# 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 ${T}$.