Absolute continuity characterization sets
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- by William D. L. Appling PDF
- Proc. Amer. Math. Soc. 55 (1976), 52-56 Request permission
Abstract:
Suppose $U$ is a set, ${\mathbf {F}}$ is a field of subsets of $U,{\mathfrak {p}_{AB}}$ is the set of all real-valued, bounded finitely additive functions on ${\mathbf {F}}$, and for each $\rho$ in ${\mathfrak {p}_{AB}},{\mathcal {A}_\rho }$ is the set of all elements of ${\mathfrak {p}_{AB}}$ absolutely continuous with respect to $\rho ,\mathfrak {p}_A^ +$ is the set of all nonnegative-valued elements of ${\mathfrak {p}_{AB}}$, and ${\mathfrak {p}_B}$ is the set of all functions from ${\mathbf {F}}$ into $\exp ({\mathbf {R}})$ with bounded range union. An extension of a previous absolute continuity characterization theorem of the author (Proc. Amer. Math. Soc. 18 (1967), 94-99) is given in the form of a characterization of those subsets $S$ of ${\mathfrak {p}_{AB}}$ having the property that if each of $\xi$ and $\mu$ is in $\mathfrak {p}_A^ +$, then $\xi$ is in ${\mathcal {A}_\mu }$ iff it is true that if $\alpha$ is in ${\mathfrak {p}_B},{\smallint _U}\alpha (I)\mu (I)$ and ${\smallint _U}\alpha (I)\xi (I)$ exist and the function $\smallint \alpha \mu$ is in $S$, then $\smallint \alpha \xi$ is in $S$.References
- William D. L. Appling, Interval functions and Hellinger integral, Duke Math. J. 29 (1962), 515–520. MR 140659
- William D. L. Appling, Some integral characterizations of absolute continuity, Proc. Amer. Math. Soc. 18 (1967), 94–99. MR 204607, DOI 10.1090/S0002-9939-1967-0204607-7
- William D. L. Appling, Two inclusion theorems for real-valued summable set functions, Rend. Circ. Mat. Palermo (2) 18 (1969), 293–306. MR 297955, DOI 10.1007/BF02849492
- 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, Untersuchen über den Integralbegriff, Math. Ann. 103 (1930), 654-696.
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 55 (1976), 52-56
- MSC: Primary 28A25
- DOI: https://doi.org/10.1090/S0002-9939-1976-0404566-3
- MathSciNet review: 0404566