A mapping theorem for logarithmic and integrationbyparts operators
Author:
William D. L. Appling
Journal:
Proc. Amer. Math. Soc. 65 (1977), 8588
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 realvalued 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 ``integrationbyparts'' equation holds: Let denote the set to which 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 oneone mapping from onto .
 [1]
William
D. L. Appling, Interval functions and Hellinger integral, Duke
Math. J. 29 (1962), 515–520. MR 0140659
(25 #4075)
 [2]
William
D. L. Appling, Set functions, finite additivity and distribution
functions, Ann. Mat. Pura Appl. (4) 96 (1972),
265–287. MR 0330390
(48 #8727)
 [3]
A.
Kolmogoroff, Untersuchungen über denIntegralbegriff,
Math. Ann. 103 (1930), no. 1, 654–696 (German).
MR
1512641, http://dx.doi.org/10.1007/BF01455714
 [1]
 W. D. L. Appling, Interval functions and the Hellinger integral, Duke Math. J. 29 (1962), 515520. MR 25 #4075. MR 0140659 (25:4075)
 [2]
 , Set functions, finite additivity and distribution functions, Ann. Mat. Pura. Appl. (4) 96 (1973), 265287. MR 48 #8727. MR 0330390 (48:8727)
 [3]
 A. Kolmogoroff, Untersuchungen über den Integralbegriff, Math. Ann. 103 (1930), 654696. MR 1512641
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DOI:
http://dx.doi.org/10.1090/S00029939197704475027
PII:
S 00029939(1977)04475027
Keywords:
Set function integral,
integrationbyparts operator,
logarithmic operator,
oneone mapping
Article copyright:
© Copyright 1977
American Mathematical Society
