Abstract
For a metrizable space X and a finite measure space (Ω, \(\mathfrak{M}\), µ), the space M µ(X) 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, equipped with the topology of convergence in measure, and some of its subspaces are studied. In particular, it is shown that M µ(X) is homeomorphic to a Hilbert space provided µ is (nonzero) nonatomic and X is completely metrizable and has more than one point.
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Niemiec, P. Spaces of measurable functions. centr.eur.j.math. 11, 1304–1316 (2013). https://doi.org/10.2478/s11533-013-0236-6
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DOI: https://doi.org/10.2478/s11533-013-0236-6