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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Integral representation without additivity


Author: David Schmeidler
Journal: Proc. Amer. Math. Soc. 97 (1986), 255-261
MSC: Primary 28C05; Secondary 47H07, 60A10, 62C10
MathSciNet review: 835875
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Abstract: Let $ I$ be a norm-continuous functional on the space $ B$ of bounded $ \Sigma $-measurable real valued functions on a set $ S$, where $ \Sigma $ is an algebra of subsets of $ S$. Define a set function $ v $ on $ \Sigma $ by: $ v (E)$ equals the value of $ I$ at the indicator function of $ E$. For each $ a$ in $ B$ let

$\displaystyle J(a) = \int_{ - \infty }^0 {(v (a \geq \alpha ) - v (S))d\alpha + \int_0^\infty {v (a \geq \alpha )d\alpha .} } $

Then $ I = J$ on $ B$ if and only if $ I(b + c) = I(b) + I(c)$ whenever $ (b(s) - b(t))(c(s) - c(t)) \geqslant 0$ for all $ s$ and $ t$ in $ S$.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0835875-8
PII: S 0002-9939(1986)0835875-8
Article copyright: © Copyright 1986 American Mathematical Society