Survey of Measurable Selection Theorems

1977 ◽  
Vol 15 (5) ◽  
pp. 859-903 ◽  
Author(s):  
Daniel H. Wagner
Keyword(s):  
Measurable Selection ◽  
Selection Theorems

1.—On Measurable Selections

1974 ◽  
Vol 72 (1) ◽  
pp. 1-7 ◽  
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.

scholarly journals Nonlinear random operator equations and inequalities in Banach spaces

1992 ◽  
Vol 15 (1) ◽  
pp. 111-118 ◽  
Author(s):  
Antonios Karamolegos ◽  
Dimitrios Kravvaritis
Keyword(s):  
Banach Spaces ◽  
Existence Theorems ◽  
Random Operator ◽  
Measurable Selection ◽  
Hammerstein Integral Equation ◽  
Operator Equations ◽  
Deterministic Theory ◽  
Measurable Multifunctions ◽  
Selection Theorems ◽  
Monotone Type

In this paper we give some new existence theorems for nonlinear random equations and inequalities involving operators of monotone type in Banach spaces. A random Hammerstein integral equation is also studied. In order to obtain random solutions we use some results from the existing deterministic theory as well as from the theory of measurable multifunctions and, in particular, the measurable selection theorems of Kuratowski/Ryll-Nardzewski and of Saint-Beuve.

scholarly journals On zero-dimensional Milutin maps and Michael selection theorems

1993 ◽  
Vol 54 (1-3) ◽  
pp. 77-83 ◽  
Author(s):  
D. Repovš ◽  
P.V. Semenov ◽  
E.V. Ščepin
Keyword(s):  
Selection Theorems

scholarly journals A Generalization of the Blackwell–Ryll-Nardzewski Measurable Selection Theorem

2021 ◽  
Vol 134 (1) ◽  
pp. 35-41
Author(s):  
K.A. Afonin ◽  
◽  
Keyword(s):  
Metric Spaces ◽  
Transition Probability ◽  
Measurable Space ◽  
Measurable Selection ◽  
Selection Theorem ◽  
Borel Set ◽  
Borel Probability ◽  
Borel Mapping ◽  
The Given ◽  
Selection Of

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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