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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A finitely additive generalization of the Fichtenholz-Lichtenstein theorem


Author: George Edward Sinclair
Journal: Trans. Amer. Math. Soc. 193 (1974), 359-374
MSC: Primary 28A35
DOI: https://doi.org/10.1090/S0002-9947-1974-0417371-1
MathSciNet review: 0417371
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Abstract: Let $\mu$ and $\nu$ be bounded, finitely additive measures on algebras over sets X and Y, respectively. Conditions are determined for a bounded function $f:X \times Y \to {\mathbf {R}}$, without assuming bimeasurability, so that the iterated integrals $\smallint _X {\smallint _Y {fd\mu d\mu } }$ and $\smallint _Y {\smallint _X {fd\mu d\nu } }$ exist and are equal. This result is then used to construct a product algebra and finitely additive product measure for $\mu$ and $\nu$. Finally, a simple Fubini theorem with respect to this product algebra and product measure is established.


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Keywords: A-continuous, Dunford-Schwartz integral, finitely additive product measure, Fubini theorem, Pták lemma, Pták theorem, separately continuous, Stone-Čech compactification, Stone space
Article copyright: © Copyright 1974 American Mathematical Society