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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Some open mapping theorems for marginals


Author: Larry Q. Eifler
Journal: Trans. Amer. Math. Soc. 211 (1975), 311-319
MSC: Primary 28A35; Secondary 60B05
MathSciNet review: 0387533
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Abstract: Let S and T be compact Hausdorff spaces and let $ P(S),P(T)$ and $ P(S \times T)$ denote the collection of probability measures on S, T and $ S \times T$, respectively. Given a probability measure $ \mu $ on $ S \times T$, set $ \pi \mu = (\alpha ,\beta )$ where $ \alpha $ and $ \beta $ are the marginals of $ \mu $ on S and T. We prove that the mapping $ \pi :P(S \times T) \to P(S) \times P(T)$ is norm open and $ {\text{weak}^\ast}$ open. An analogous result for $ {L_1}(X \times Y,\mu \times \nu )$ where $ (X,\mu )$ and $ (Y,\nu )$ are probability spaces is established.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0387533-1
PII: S 0002-9947(1975)0387533-1
Keywords: Marginals, open mapping theorems, probability measures
Article copyright: © Copyright 1975 American Mathematical Society