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Article 30

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 ).