What is Measurable Selection Theorem?

Measurable Selection Theorem has wide applications on stochastic control theory, and other fields. We will describe this theorem briefly, see details in [Wag77] (Download).

Suppose {(T, \mathcal{M})} is a measurable space, {(X, \tau)} is a topological space, and {\emptyset \neq F(t) \subset X} for {t\in T}. Denote {Gr F = \{(t,x): x\in F(t)\}}. The problems is that of existence of {\mathcal{M}}-measurable {f:T\mapsto X} such that {f(t) \in F(t)} for all {t\in T}.

The principle conditions that yield such {f} are one of the following two conditions:

  1. {X} is Polish, each {F(t)} is closed, and {\{t:F(t)\cap U \neq \emptyset\} \in \mathcal{M}} whenever {U\in \tau}.
  2. {T} is a Hausdorff space, {Gr F} is a continuous image of a Polish space, and {\mathcal{M}} is the {\sigma}-algebra of sets measurable with respect to an outer measure, among which are the open sets of {T}.

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