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