# Convergence and approximate differentiation

### Real Analysis Exchange 23 (1997/8), no. 1, 141-160.

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

## Abstract

The main result of this paper is Theorem 1, which states the following:

Let F, F_{n}:[a,b] → R, n = 1,2,... be Lebesgue measurable functions such that {F_{n}}_{n} converges pointwise to F on [a,b]. If

- each F
_{n} is approximately derivable a.e. on [a,b],
- {F
_{n}}_{n} is uniformly absolutely continuous on a set P ⊂ [a,b], and
- {(F
_{n})'_{ap}}_{n} converges in measure to a measurable function g, finite a.e. on [a,b],

then F is approximately derivable a.e. on P and F'_{ap}(x) = g(x) a.e. on P.

An immediate consequence of this result is the famous theorem of Dzvarseisvili on the passage to the limit for the Denjoy and Denjoy* integrals (see V.G. Celidze and A.G. Dzvarseisvili , *The theory of the Denjoy integral and some applications*, World Scientific, 1978, Theorem 47, p.40).

As it was pointed out by Bullen(see the same book p. 309), "the Denjoy* integral case of Theorem 47 was rediscovered by Lee P. Y.'' (see also P. Y. Lee, *Lanzhou lectures on Henstock integration*, World Scientific, Singapore, 1989, Theorem7.6 ).