Spaces of measurable functions

Piotr Niemiec

Open Mathematics (2013)

  • Volume: 11, Issue: 7, page 1304-1316
  • ISSN: 2391-5455

Abstract

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For a metrizable space X and a finite measure space (Ω, 𝔐 , µ), the space M µ(X) of all equivalence classes (under the relation of equality almost everywhere mod µ) of 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.

How to cite

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Piotr Niemiec. "Spaces of measurable functions." Open Mathematics 11.7 (2013): 1304-1316. <http://eudml.org/doc/269115>.

@article{PiotrNiemiec2013,
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.},
author = {Piotr Niemiec},
journal = {Open Mathematics},
keywords = {Measurable function; Absolute retract; Infinite-dimensional manifold; Nonatomic measure; Functor of extension; measurable function; absolute retract; infinite-dimensional manifold; nonatomic measure; functor of extension; convergence in measure; Hilbert space},
language = {eng},
number = {7},
pages = {1304-1316},
title = {Spaces of measurable functions},
url = {http://eudml.org/doc/269115},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Piotr Niemiec
TI - Spaces of measurable functions
JO - Open Mathematics
PY - 2013
VL - 11
IS - 7
SP - 1304
EP - 1316
AB - 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.
LA - eng
KW - Measurable function; Absolute retract; Infinite-dimensional manifold; Nonatomic measure; Functor of extension; measurable function; absolute retract; infinite-dimensional manifold; nonatomic measure; functor of extension; convergence in measure; Hilbert space
UR - http://eudml.org/doc/269115
ER -

References

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