Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On images of Borel measures under Borel mappings

Author: Dimitris Gatzouras
Journal: Proc. Amer. Math. Soc. 130 (2002), 2687-2699
MSC (2000): Primary 28A33, 46E27, 60B05, 60B10; Secondary 26A21, 28C15, 54E70, 54H05
Published electronically: March 29, 2002
MathSciNet review: 1900877
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Abstract: Let $X$ and $Y$ be metric spaces. We show that the tight images of a (fixed) tight Borel probability measure $\mu$ on $X$, under all Borel mappings $f\colon X\to Y$, form a closed set in the space of tight Borel probability measures on $Y$ with the weak$^*$-topology. In contrast, the set of images of $\mu$ under all continuous mappings from $X$ to $Y$ may not be closed. We also characterize completely the set of tight images of $\mu$ under Borel mappings. For example, if $\mu$ is non-atomic, then all tight Borel probability measures on $Y$ can be obtained as images of $\mu$, and as a matter of fact, one can always choose the corresponding Borel mapping to be of Baire class 2.

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Additional Information

Dimitris Gatzouras
Affiliation: Department of Mathematics, University of Crete, Leoforos Knossou, 714 09 Iraklion, Crete, Greece
Address at time of publication: Department of Mathematics, Agricultural University of Athens, Iera Odos 75, 118 55 Athens, Greece

Keywords: Convergence of a sequence of images of a measure, tight measure, Prohorov's theorem, characterization of images of a tight measure, Baire class 2 mapping
Received by editor(s): November 15, 1999
Received by editor(s) in revised form: April 19, 2001
Published electronically: March 29, 2002
Additional Notes: This research was supported by the European Commission as part of the programmes E$Π$ET and K$ΠΣ$
Communicated by: David Preiss
Article copyright: © Copyright 2002 American Mathematical Society