Spaces of measurable functions

For a metrizable space $X$ and a finite measure space $(\mathrm{\Omega },\mathfrak{M},\mu )$ let $M_{\mu }(X)$ and $M_{\mu }^f(X)$ be the spaces of all equivalence classes (under the relation of equality almost everywhere mod $\mu$) of $mathfrakM$-measurable functions from $\mathrm{\Omega }$ 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 $M_{\mu }^f(A)$ 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.