Topological Spaces with Skorokhod Representation Property

Банах, Тарас; Богачев, В. И.; Kolesnikov, Alexander V.

doi:10.1007/s11253-006-0002-z

Cited by 4 publications

(8 citation statements)

References 22 publications

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“…Combining Theorem 5.3 with the second item of Theorem 5.9, we get Corollary 5. 10. Under (CH), every regular k * -metrizable k-space satisfies the equality knw(X) = d(X).…”

Section: Cardinal Invariants Of K * -Metrizable Spacesmentioning

confidence: 99%

“…8, 12.9 give examples of infinite-dimensional Banach spaces X which being endowed with the weak topology are k * -metrizable spaces and hence have the weak Skorohod property. In contrast, the space (X, weak) has the strong Skorohod property if and only if X is finite-dimensional, see [10]. 13.…”

Section: The K * -Metrizability Of the Weak Topology On Banach Spacesmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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k*-Metrizable spaces and their applications

2008

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In this paper we introduce and study so-called k * -metrizable spaces forming a new class of generalized metric spaces, and display various applications of such spaces in topological algebra, functional analysis, and measure theory.By definition, a Hausdorff topological space X is k * -metrizable if X is the image of a metrizable space M under a continuous map f : M → X having a section s : X → M that preserves precompact sets in the sense that the image s(K) of any compact set K ⊂ X has compact closure in X.Contents 42 13. The structure of sequential k * -metrizable groups 47 References 49 1 Proposition 1.3. Let X, Y be topological spaces such that X is µ-complete and each compact subset of Y is sequentially compact. A function s : Y → X is precompact-preserving if and only if s is cs * -continuous.Proof. The "only if" part is trivial and holds without any assumptions on X and Y . To prove the "if" part, assume that s : Y → X is a cs * -continuous function from a space Y whose all compact subsets are sequentially compact into a µ-complete space X. To show that s is precompact-preserving, take any compact subset K ⊂ Y . We claim that the image s(K) is precompact. Assuming that it is not so, we get that s(K) is not bounded by the µ-completeness of X. Consequently there is an infinite locally finite family U = {U n : n ∈ ω}

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“…Combining Theorem 5.3 with the second item of Theorem 5.9, we get Corollary 5. 10. Under (CH), every regular k * -metrizable k-space satisfies the equality knw(X) = d(X).…”

Section: Cardinal Invariants Of K * -Metrizable Spacesmentioning

confidence: 99%

Section: The K * -Metrizability Of the Weak Topology On Banach Spacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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k*-Metrizable spaces and their applications

2008

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“…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.…”

Section: The Skorohod Parametrization With a Parametermentioning

confidence: 86%

Kantorovich problems and conditional measures depending on a parameter

Богачев

Malofeev

2020

Journal of Mathematical Analysis and Applications

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We study measurable dependence of measures on a parameter in the following two classical problems: constructing conditional measures and the Kantorovich optimal transportation. For parametric families of measures and mappings we prove the existence of conditional measures measurably depending on the parameter. A particular emphasis is made on the Borel measurability (which cannot be always achieved). Our second main result gives sufficient conditions for the Borel measurability of optimal transports and transportation costs with respect to a parameter in the case where marginal measures and cost functions depend on a parameter. As a corollary we obtain the Borel measurability with respect to the parameter for disintegrations of optimal plans. Finally, we show that the Skorohod parametrization of measures by mappings can be also made measurable with respect to a parameter.

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“…We also point out that topological spaces with a Skorohod representation property behave delicately: they are not known to be closed for topological products; counterexamples show that, even for a weakly convergent sequence of probabilities, Skorohod representation may only work for a subsequence, etc. We refer the interested reader to [2] for details.…”

Section: A Representation Theoremmentioning

confidence: 99%

Skorohod's Representation Theorem and Optimal Strategies for Markets with Frictions

Chau¹,

Rásonyi²

2017

SIAM J. Control Optim.

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We prove the existence of optimal strategies for agents with cumulative prospect theory preferences who trade in a continuous-time illiquid market, transcending known results which pertained only to risk-averse utility maximizers. The arguments exploit an extension of Skorohod's representation theorem for tight sequences of probability measures. This method is applicable in a number of similar optimization problems.

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Topological Spaces with Skorokhod Representation Property

Abstract: 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.

Cited by 4 publications

References 22 publications

k*-Metrizable spaces and their applications

k*-Metrizable spaces and their applications

Kantorovich problems and conditional measures depending on a parameter

Skorohod's Representation Theorem and Optimal Strategies for Markets with Frictions

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