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

 

A mapping theorem for logarithmic and integration-by-parts operators


Author: William D. L. Appling
Journal: Proc. Amer. Math. Soc. 65 (1977), 85-88
MSC: Primary 26A42
MathSciNet review: 0447502
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Abstract: Suppose U is a set, F is a field of subsets of U, $ {\mathfrak{p}_{AB}}$ is the set of all bounded real-valued finitely additive functions defined on F, and W is a collection of functions from F into $ \exp ({\mathbf{R}})$, closed under multiplication, each element of which has range union bounded and bounded away from 0. Let $ \mathcal{P}$ denote the set to which T belongs iff T is a function from W into $ {\mathfrak{p}_{AB}}$ such that if each of $ \alpha $ and $ \beta $ is in W and V is in F, then the following integrals exist and the following ``integration-by-parts'' equation holds:

$\displaystyle \int_V \alpha (I)T(\beta )(I) + \int_V {\beta (I)T(\alpha )(I) = T(\alpha \beta )(V).} $

Let $ \mathfrak{L}$ denote the set to which S belongs iff S is a function from W into $ {\mathfrak{p}_{AB}}$ such that if each of $ \alpha $ and $ \beta $ is in W, then the integral $ \smallint_U {\alpha (I)S(\beta )(I)} $ exists and the following ``logarithmic'' equation holds: $ S(\alpha \beta ) = S(\alpha ) + S(\beta )$. It is shown that $ \{ (T,S):T\;{\text{in}}\;\mathcal{P},\;S = \{ (\alpha ,\;\smallint {(1/\alpha )T(\alpha )):\;\alpha \;{\text{in}}\;W\} \} } $ is a one-one mapping from $ \mathcal{P}$ onto $ \mathfrak{L}$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0447502-7
PII: S 0002-9939(1977)0447502-7
Keywords: Set function integral, integration-by-parts operator, logarithmic operator, one-one mapping
Article copyright: © Copyright 1977 American Mathematical Society