A representation characterization theorem
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- by William D. L. Appling PDF
- Proc. Amer. Math. Soc. 50 (1975), 317-321 Request permission
Abstract:
Given a field ${\mathbf {F}}$ of subsets of a set $U$ and a real-valued function $T$ defined on a set $S$ of functions from ${\mathbf {F}}$ into $\exp ({\mathbf {R}})$ with bounded range unions, a necessary and sufficient condition is given in order that there be a bounded finitely additive function $\theta$ from ${\mathbf {F}}$ into ${\mathbf {R}}$ such that if $\alpha$ is in $S$, then the integral $\int _U {\alpha (I)\theta (I)}$, as a variational integral, i.e., a refinement-wise limit of appropriate sums over (finite) subdivisions, exists and is $T(\alpha )$.References
- William D. L. Appling, Interval functions and Hellinger integral, Duke Math. J. 29 (1962), 515–520. MR 140659
- William D. L. Appling, Interval functions and continuity, Rend. Circ. Mat. Palermo (2) 11 (1962), 285–290. MR 166323, DOI 10.1007/BF02843876
- William D. L. Appling, Summability of real-valued set functions, Riv. Mat. Univ. Parma (2) 8 (1967), 77–100. MR 251187
- William D. L. Appling, Set functions, finite additivity and distribution functions, Ann. Mat. Pura Appl. (4) 96 (1972), 265–287. MR 330390, DOI 10.1007/BF02414845
- A. Kolmogoroff, Untersuchungen über denIntegralbegriff, Math. Ann. 103 (1930), no. 1, 654–696 (German). MR 1512641, DOI 10.1007/BF01455714
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 50 (1975), 317-321
- MSC: Primary 28A10
- DOI: https://doi.org/10.1090/S0002-9939-1975-0369643-3
- MathSciNet review: 0369643