Absolute continuity characterization sets

Author:
William D. L. Appling

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

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Abstract: Suppose is a set, is a field of subsets of is the set of all real-valued, bounded finitely additive functions on , and for each in is the set of all elements of absolutely continuous with respect to is the set of all nonnegative-valued elements of , and is the set of all functions from into 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 of having the property that if each of and is in , then is in iff it is true that if is in and exist and the function is in , then is in .

**[1]**W. D. L. Appling,*Interval functions and the Hellinger integral*, Duke Math. J.**29**(1962), 515-520. MR**25**#4075. MR**0140659 (25:4075)****[2]**-,*Some integral characterizations of absolute continuity*, Proc. Amer. Math. Soc.**18**(1967), 94-99; Addendum, ibid.**24**(1970), 788-793. MR**34**#4446;**41**#406. MR**0204607 (34:4446)****[3]**-,*Two inclusion theorems for real-valued summable set functions*, Rend. Circ. Mat. Palermo (2)**18**(1969), 293-306. MR**45**#7007. MR**0297955 (45:7007)****[4]**-,*Set functions, finite additivity and distribution functions*, Ann. Mat. Pura Appl.**96**(1973), 265-287. MR**0330390 (48:8727)****[5]**A. Kolmogoroff,*Untersuchen über den Integralbegriff*, Math. Ann.**103**(1930), 654-696.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0404566-3

Keywords:
Set function with bounded range union,
bounded finitely additive set function,
set function integral (variational integral),
absolute continuity characterization set

Article copyright:
© Copyright 1976
American Mathematical Society