On images of Borel measures under Borel mappings |
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Authors: | Dimitris Gatzouras |
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Affiliation: | Department of Mathematics, University of Crete, Leoforos Knossou, 714 09 Iraklion, Crete, Greece |
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Abstract: | Let and be metric spaces. We show that the tight images of a (fixed) tight Borel probability measure on , under all Borel mappings , form a closed set in the space of tight Borel probability measures on with the weak-topology. In contrast, the set of images of under all continuous mappings from to may not be closed. We also characterize completely the set of tight images of under Borel mappings. For example, if is non-atomic, then all tight Borel probability measures on can be obtained as images of , and as a matter of fact, one can always choose the corresponding Borel mapping to be of Baire class 2. |
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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 |
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