Spaces of measurable functions

TL;DR

This paper studies the topological structure of spaces of measurable functions from a measure space to a metrizable space, showing they are absolute retracts and, in certain cases, homeomorphic to infinite-dimensional Hilbert spaces.

Contribution

It proves that these function spaces are absolute retracts and characterizes their topology, including conditions under which they are homeomorphic to Hilbert spaces.

Findings

$M_{}(X)$ is homotopy dense in $M_{}(X)$ for dense subsets $A$ of $X$.

If $X$ is completely metrizable, then $M_{}(X)$ is homeomorphic to an infinite-dimensional Hilbert space.

$M_{}(X)$ is a noncompact absolute retract under the given conditions.

Abstract

For a metrizable space $X$ and a finite measure space $(\Omega,\mathfrak{M},\mu)$ let $M_{\mu}(X)$ and $M^f_{\mu}(X)$ be the spaces of all equivalence classes (under the relation of equality almost everywhere mod $\mu$) of $mathfrak{M}$-measurable functions from $\Omega$ to $X$ whose images are separable and finite, respectively, equipped with the topology of convergence in measure. The main aim of the paper is to prove the following result: if $\mu$ is (nonzero and) nonatomic and $X$ has more than one point, then the space $M_{\mu}(X)$ is a noncompact absolute retract and $M^f_{\mu}(A)$ is homotopy dense in $M_{\mu}(X)$ for each dense subset $A$ of $X$. In particular, if $X$ is completely metrizable, then $M_{\mu}(X)$ is homeomorphic to an infinite-dimensional Hilbert space.