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 Omega-Polish spaces and measurable selections

1977 ◽  
Vol 23 (3) ◽  
pp. 257-265 ◽  
Author(s):  
Le Van Tu
Keyword(s):  
Large Class ◽  
Banach Spaces ◽  
Polish Space ◽  
Measurable Space ◽  
Selection Theorem ◽  
Polish Spaces ◽  
Measurable Selections

AbstractIn this paper, the author introduces the notion of Ω-Polish spaces (which includes the Polish spaces and a large class of Banach spaces) and extends Castaing's selection theorem (1966) for closed-valued measurable thin multifunctions from a measurable space into an Ω-Polish space. He also extends Robertson's theorem (1974) in the same way.

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.

Measures Defined by Gages

1992 ◽  
Vol 44 (6) ◽  
pp. 1303-1316 ◽  
Author(s):  
Washek F. Pfeffer ◽  
Brian S. Thomson
Keyword(s):  
Set Theory ◽  
Measure Space ◽  
Hausdorff Space ◽  
Outer Measure ◽  
Riesz Representation Theorem ◽  
Locally Compact ◽  
Riemann Integral ◽  
Riesz Representation ◽  
Intuitive Concept ◽  
Locally Compact Hausdorff Space

AbstractUsing ideas of McShane ([4, Example 3]), a detailed development of the Riemann integral in a locally compact Hausdorff space X was presented in [1]. There the Riemann integral is derived from a finitely additive volume v defined on a suitable semiring of subsets of X. Vis-à-vis the Riesz representation theorem ([8, Theorem 2.141), the integral generates a Riesz measure v in X, whose relationship to the volume v was carefully investigated in [1, Section 7].In the present paper, we use the same setting as in [1] but produce the measure directly without introducing the Riemann integral. Specifically, we define an outer measure by means of gages and introduce a very intuitive concept of gage measurability that is different from the usual Carathéodory définition. We prove that if the outer measure is σ-finite, the resulting measure space is identical to that defined by means of the Carathéodory technique, and consequently to that of [1, Section 7]. If the outer measure is not σ-finite, we investigate the gage measurability of Carathéodory measurable sets that are σ-finite. Somewhat surprisingly, it turns out that this depends on the axioms of set theory.

A measurable selection theorem for compact-valued maps

1979 ◽  
Vol 27 (4) ◽  
pp. 341-352 ◽  
Author(s):  
Siegfried Graf
Keyword(s):  
Measurable Selection ◽  
Selection Theorem

An effective selection theorem

1982 ◽  
Vol 47 (2) ◽  
pp. 388-394 ◽  
Author(s):  
Ashok Maitra
Keyword(s):  
Polish Space ◽  
Local Version ◽  
Compact Set ◽  
Selection Theorem ◽  
Analytic Subset ◽  
Effective Version ◽  
Basis Theorem ◽  
Measurable Selector ◽  
Effective Selection ◽  
Borel Measurable

A recent result of J.P. Burgess [1] states:Theorem 0. Let F be a multifunction from an analytic subset T of a Polish space to a Polish space X. If F is Borel measurable, Graph(F) is coanalytic in T × X and F(t) is nonmeager in its closure for each t Є T, then F admits a Borel measurable selector.The above result unifies and significantly extends earlier results of H. Sarbadhikari [8], S.M. Srivastava [9] and G. Debs (unpublished). The reader is referred to [1] for details.The aim of this article is to give an effective version of Theorem 0. We do this by proving a basis theorem for Π11 sets which are nonmeager in their closure and satisfy a local version of the measurability condition in Theorem 0. Our basis theorem generalizes a well-known result of P.G. Hinman [4] and S.K. Thomason [10] (see also [5] and [7, 4F.20]). Our methods are similar to those used by A. Louveau to prove that a , σ-compact set is contained in a , σ-compact set (see [7, 4F.18]).The paper is organized as follows. §2 is devoted to preliminaries. In §3, we prove the basis theorem and deduce as a consequence an effective version of Theorem 0. We show in §4 how our methods can be used to give alternative proofs of some known results.Discussions with R. Barua, B.V. Rao and V.V. Srivatsa are gratefully acknowledged. I am indebted to J.P. Burgess for drawing my attention to an error in an earlier draft of this paper.

scholarly journals Stationary countable dense random sets

2000 ◽  
Vol 32 (01) ◽  
pp. 86-100 ◽  
Author(s):  
Wilfrid S. Kendall
Keyword(s):  
Probability Theory ◽  
Polish Space ◽  
Measurable Subset ◽  
Locally Compact Group ◽  
Random Sets ◽  
Random Set ◽  
Locally Compact ◽  
Polish Spaces ◽  
Probability Zero ◽  
Group Of Symmetries

We study the probability theory of countable dense random subsets of (uncountably infinite) Polish spaces. It is shown that if such a set is stationary with respect to a transitive (locally compact) group of symmetries then any event which concerns the random set itself (rather than accidental details of its construction) must have probability zero or one. Indeed the result requires only quasi-stationarity (null-events stay null under the group action). In passing, it is noted that the property of being countable does not correspond to a measurable subset of the space of subsets of an uncountably infinite 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 Wadge-like reducibilities on arbitrary quasi-Polish spaces

2014 ◽  
Vol 25 (8) ◽  
pp. 1705-1754 ◽  
Author(s):  
LUCA MOTTO ROS ◽  
PHILIPP SCHLICHT ◽  
VICTOR SELIVANOV
Keyword(s):  
Real Line ◽  
Polish Space ◽  
Baire Space ◽  
Topological Spaces ◽  
Polish Spaces ◽  
The Real ◽  
Degree Structure ◽  
Wadge Hierarchy

The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well ordered), but for many other natural nonzero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called Δ0α-reductions, and try to find for various natural topological spaces X the least ordinal αX such that for every αX ⩽ β < ω1 the degree-structure induced on X by the Δ0β-reductions is simple (i.e. similar to the Wadge hierarchy on the Baire space). We show that αX ⩽ ω for every quasi-Polish space X, that αX ⩽ 3 for quasi-Polish spaces of dimension ≠ ∞, and that this last bound is in fact optimal for many (quasi-)Polish spaces, including the real line and its powers.

A measurable-selection theorem

2016 ◽  
pp. 407-411
Author(s):  
James C. Robinson ◽  
Jose L. Rodrigo ◽  
Witold Sadowski
Keyword(s):  
Measurable Selection ◽  
Selection Theorem
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