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, Fn:[a,b] → R, n = 1,2,... be Lebesgue measurable functions such that {Fn}n converges pointwise to F on [a,b]. If
- each Fn is approximately derivable a.e. on [a,b],
- {Fn}n is uniformly absolutely continuous on a set P ⊂ [a,b], and
- {(Fn)'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 ).
