A chain rule for the approximate derivative and change of variables for the D-integral (Q799826)

The main result of the paper is the following theorem: if g is a finite valued function defined on [a,b] and if \(A=\{x:g'_{ap}(x)=0\}\) and \(B=\{x:g'_{ap}(x)\) does not exist\}, and F is defined on an interval containing the range of g and satisfies a Lusin condition (N) on \(g(A\cup B\}\) and is approximately derivable at almost every point of g(B), then \((F\circ g)^*_{ap}(x)=(F^*_{ap}\circ g)(x)\cdot g^*_{ap}(x)\) holds at almost every point of [a,b]. Examples are given to show that none of the hypotheses in this theorem can be dropped. A symbol \(F^*_{ap}(x)\) means \(F'_{ap}(x)\) if it exists and is finite, otherwise it is 0. As a corollary a theorem on change of variables for Denjoy integral is proved.