A measurable selection theorem for compact-valued maps

One of the main forms of the measurable selection theorem is connected with the existence of the graph of a measurable mapping in a given measurable set 𝑆 in the product of two measurable spaces 𝑋 and 𝑌 . Such a graph enables one to pick a point in the section 𝑆𝑥 for each 𝑥 in 𝑋 such that the obtained mapping will be measurable. The indicated selection is called a measurable selection of the multi-valued mapping associating to the point 𝑥 the section 𝑆𝑥 , which is a set in 𝑌 . The classical theorem of Blackwell and Ryll-Nardzewski states that a Borel set 𝑆 in the product of two complete separable metric spaces contains the graph of a Borel mapping (hence admits a Borel selection) provided that there is a transition probability on this product with positive measures for all sections of 𝑆 . The main result of this paper gives a generalization to the case where only one of the two spaces is complete separable and the other one is a general measurable space whose points parameterize a family of Borel probability measures on the first space such that the sections of the given set 𝑆 in the product have positive measures.

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1.—On Measurable Selections

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0080454100009365 ◽

1974 ◽

Vol 72 (1) ◽

pp. 1-7 ◽

Cited By ~ 5

Author(s):

A. P. Robertson

Keyword(s):

Measure Space ◽

Hausdorff Space ◽

Polish Space ◽

Measurable Selection ◽

Selection Theorem ◽

Polish Spaces ◽

Measurable Multifunction ◽

Selection Theorems ◽

Suitable Measure ◽

Separable Metric Space

SynopsisMeasurable selection theorems are proved, for a compact-valued measurable multifunction into a Hausdorff space that is the continuous image of a separable metric space, and for a closed-valued measurable multifunction from a suitable measure space to a regular Souslin space. The connection between Polish spaces and certain subsets of the real line is related to a measurable selection theorem for multifunctions into a Polish space.

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A measurable-selection theorem

The Three-Dimensional Navier-Stokes Equations ◽

10.1017/cbo9781139095143.026 ◽

2016 ◽

pp. 407-411

Author(s):

James C. Robinson ◽

Jose L. Rodrigo ◽

Witold Sadowski

Keyword(s):

Measurable Selection ◽

Selection Theorem

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A measurable selection theorem

Fundamenta Mathematicae ◽

10.4064/fm-110-2-91-100 ◽

1980 ◽

Vol 110 (2) ◽

pp. 91-100 ◽

Cited By ~ 1

Author(s):

John Burgess

Keyword(s):

Measurable Selection ◽

Selection Theorem

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On a class of generalized stochastic Browder mixed variational inequalities

Journal of Inequalities and Applications ◽

10.1186/s13660-019-2278-1 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Chao Min ◽

Fei-fei Fan ◽

Zhao-zhong Yang ◽

Xiao-gang Li

Keyword(s):

Variational Inequalities ◽

Existence Of Solutions ◽

Uniqueness Of Solution ◽

Measurable Selection ◽

Stochastic Variational Inequalities ◽

Selection Theorem ◽

Solution Sets ◽

Mixed Variational Inequalities

AbstractIn this paper, we introduce a class of stochastic variational inequalities generated from the Browder variational inequalities. First, the existence of solutions for these generalized stochastic Browder mixed variational inequalities (GS-BMVI) are investigated based on FKKM theorem and Aummann’s measurable selection theorem. Then the uniqueness of solution for GS-BMVI is proved based on strengthening conditions of monotonicity and convexity, the compactness and convexity of the solution sets are discussed by Minty’s technique. The results of this paper can provide a foundation for further research of a class of stochastic evolutionary problems driven by GS-BMVI.

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Stability of stochastic differential equations driven by multifractional Brownian motion

Random Operators and Stochastic Equations ◽

10.1515/rose-2021-2055 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Oussama El Barrimi ◽

Youssef Ouknine

Keyword(s):

Brownian Motion ◽

Differential Equations ◽

Stochastic Differential Equations ◽

Stationary Process ◽

Strong Stability ◽

Pathwise Uniqueness ◽

Uniqueness Property ◽

Selection Theorem ◽

Multifractional Brownian Motion ◽

Stability Results

Abstract Our aim in this paper is to establish some strong stability results for solutions of stochastic differential equations driven by a Riemann–Liouville multifractional Brownian motion. The latter is defined as a Gaussian non-stationary process with a Hurst parameter as a function of time. The results are obtained assuming that the pathwise uniqueness property holds and using Skorokhod’s selection theorem.

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