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Article 21

A generalization of the Banach-Zarecki theorem

Real Analysis Exchange 20 (1995), no. 2, 639-646.

Mathematical reviews subject classification: 26A24, 26A39.



Starting from Foran's condition A(N) and B(N) (see An extension of the Denjoy integral, Proc. Amer. Math. Soc., 49 (1975), 359-365), we introduce the more general conditions AC and VB.

Note that A(1) is identical with AC (absolute continuity) and B(1) is identical with VB (bounded variation).

We show the following result:

A bounded measurable function F:P → R is AC∞  on the measurable set P if and only if F is VB and satisfy Lusin's condition (N) on P.



This is of course generalization of the classical Banach-Zarecki Theorem (see S. Saks, Theory of the integral, 2nd. rev. ed., vol. PWN, Monografie  Matematyczne, Warsaw, 1937):

A continuous function F:[a,b] → R is AC on subset P of [a,b] if and only if it is VB and satisfy Lusin's condition (N) on P.