strong Skorokhod property
for them is inherited by arbitrary subspaces (a simple verification is found in [11]).
Lemma 2.1. Let X be a space with the strong Skorokhod property for Radon mea-
sures. Then every subset Y of X has this property as well. If Y is universally mea-
surable, then the analogous assertion is valid for the strong Skorokhod property for Borel
measures on the spaces X and Y.
The following trivial lemma from [11] enables one to reduce the case of a general
universally measurable set in a Polish space to the that of unit interval [0, 1]. The question
whether such a reduction is possible has been posed by A. N. Shiryaev [20].
Given a continuous mapping f : X → Y between completely regular spaces, we say
that the corresponding mapping
f̂ : Pr(X) → Pr(Y ), µ �→ µ ◦ f−1,
has a continuous right inverse in the weak topology if there exists a mapping Ψ: Pr(Y ) →
→ Pr(X) continuous in the weak topology such that f̂
(
Ψ(ν)
)
= ν for all ν ∈ Pr(Y ).
The idea is this: suppose a space Y is obtained as the image of a subspace X ⊂ [0, 1]
under a continuous mapping f. This mapping induces the mapping f̂ between Pr(X)
and Pr(Y ). If f̂ admits a continuous right inverse Ψ, then one can use a parameteriza-
tion µ �→ ξµ of Pr(X) to obtain a parameterization of Pr(Y ) in the form ν �→ ξΨ(ν).
Lemma 2.2. Let f be a continuous mapping from a topological space X with the
strong Skorokhod property for Radon measures onto a topological space Y such that
the corresponding mapping f̂ : Pr(X) → Pr(Y ) is surjective and has a continuous
right inverse in the weak topology. Then Y has the strong Skorokhod property for Radon
measures.
In order to employ this lemma, we need, of course, to know when a continuous
right inverse exists. This question has been investigated in [11]. Here we need only
the following lemma, which is deduced in [11] (see also [16], § 8.5) from a well-known
functional-analytic result due to Milyutin. For the reader’s convenience we recall this
result. Let f : X → Y be a continuous surjection of compact spaces X and Y. A lin-
ear operator U : C(X) → C(Y ) is called a regular operator of averaging for f if
Uψ ≥ 0 whenever ψ ≥ 0 and U(ϕ ◦ f) = ϕ for all ϕ ∈ C(Y ). Such an oper-
ator is automatically continuous and has unit norm. It is readily seen that the operator
V = U∗ : Mr(Y ) = C(Y )∗ →Mr(X) = C(X)∗ maps Pr(Y ) to Pr(X) and f̂ ◦ V
is the identity mapping on M(Y ), i.e., V is a continuous right inverse for F̂ . Indeed,
for all ν ∈Mr(Y ) we have∫
Y
ϕ(y)
[
f̂ ◦ V (ν)
]
(dy) =
∫
X
ϕ
(
f(x)
)
V (ν)(dx) =
=
∫
Y
U(ϕ ◦ f)(y) ν(dy) =
∫
Y
ϕ(y) ν(dy) ∀ϕ ∈ C(Y ).
A compact space S is called a Miljutin space if, for some cardinality τ, there exists
a continuous surjection f : {0, 1}τ → S, where {0, 1} is the two point space, having
a regular operator of averaging. The Miljutin lemma (see [21], Theorem 5.6) states that
a closed interval is a Miljutin space. It follows that every metrizable compact space S
is a Miljutin space, and the set N can be taken for τ in the above definition. Since the
space {0, 1}N is homeomorphic to the classical Cantor set C ⊂ [0, 1] that consists of all
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
TOPOLOGICAL SPACES WITH THE SKOROKHOD REPRESENTATION PROPERTY 1175
points in the closed interval [0, 1], whose triadic decomposition does not contain 1, we
conclude that for every nonempty metrizable compact space S, there exists a continuous
surjection f : C → S that has a regular operator of averaging. Therefore, the following
assertion is true.
Lemma 2.3. Let S be a nonempty metrizable compact space. Then there exists a
continuous surjection f : C → S such that the mapping f̂ has a linear continuous right
inverse.
It is readily verified by using the distribution functions that [0, 1] has the SSP (see
[16], § 8.5). Therefore, we arrive at the following assertion.
Theorem 2.2. Let X be a set in a Polish space. Then X has the SSP for Radon
measures.
Let us mention a number of assertions related to Lemma 2.3, although we do not
employ them below. See [11] and [22] for proofs and related comments and references.
Theorem 2.3. (i) Let X and Y be completely regular spaces and let f : X → Y
be an open surjective mapping that satisfies the following local conservativity condition:
for every open set V ⊂ X and every Radon probability measure ν on f(V ), there
exists a Radon probability measure µ on V with µ ◦ f−1 = ν. Then
f̂ : Pr(X) → Pr(Y )
is an open surjection.
(ii) Let X and Y be completely regular Souslin spaces and let f : X → Y be an
open surjective mapping. Then the mapping f̂ : Pr(X) → Pr(Y ) is an open surjection
in the weak topology.
(iii) Let f : X → Y be an open surjective mapping of complete metric spaces. Then
the mapping f̂ : Pr(X) → Pr(Y ) is an open surjection and has a continuous right
inverse in the weak topology.
(iv) For every set Y in a Polish space, there exist a subset X of the space R of the
irrational numbers in [0, 1] and a continuous surjective mapping f : X �→ Y such that
the mapping f̂ : Pr(X) → Pr(Y ) has a continuous right inverse in the weak topology.
If Y is universally measurable, then X is universally measurable as well.
(v) For every set Y in a Polish space, there exist a subset X of the Cantor set C
and a continuous surjective mapping f : X �→ Y such that the mapping
f̂ : Pr(X) → Pr(Y )
has a continuous right inverse in the weak topology. In the case of compact Y, the
set X can be chosen compact. If Y is universally measurable, then X is universally
measurable as well.
Remark 2.1. (i) It is to be noted that in Blackwell, Dubins [3], there is a very short
sketch of the proof of their theorem (moreover, as noted in [23], there is a flaw in this
sketch), but a detailed and correct proof on this way with the verification of all details
is not that short as one can see in Fernique [5] and Lebedev [24] (Ch. 5). Unlike the
topological approach explained above, the method of proof in the works cited develops
the original method of Skorokhod.
There is an interesting approach to parameterization of measures by mappings con-
nected with the Monge – Kantorovich problem and other extremal problems for measures
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1176 T. O. BANAKH, V. I. BOGACHEV, A. V. KOLESNIKOV
with given marginals. Its main idea is to obtain mappings parameterizing measures as
solutions of certain variational problems; for example, if we consider Borel probability
measures on the cube V = [0, 1]d in R
d and denote by λ Lebesgue measure on V,
then, for any Borel probability measure µ on V, there is a Borel mapping ξµ at which
the expression ∫
V
|ξ(x)− x|2 λ(dx)
attains its minimum over the class of Borel mappings ξ with λ ◦ ξ−1 = µ. This ap-
proach is discussed by Cuesta – Albertos and Matrán-Bea [18], Krylov [25], Tuero [19].
In particular, Krylov [25] considered measures on R
n and obtained a parameterization
with certain differentiability properties. However, there are other natural generaliza-
tions of increasing functions that parameterize Borel probability measures, but cannot
be taken as Skorokhod parameterizations. For example, the so called increasing trian-
gular mappings (see [26]), i.e., Borel mappings T on V such that T = (T1, . . . , Td),
where every component Tk depends only on the variables x1, . . . , xk and the functions
xk �→ Tk(x1, . . . , xk) is increasing, have the property that every Borel probability mea-
sure µ on V is the image of λ under a unique (up to a modification) increasing trian-
gular Borel mapping Tµ. However, these mappings lack the desired continuity property.
(ii) Theorem 2.3 shows that it would be also enough to prove that the Hilbert cube
Q = [0, 1]∞ has the strong Skorokhod property, since by the Uryson theorem, every
separable metric space is homeomorphic to a subset of Q. Hence it suffices to have at
least one compact metric space Z with the strong Skorokhod property such that Z can
be mapped onto Q by the aid of an open mapping. Since one can deduce from [25]
that the space R
d has the strong Skorokhod property, it remains to find a compact set in
R
d that can be mapped onto Q by an open mapping. Such compacta exist indeed [27];
moreover, one can find a compact set Z ⊂ R
3 with the desired property (see [28] (Ch. 4,
§ 4) or [27]).
(iii) Although the proof of Theorem 2.2 in the case X = [0, 1] is straightforward, the
statement is yet surprising in view of the fact that the space B of all Borel functions from
[0, 1] to [0, 1] has neither metric nor topology in which the convergent sequences are
exactly those that converge almost everywhere (the same is true even for the subspace of
continuous functions, see [16], Exercise 2.12.62). Theorem 2.2 says that the factor-space
of B by the equivalence relation determined by the equality of the images of Lebesgue
measure under the corresponding functions is metrizable (by a metric that metrizes the
weak topology). If (X, d) is a complete separable metric space, then the subset ΛX
in the space B([0, 1], X) of Borel mappings from [0, 1] to X formed by the parame-
terezing mappings ξµ, µ ∈ Pr(X), can be equipped with the metric of convergence in
measure, i.e.,
d0(f, g) =
1∫
0
d(f(t), g(t))
d(f(t), g(t)) + 1
dt.
Then a sequence in ΛX converges in this metric if and only if it converges almost ev-
erywhere (in order to see this, it suffices to observe that convergence in measure implies
weak convergence of induced measures). Such a metric can be also used for metrization
of the weak topology on Pr(X).
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
TOPOLOGICAL SPACES WITH THE SKOROKHOD REPRESENTATION PROPERTY 1177
The case of a general metric space follows by the fact that for any complete metric
space Y, there is a cardinality τ such that Y is the image of the space B(τ) that is the
countable power of a discrete space of cardinality τ under an open mapping. It is verified
in [11] that B(τ) has the SSPr. This yields the following assertion.
Theorem 2.4. Every metric space has the SSPr.
A powerful tool in the study of open mappings is the following classical result, called
Michael’s selection theorem (see [29] or [30, p. 190]). Let M be a metrizable space, let
P be a complete metrizable closed subset of locally convex space L, and let Φ: M →
→ 2P be a lower semicontinuous mapping with values in the set of nonempty convex
closed subsets of P. Then, there exists a continuous mapping f : M → P such that
f(x) ∈ Φ(x) for all x. For our purposes, it is enough to deal with the case where L is
a normed space; a short proof for this case can be found in [30] (A§ 1) (note that Filippov
[31] constructed an example showing that one cannot omit the requirement that P is
closed even if P is a Gδ-set in a Hilbert space). Namely, we deal with the situation where
M = Pr(Y ) and P = Pr(X) for some Polish spaces X and Y ; the weak topology on
these sets is generated by the Kantorovich – Rubinstein norm on Mr(X) and Mr(Y ).
A typical application of this theorem is this: let T : P → M be a continuous open
affine mapping of a complete metrizable convex closed set P in a locally convex space
L to a metrizable set M in a locally convex space. Then Φ(x) = T−1(x) satisfies the
hypotheses of Michael’s theorem, therefore, T has a continuous right inverse. In our
situation, an open mapping from X to Y generates an open affine mapping between
P := Pr(X) and M := Pr(Y ). This demonstrates yet another interesting link between
Skorokhod’s representation and general topology.
3. Nonmetrizable spaces with the strong Skorokhod property. Now we discuss
what happens if we leave the area of metric spaces. As shown in [11], the space R
∞
0
of all finite real sequences with its natural topology of the inductive limit of the spaces
R
n fails to have even the uniformly tight strong Skorokhod property; moreover, one can
find a weakly convergent uniformly tight sequence of probability measures on R
∞
0 that
does not admit a Skorokhod parameterization by mappings. It is worth noting that R
∞
0
has the weak Skorokhod property considered in Section 5. However, we shall see that
the class of spaces with the SSP is wider than the class of metric spaces and includes
all almost metrizable sequentially Prokhorov spaces. In particular, we construct a class
of nonmetrizable topological spaces with the strong Skorokhod property for Radon mea-
sures. Our simplest example is a countable set which is the set of natural numbers with
an extra point from its Stone – Čech compactification. This example will be justified by
more general facts.
First we note that the uniformly tight strong Skorokhod property for Radon measures
is preserved by bijective continuous proper mappings. In particular, the strong Skorokhod
property for Radon measures is preserved by bijective continuous proper mappings onto
sequentially Prokhorov spaces. We recall that a mapping f : X → Y between topologi-
cal spaces is called proper if f−1(K) is compact for every compact subspace K ⊂ Y.
Theorem 3.1. Let X and Y be two topological spaces such that there exists a
bijective continuous proper mapping F : X → Y. Assume that X has the uniformly
tight strong Skorokhod property for Radon measures. Then Y possesses this property
as well. In particular, if Y is sequentially Prokhorov, then Y has the strong Skorokhod
property for Radon measures.
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1178 T. O. BANAKH, V. I. BOGACHEV, A. V. KOLESNIKOV
Remark 3.1. For a bijective continuous proper mapping f : X → Y and a uni-
formly tight weakly convergent sequence µn ⇒ µ of probability Radon measures on Y,
the sequence of measures µn ◦ f−1 on X is weakly convergent to µ ◦ f−1.
We define a topological space X to be almost metrizable if there exists a bijective
continuous proper mapping f : M → X from a metrizable space M. If M is discrete,
then X is called almost discrete.
One can easily show by examples that an almost metrizable space may not be metriz-
able (such examples are given below). On the other hand, each almost metrizable k-space
is metrizable. We recall that a topological space X is a k-space if a subset Z ⊂ X
is closed in X precisely when Z ∩ K is closed for every compact subset K ⊂ X
(see [32]).
One can readily show that almost metrizable spaces and almost discrete spaces have
the following properties.
Proposition 3.1. (i) Any subspace of an almost metrizable space is almost metriz-
able.
(ii) A topological space is metrizable if and only if it is an almost metrizable k-space.
(iii) A topological space X is almost metrizable if and only if the strongest topology
inducing the original topology on each compact subset of X is metrizable.
(iv) A topological space is almost discrete if and only if it contains no infinite compact
subspaces.
(v) A countable product of almost metrizable spaces is almost metrizable.
(vi) The classes of almost metrizable and almost discrete spaces are stable under
formation of arbitrary topological sums.
(vii) The images of almost metrizable and almost discrete spaces under continuous
bijective proper mappings belong to the respective classes.
The following theorem characterizes almost metrizable spaces with the strong Sko-
rokhod property for Radon measures.
Theorem 3.2. Any almost metrizable space has the uniformly tight strong Skorokhod
property for Radon measures. Moreover, an almost metrizable space has the strong Sko-
rokhod property for Radon measures if and only if it is sequentially Prokhorov.
Since the countable product of sequentially Prokhorov spaces is also sequentially
Prokhorov (see, e.g., [33], § 8.3), we arrive at the following statement.
Corollary 3.1. The countable product of almost metrizable spaces with the SSPr
has this property as well.
For almost discrete spaces we have even stronger results. We recall that a topological
space X is called sequentially compact if each sequence in X contains a convergent
subsequence, see [32] (§ 3.10).
Theorem 3.3. For a topological space X the following conditions are equivalent:
(i) X is an almost discrete space.
(ii) Each compact subset of X is sequentially compact and each uniformly tight
sequence of Radon probability measures µn on X that converges weakly to a Radon
measure is convergent in the variation norm (equivalently, one has convergence µn(x) →
→ µ(x) for each x ∈ X ).
Corollary 3.2. Let X be an almost discrete sequentially Prokhorov space, let E
be a completely regular space, and let a sequence of Radon probability measures µn on
X × E converge weakly to a Radon probability measure µ. Then, for each x ∈ X, the
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TOPOLOGICAL SPACES WITH THE SKOROKHOD REPRESENTATION PROPERTY 1179
restrictions of the measures µn to the set x×E converge weakly to the restriction of µ,
i.e., one has µn|x×E ⇒ µ|x×E .
Almost metrizable spaces need not be metrizable (we shall encounter below even
countable almost metrizable nonmetrizable spaces). A somewhat unexpected example is
the Banach space l1 endowed with the weak topology. The space l1 is known to have
the Shur property. We recall that a Banach space X has the Shur property if each weakly
convergent sequence in X is norm convergent.
Theorem 3.4. Let X be a Banach space with the Shur property and let τ be an in-
termediate topology between the norm and weak topologies on X. Then the space (X, τ)
is almost metrizable and has the uniformly tight strong Skorokhod for Radon measures.
It should be noted that, as shown in [34, p. 127], the space l1 with the weak topol-
ogy (as well as any infinite dimensional Banach space with the weak topology) is not
sequentially Prokhorov.
We shall now give an example of an almost discrete space without the strong Sko-
rokhod property.
Example 3.1. Let Xn, n ∈ N, be pairwise disjoint finite sets in N with
Card (Xn) < Card (Xn+1) for each n. Fix any point ∞ /∈
⋃
n∈N
Xn and define a
topology on the union X = {∞} ∪
⋃∞
n=1Xn as follows. All points except for ∞ are
isolated and the neighborhood base of a unique nonisolated point ∞ is formed by the
sets X \ F, where F ⊂
⋃
n∈N
Xn is a subset for which there is m ∈ N such that
Card (F ∩Xn) ≤ m for every n. It can be shown that the space X is almost discrete
and fails to have the SSPr. To this end, it suffices to note that X has no nontrivial con-
vergent sequences. On the other hand, the sequence of measures µn, where each µn
is concentrated on Xn and assigns equal values [Card (Xn)]−1 to the points of Xn,
converges weakly to Dirac’s measure at ∞. The existence of a Skorokhod parameteriza-
tion of a subsequence of µn would give a nontrivial convergent sequence. For the same
reason, no subsequence in {µn} is uniformly tight (otherwise such a subsequence could
be Skorokhod parameterized by Remark 3.1).
Finally, we shall see that nonmetrizable almost metrizable spaces with the strong Sko-
rokhod property for Radon measures exist indeed. Such spaces will be constructed as sub-
spaces of extremally disconnected spaces. A topological space X is called
extremally disconnected if the closure U of any open subset U of X is open, see,
e.g. [32]. A standard example of an extremally disconnected space is βN, the Stone –
Čech compactification of N. More generally, the Stone – Čech compactification βX of
a Tykhonoff space X is extremally disconnected if and only if the space X is extremally
disconnected [32] (§ 6.2).
Theorem 3.5. Any countable subspace X of an extremally disconnected completely
regular space K is almost discrete and has the strong Skorokhod property for Radon
measures.
Corollary 3.3. For every p ∈ βN \ N, the space X = {p} ∪ N with the induced
topology is a nonmetrizable almost discrete space with the strong Skorokhod property for
Radon measures.
It turns out that Theorem 3.5 holds true for a wider class of spaces. We say that
a completely regular space X is a Grothendieck space if the space Cb(X) with the
sup-norm is a Grothendieck Banach space. We recall that a Banach space E is said to
be a Grothendieck Banach space if ∗-weak convergence of countable sequences in E∗
is equivalent to weak convergence (i.e., convergence in the topology σ(E∗, E∗∗)). As
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1180 T. O. BANAKH, V. I. BOGACHEV, A. V. KOLESNIKOV
show by Grothendieck, each extremally disconnected Tykhonoff space is Grothendieck
(see also [35] for a discussion and further generalizations of the Grothendieck theorem).
Theorem 3.6. Any countable subspace X of a Grothendieck space K is almost
discrete and has the strong Skorokhod property.
Corollary 3.4. A subspace X of a Grothendieck space K is almost discrete and
has the SSPr if and only if all compact subsets of X are metrizable (equivalently, finite).
In particular, this is true if K is an extremally disconnected completely regular space.
The space {p}∪N, p ∈ βN\N, is probably the simplest example of a nonmetrizable
space with the strong Skorokhod property. The fact that it is not metrizable is seen from
the property that p belongs to the closure of N, but there are no infinite convergent
sequences with elements from N (if such a sequence {ni} converges, then the function
f(n2i) = 0, f(n2i+1) = 1 has no continuous extensions to βN ).
It should be noted that although weak convergence of countable sequences of proba-
bility measures on the space X in Corollary 3.3 is the same one that corresponds to the
discrete metric on X, the two weak topologies on the space of probability measures are
different (otherwise X would be metrizable in the topology from βN ).
Thus, in the class of countable spaces with a unique nonisolated point, there are almost
metrizable nonmetrizable spaces which have (or have not) the strong Skorokhod property.
On the other hand, all countable spaces with a unique nonisolated point have the
strong Skorokhod property for uniformly tight families of Radon measures.
Proposition 3.2. Any uniformly tight family of probability measures on a countable
space with a unique nonisolated point has the strong Skorokhod property.
Note that the nonmetrizable spaces with the strong Skorokhod property which have
been constructed so far are not topologically homogeneous. There exist also nonmetriz-
able topologically homogeneous spaces with the strong Skorokhod property for Radon
measures. A topological space X is called topologically homogeneous if for every pair
of points x, y ∈ X, there is a homeomorphism h : X → X such that h(x) = y. As
a rule, pathological examples of topologically homogeneous spaces are constructed by
using the technique of left topological groups. We recall that a left topological group is
a group (G, ∗) endowed with a left invariant topology, i.e., a topology τ such that for
each g ∈ G, the left shift lg : x �→ g ∗ x is continuous on (G, τ). A rich theory of
left topological groups has been developed by I. V. Protasov [36, 37]. According to [37]
(Theorem 4.1), each infinite group G admits a nondiscrete regular extremally discon-
nected left invariant topology. The theorem cited together with Theorem 3.6 implies the
following assertion.
Corollary 3.5. Each countable group G admits a left invariant topology τ such
that (G, τ) is a nonmetrizable countable almost discrete topologically homogeneous ex-
tremally disconnected space with the strong Skorokhod property for Radon measures.
Since compact subsets of almost metrizable spaces are metrizable, we conclude that
each Radon measure µ on an almost metrizable space X is concentrated on a σ-compact
space C ⊂ X with a countable network in the sense that µ(C) = 1. We recall that a
space X has a countable network if there is a countable family N of subsets of X such
that, for every point x ∈ X and every neighborhood U ⊂ X of x, there is an element
N ∈ N with x ∈ N ⊂ U.
4. The strong Skorokhod property in linearly ordered and compact spaces. Any
linear order ≤ on a set X generates two natural topologies on X. The usual interval
topology is generated by the pre-basis consisting of the rays
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TOPOLOGICAL SPACES WITH THE SKOROKHOD REPRESENTATION PROPERTY 1181
(←, a) = {x ∈ X : x < a}
and
(a,→) = {x ∈ X : x > a}, where a ∈ X.
The Sorgenfrey topology on X is generated by the pre-basis consisting of the rays (a,→)
and (←, a] = {x ∈ X : x ≤ a}, where a ∈ X. The space X endowed with the inter-
val topology will be denoted by
(
X, (≤)
)
. The space X endowed with the Sorgenfrey
topology will be denoted by
(
X, (≤]
)
. According to [32] (2.7.5) the space
(
X, (≤)
)
is
hereditarily normal, and the space
(
X, (≤]
)
is Tykhonoff and zero-dimensional.
We recall that a Souslin line is a linearly ordered nonseparable compact space with
countable cellularity, see [32] (2.7.9). It is known that the existence of a Souslin line is
independent of the ZFC axioms.
It is worth noting that the Sorgenfrey line can be topologically embedded into the
product A = [0, 1]×{0, 1} endowed with the interval topology generated by the lexico-
graphic order ≤: (x1, t1) ≤ (x2, t2) if and only if x1 < x2 or x1 = x2 and t1 ≤ t2.
The space A (called the Alexandroff two arrows space) is a nonmetrizable separable first
countable compact space.
A topological space X is called linearly ordered if it carries the interval topology
generated by some liner order on X. Standard examples of linearly ordered spaces are
the real line and intervals of ordinals.
Theorem 4.1. If ≤ is a linear order on a set X, then the spaces
(
X, (≤)
)
and(
X, (≤]
)
have the strong Skorokhod property for discrete probability measures.
Corollary 4.1. If each Radon probability measure on a linearly ordered topological
space X is discrete, then the space X has the strong Skorokhod property for Radon
measures.
A topological space X is called scattered if each subspace E of X has an isolated
point. It is well known that each Radon measure on a scattered space is discrete, see
[38] (Lemma 294).
Corollary 4.2. (i) Each scattered linearly ordered space has the SSPr.
(ii) For every ordinal α, the interval [0, α] endowed with the usual order topology
has the SSPr.
(iii) For any linear order ≤ on a set X, the space
(
X, (≤]
)
has the SSPr.
(iv) The Sorgenfrey line
(
R, (≤]
)
has the SSPr.
(v) The Alexandroff two arrows space A has the strong Skorokhod property for dis-
crete probability measures.
Since the space A admits a continuous surjection onto the interval [0, 1], it carries
nondiscrete probability measures, e.g., the measure whose projection is Lebesgue mea-
sure on [0, 1] (there is only one such measure on A ).
Now we briefly discuss the strong Skorokhod property in nonmetrizable compact
topological spaces. We already know that the ordinal interval [0, α] has the SSPr.
In particular, there exist nonmetrizable compact spaces with this property. However, it
seems that such examples are not numerous. One of the simplest examples of nonmetriz-
able compacta is the Alexandroff supersequence, which is the one point compactification
αℵ1 of a discrete space of the smallest uncountable size.
Example 4.1. The Alexandroff supersequence αℵ1 fails to have the strong Sko-
rokhod property even for probability measures with two-point supports. The same is true
for any topological space containing a copy of the Alexandroff supersequence αℵ1.
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1182 T. O. BANAKH, V. I. BOGACHEV, A. V. KOLESNIKOV
We recall that a space is dyadic if it is the image of certain power {0, 1}τ under
a continuous mapping. In the class of dyadic compacta only metrizable ones enjoy the
strong Skorokhod property.
Theorem 4.2. Each dyadic compact with the strong Skorokhod property for proba-
bility measures with two-point supports is metrizable. In particular, if a dyadic compact
has the strong Skorokhod property for probability measures with two-point supports, then
it has the strong Skorokhod property for Radon measures.
Besides the class of dyadic compact spaces, there are many interesting classes of
compact spaces for which their relation to the class of spaces with the strong Skorokhod
property has not yet been clarified. In this respect, it would be interesting to investigate
the classes of Eberlein, Corson, and Rosenthal compacta (see, e.g., [39, 40]). We recall
that a compact space K is defined to be
an Eberlein compact if K is homeomorphic to a compact subset of the Σ∗-product
Σ∗(τ) =
{
(xi)i∈τ ∈ R
τ : ∀ε > 0 the set {i ∈ τ : |xi| > ε} is finite
}
for some
cardinal τ ;
a Corson compact if K is homeomorphic to a compact subset of the Σ-product
Σ(τ) =
{
(xi)i∈τ ∈ R
τ : the set {i ∈ τ : xi �= 0} is countable
}
for some cardi-
nal τ ;
a Rosenthal compact if K is homeomorphic to a compact subset of the space
B1(P ) ⊂ R
P of all functions of the first Baire class on a Polish space P.
It is clear that any Eberlein compact is Corson. It is known that each separable Corson
compact as well as each Eberlein compact with countable cellularity is metrizable. The
Alexandroff two arrows space is a standard example of a Rosenthal compact which is not a
Corson compact. The Alexandroff supersequence is both an Eberlein and Rosenthal com-
pact. Thus, there exist Eberlein and Rosenthal compacta without the strong Skorokhod
property. The space [0, ω1] is neither Corson nor Rosenthal, see [41, p. 256, 259].
5. The weak Skorokhod property.
Definition 5.1. (i) We shall say that a topological space X has the weak Sko-
rokhod property if to every Radon probability measure µ on X, one can associate a
µ -measurable mapping ξµ : [0, 1] → X on ([0, 1],B([0, 1]), λ) with λ ◦ ξ−1
µ = µ in
such a way that for every uniformly tight sequence of Radon probability measures µn
on X, there is a subsequence in {ξµn} that converges almost everywhere.
(ii) We shall say that a topological space X has the weak Skorokhod property for
uniformly tight sequences if, for every uniformly tight sequence {µn} of Radon prob-
ability measures on X, there exist a subsequence {µnk
} ⊂ {µn}, and a sequence of
λ-measurable mappings ξk : [0, 1] → X with λ ◦ ξ−1
nk
= µnk
that converges almost
everywhere.
In both cases one can consider the extended WSP when [0, 1] with Lebesgue measure
is replaced by an arbitrary probability space (Ω,F , P ). We shall abbreviate the weak
Skorokhod property as WSP. The reader is warned that for simplicity of terminology we
alter the terminology from [11] where the WSP has been definied as the extended WSP
above.
Lemma 5.1. Suppose that there is a sequence of continuous functions ψj separat-
ing the points in X. Let (Ω,F , P ) be a probability space and let ξn : Ω → X be
(F ,B(X))-measurable mappings convergent almost everywhere to a mapping ξ such
that the sequence of measures P ◦ ξ−1
n is uniformly tight. Then the mapping ξ is
(FP ,B(X))-measurable, where FP is the completion of F with respect to P.
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TOPOLOGICAL SPACES WITH THE SKOROKHOD REPRESENTATION PROPERTY 1183
Proof. Let us take increasing compact sets Ki ⊂ X such that P ◦ξ−1
n (Ki) ≥ 1−2−i
for all n and i. Set ψ(x) := (ψ1(x), ψ2(x), . . .). The sets Ki are metrizable, since ψ
separates their points. Hence ψ is a Borel isomorphism between E :=
⋃∞
i=1Ki and
ψ(E) ⊂ R
∞. The mapping ψ ◦ ξ with values in R
∞ is defined P-a.e. and is P-
measurable. We have ψ(ξ(ω)) ∈ ψ(E) for P-a.e. ω, since P ◦ (ψ ◦ ξn)−1(ψ(Ki)) ≥
≥ 1 − 2−i for all n and i, which yields P ◦ (ψ ◦ ξ)−1(ψ(Ki)) ≥ 1 − 2−i for all i.
Therefore, ξ(ω) ∈ E for P-a.e. ω. Let us fix any x0 ∈ X. The mapping h : R
∞ → X
defined by h = g−1 on ψ(E) and h(y) = x0 if y �∈ ψ(E) is Borel measurable,
because ψ(E) is a countable union of compact sets. It remains to note that ξ(ω) =
= h(ψ(ξ(ω))) for P-a.e. ω.
We recall that in general a pointwise limit of Borel mappings with values in a topo-
logical space may fail to be Borel measurable.
Theorem 5.1. (Jakubowki [6]). Every space that possesses a countable family of
continuous functions separating the points has the weak Skorokhod property for uniformly
tight sequences.
This is a very useful general result. In particular, it applies to all completely regular
Souslin spaces.
The following result is proved in [11].
Proposition 5.1. (i) Let X be a topological space that has a continuous injection
into R
∞ and let Φ: X → Y be a Borel mapping with values in a metrizable Souslin
space Y such that the sets Φ(A) and Φ−1(B) have compact closure for all compact
sets A and B. Then X has the weak Skorokhod property.
(ii) Let f be a continuous surjection of a space X with the weak Skorokhod property
onto a space Y such that the preimages of compact sets are compact. Then Y has the
weak Skorokhod property.
Let us consider examples of spaces with the weak Skorokhod property. More ex-
amples can be produced by using the fact that every subset of a space with the weak
Skorokhod property has this property as well.
Example 5.1. (i) Let X =
∏∞
m=1X
m, where each space Xm admits a continuous
injection into R
∞ and has a fundamental sequence {Km
n } of compacts sets (i.e., every
compact subset of Xm is contained in one of the sets Km
n ). Then X has the weak
Skorokhod property.
(ii) Let a locally convex space X be the strict inductive limit of an increasing se-
quence of its closed metrizable Souslin linear subspaces Xj . Then X has the weak
Skorokhod property. The same is true for any countable product of such inductive limits.
Note that the space of tempered distributions S ′(Rd) has a fundamental sequence of
compact sets; the same is true for the duals to separable Banach spaces with the weak-∗
topology. The above example covers also the space of distributions D′(Rd) (which is
homeomorphic to a closed subspace of a countable product of spaces having fundamental
sequences of compact sets). We observe that part (ii) of the example covers the space
D(Rd) of smooth compactly supported functions on R
d (with its natural topology of the
inductive limit of the spaces C∞
0 (Un), where Un is the centered open ball of radius n ).
The next theorem is proved in [15]. A mapping f : X → Y between topological
spaces is called compact-covering if for every compact set K ⊂ Y there is a compact set
l(K) ⊂ X such that K = f(l(K)). If one can choose the sets l(K) in such a way that
l(K) ⊂ l(K ′) whenever K ⊂ K ′, then f is called monotone compact-covering.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 9
1184 T. O. BANAKH, V. I. BOGACHEV, A. V. KOLESNIKOV
Theorem 5.2. A completely regular space Y has the WSP if and only if there is a
monotone compact-covering mapping f : X → Y from a space X possessing the WSP.
In particular, a completely regular space that is the image of a metrizable space under
a monotone compact-covering mapping has the WSP.
The last result justifies the following definition.
Definition 5.2. A Hausdorff topological space X is called a monotone ℵ -space if
X is the image of a metrizable space M under a monotone compact-covering mapping
f : M → X. If M is separable, then X is called a monotone ℵ0-space.
Thus, any completely regular monotone ℵ -space possesses the WSP.
Monotone ℵ-spaces are thoroughly investigated in Sections 3 and 4 in [15]. Here we
mention only the following result. A topological space is called submetrizable if it admits
a continuous bijective mapping onto a metrizable space.
Theorem 5.3. (i) Let X be a submetrizable sequentially Prokhorov completely reg-
ular monotone ℵ-space. Then the space Pr(X) with the weak topology is a monotone
ℵ -space, hence has the WSP.
(ii) Let X be a sequentially Prokhorov completely regular monotone ℵ0-space. Then
the space Pr(X) is a monotone ℵ0 -space.
Note also that if X is a separable Banach space, then X∗ with the topology σ(X∗, X)
is a monotone ℵ0 -space. If also X∗ is separable, then the same is true for X with the
weak topology.
6. Open problems. Here we mention a number of open questions that are closely
related to the above discussion and have simple formulations.
Question 1. Is there an infinite extremally disconnected compact space with the strong
Skorokhod property for Radon measures? In particular, does βN possess the strong
Skorokhod property for Radon measures?
Let us give the definitions of two interesting countable spaces. The Fréchet – Urysohn
fan V is defined as follows (cf. [32], 1.6.18). Let
V := {k + (n+ 1)−1 : k, n ∈ N} ∪ {0}
be endowed with the following topology: every point k+ (n+ 1)−1 has its usual neigh-
borhoods from the space V \{0} and the point 0 has an open base formed by the sets
Un1,...,nj ,... := {k + (n+ 1)−1 : k ∈ N, n ≥ nk} ∪ {0},
where {nj} is a sequence of natural numbers. The Arens fan A2 is the space
A2 = {(0, 0), (1/i, 0), (1/i, 1/j) : 1 ≤ i ≤ j <∞}
endowed with the strongest topology inducing the original topology on each compact
Kn = {(0, 0), (1/k, 0), (1/i, 1/j) : k ∈ N, 1 ≤ i ≤ n, i ≤ j <∞}.
Question 2. Do the Fréchet – Urysohn and Arens fans have the strong Skorokhod
property?
Let us note that according to Proposition 3.2 the Fréchet – Urysohn fan has the strong
Skorokhod property for uniformly tight families of probability measures.
Question 3. Does the product [0, 1]× [0, ω1] have the strong Skorokhod property for
Radon measures or for discrete probability measures?
Question 4. Does every linearly ordered compact space have the strong Skorokhod
property for Radon measures? In particular, does the Souslin line have the strong Sko-
rokhod property for Radon measures?
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TOPOLOGICAL SPACES WITH THE SKOROKHOD REPRESENTATION PROPERTY 1185
Question 5. Is it true that the Alexandroff two arrows space A possess the strong
Skorokhod property for Radon measures?
Question 6. Is every Eberlein (Corson, Rosenthal) compact with the strong Sko-
rokhod property metrizable?
It is worth noting that each countable family M of probability measures on an Eber-
lein compact E admits a Skorokhod representation. Indeed, one can show that there is a
metrizable compact subset K ⊂ E with µ(K) = 1 for all µ ∈M.
Question 7. Suppose a compact K has the strong Skorokhod property and admits a
continuous finite-to-one map onto a metrizable compact. Is K metrizable?
Question 8. Stability of the class of spaces with the strong Skorokhod property (or
with the other related properties mentioned above) with respect to formation of finite and
countable products.
In particular, it would be interesting to study whether the product X × Y, where X
has the strong property and Y is separable metric (say, Y = N
∞), retains the same
Skorokhod property. We do not know an answer even for finite or countable Y (in
particular, we do not know whether the topological sum of two spaces with the strong
Skorokhod property has that property).
Question 9. Suppose that a completely regular space X admits a continuous injec-
tion into a Polish space. Does X have the weak Skorokhod property?
It would be also interesting to study a version of Skorokhod’s representation in which
convergence of mappings almost everywhere is replaced by convergence in measure,
which makes sense for subspaces of locally convex spaces (then one has to consider
convergence in measure with respect to every fixed seminorm defining the topology of
a given space) or for uniform spaces.
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