# A generalization of the Banach-Zarecki theorem

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

#### Mathematical reviews subject classification: 26A24, 26A39.

## Abstract

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.