Spaces of measurable functions

For a metrizable space $X$ and a finite measure space $(Ω,\mathfrak{M},μ)$ let $M_μ(X)$ and $M^f_μ(X)$ be the spaces of all equivalence classes (under the relation of equality almost everywhere mod $μ$) of $mathfrak{M}$-measurable functions from $Ω$ 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 $μ$ is (nonzero and) nonatomic and $X$ has more than one point, then the space $M_μ(X)$ is a noncompact absolute retract and $M^f_μ(A)$ is homotopy dense in $M_μ(X)$ for each dense subset $A$ of $X$. In particular, if $X$ is completely metrizable, then $M_μ(X)$ is homeomorphic to an infinite-dimensional Hilbert space.

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