Spaces of measurable functions

Piotr Niemiec

Open Mathematics, Tome 11 (2013), p. 1304-1316 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2013-01-01

Zbl 1277.54021

EUDML-ID : urn:eudml:doc:269115

Bibliographie

[1] Banakh T.O., Topology of spaces of probability measures. I. The functors P τ and $\hat{P}$ , Mat. Stud., 1995, 5, 65–87 (in Russian), English translation available at http://arxiv.org/abs/1112.6161

[2] Banakh T.O., Topology of spaces of probability measures. II. Barycenters of Radon probability measures and the metrization of the functors P τ and $\hat{P}$ , Mat. Stud., 1995, 5, 88–106 (in Russian), English translation available at http://arxiv.org/abs/1206.1727

[3] Banakh T., Bessaga Cz., On linear operators extending [pseudo]metrics, Bull. Polish Acad. Sci. Math., 2000, 48(1), 35–49 | Zbl 0948.54021

[4] Banakh T.O., Radul T.N., Topology of spaces of probability measures, Sb. Math., 1997, 188(7), 973–995 http://dx.doi.org/10.1070/SM1997v188n07ABEH000241[Crossref] | Zbl 0893.28004

[5] Banakh T., Zarichnyy I., Topological groups and convex sets homeomorphic to non-separable Hilbert spaces, Cent. Eur. J. Math., 2008, 6(1), 77–86 http://dx.doi.org/10.2478/s11533-008-0005-0[WoS][Crossref] | Zbl 1202.57023

[6] Bessaga Cz., Pełczyński A., On spaces of measurable functions, Studia Math., 1972, 44(6), 597–615 | Zbl 0256.46048

[7] Chapman T.A., Deficiency in infinite-dimensional manifolds, General Topology and Appl., 1971, 1(3), 263–272 http://dx.doi.org/10.1016/0016-660X(71)90097-3[Crossref]

[8] Dobrowolski T., Toruńczyk H., Separable complete ANR’s admitting a group structure are Hilbert manifolds, Topology Appl., 1981, 12(3), 229–235 http://dx.doi.org/10.1016/0166-8641(81)90001-8[Crossref] | Zbl 0472.57009

[9] Halmos P.R., Measure Theory, Van Nostrand, New York, 1950

[10] Hartman S., Mycielski J., On the imbedding of topological groups into connected topological groups, Colloq. Math., 1958, 5, 167–169 | Zbl 0086.02601

[11] Kuratowski K., Mostowski A., Set Theory, 2nd ed., Stud. Logic Found. Math., 86, North-Holland/PWN, Amsterdam-New York-Oxford/Warsaw, 1976

[12] Maharam D., On homogeneous measure algebras, Proc. Nat. Acad. Sci. U.S.A., 1942, 28, 108–111 http://dx.doi.org/10.1073/pnas.28.3.108[Crossref] | Zbl 0063.03723

[13] Niemiec P., Functor of continuation in Hilbert cube and Hilbert space, preprint available at http://arxiv.org/abs/1107.1386 | Zbl 1307.54021

[14] Rudin W., Real and Complex Analysis, McGraw-Hill, New York-Toronto, 1966 | Zbl 0142.01701

[15] Takesaki M., Theory of Operator Algebras, I, Encyclopaedia Math. Sci., 124, Springer, Berlin, 2002

[16] Toruńczyk H., Characterization of infinite-dimensional manifolds, In: Proceedings of the International Conference on Geometric Topology, Warsaw, 1978, PWN, Warsaw, 1980, 431–437

[17] Toruńczyk H., Characterizing Hilbert space topology, Fund. Math., 1981, 111(3), 247–262 | Zbl 0468.57015

[18] Toruńczyk H., A correction of two papers concerning Hilbert manifolds: “Concerning locally homotopy negligible sets and characterization of l 2-manifolds” [Fund. Math. 101 (1978), no. 2, 93–110; MR 80g:57019] and “Characterizing Hilbert space topology” [ibid. 111 (1981), no. 3, 247–262; MR 82i:57016], Fund. Math., 1985, 125, 89–93