Open AccessA Generalization of the Blackwell–Ryll-Nardzewski Measurable Selection Theorem
Author(s) -
K.A. Afonin
Publication year - 2021
Publication title -
l.n. gumilev atyndaġy euraziâ u̇lttyk̦ universitetìnìn̦ habaršysy. matematika, informatika, mehanika seriâsy
Language(s) - English
Resource type - Journals
eISSN - 2663-1326
pISSN - 2616-7182
DOI - 10.32523/2616-678x-2020-134-1-35-41
Subject(s) - mathematics , product measure , borel set , measurable function , separable space , metric space , borel hierarchy , polish space , discrete mathematics , generalization , complete metric space , borel measure , combinatorics , probability measure , product (mathematics) , mathematical analysis , bounded function , geometry
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 in the product of two measurable spaces and . Such a graph enables one to pick a point in the section for each in 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 the section , which is a set in . The classical theorem of Blackwell and Ryll-Nardzewski states that a Borel set 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 . 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 in the product have positive measures.