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*, is the set of all bounded real-valued finitely additive functions defined on **F**, and *W* is a collection of functions from **F** into , closed under multiplication, each element of which has range union bounded and bounded away from 0. Let denote the set to which *T* belongs iff *T* is a function from *W* into such that if each of and is in *W* and *V* is in **F**, then the following integrals exist and the following ``integration-by-parts'' equation holds:

*S*belongs iff

*S*is a function from

*W*into such that if each of and is in

*W*, then the integral exists and the following ``logarithmic'' equation holds: . It is shown that is a one-one mapping from onto .

**[1]**William D. L. Appling,*Interval functions and Hellinger integral*, Duke Math. J.**29**(1962), 515–520. MR**0140659****[2]**William D. L. Appling,*Set functions, finite additivity and distribution functions*, Ann. Mat. Pura Appl. (4)**96**(1972), 265–287. MR**0330390****[3]**A. Kolmogoroff,*Untersuchungen über denIntegralbegriff*, Math. Ann.**103**(1930), no. 1, 654–696 (German). MR**1512641**, 10.1007/BF01455714

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DOI:
https://doi.org/10.1090/S0002-9939-1977-0447502-7

Keywords:
Set function integral,
integration-by-parts operator,
logarithmic operator,
one-one mapping

Article copyright:
© Copyright 1977
American Mathematical Society