Topological Spaces with the Strong Skorokhod Property

Abstract

We study topological spaces with the strong Skorokhod property, i.e., spaces on which all Radon probability measures can be simultaneously represented as images of Lebesgue measure on the unit interval under certain Borel mappings so that weakly convergent sequences of measures correspond to almost everywhere convergent sequences of mappings. We construct nonmetrizable spaces with such a property and investigate the relations between the Skorokhod and Prokhorov properties. It is also shown that a dyadic compact has the strong Skorokhod property precisely when it is metrizable.

k*-Metrizable spaces and their applications

“…It is well-known that for every continuous map f : X → Y between Hausdorff spaces the function P r (f ) : P r (X) → P r (Y ) is continuous; for an injective map f between Tychonoff spaces the map P r (f ) is injective; for a surjective map f between compact Hausdorff spaces the map P R (f ) is surjective; for a topological embedding f : X → Y of Tychonoff spaces the map P r (f ) is a topological embedding (see [17,Ch. 8,9]).…”

“…Following [18] and [8] we say that a Tychonoff space X has the strong Skorohod property if to each measure µ ∈ P r (X) one can assign a Borel function ξ µ : [0, 1] → X such that µ is the image of Lebesgue measure under the function ξ µ and for every sequence (µ n ) ⊂ P r (X) convergent to a measure µ 0 ∈ P r (X) the function sequence (ξ µn ) converges to ξ µ 0 almost surely on [0, 1]. The uniformly tight strong Skorohod property is defined by requiring the latter only for uniformly tight weakly convergent sequences (µ n ).…”

“…Thus, each metrizable space has the strong Skorohod property. The situation changes beyond the class of metrizable spaces, see [8], [9], [10]. It was observed in [18, 4.1] that the inductive limit R ∞ = lim −→ R n of finite-dimensional Euclidean spaces (in some sense, the simplest non-metrizable locally convex space) fails to have the strong Skorohod property.…”

Kantorovich problems and conditional measures depending on a parameter

“…This important result was generalized by Blackwell and Dubins [7] and Fernique [23], who proved that for every measure µ ∈ P(X) there is a Borel mapping ξ µ : [0, 1] → X such that µ is the image of Lebesgue measure λ under ξ µ and measures µ n converge weakly to µ if and only if the mappings ξ µn converge to ξ µ almost everywhere. A topological proof of this result along with some generalizations was given in [13] (see also [4], [9], and [12] on this topic). The purpose of this section is to verify that this topological proof actually yields the following result.…”