The AP-Denjoy and AP-Henstock integrals revisited

On Descriptive Characterizations of an Integral Recovering a Function from Its $$L^r$$-Derivative

“…As we have already mentioned the constructed example gives another proof of the following result obtained in [16,Theorem 14…”

“…The main aim of the present paper is to prove that any ACG rfunction is L r -differentiable almost everywhere and thereby obtain a full descriptive characterization of the HK r -integral (see Theorem 13 in Section 3). In Section 4, we construct an example of a function from a Lusin-type class, [ACG], which, even under an additional assumption of continuity, fails to have the property of being L r -differentiable almost everywhere for any r. Thereby we show that the class [ACG] is not contained in the class ACG r for any r. As a byproduct of this example we get a new proof, based on the L r -derivative, of a result, previously obtained in [16], stating that the HK r -integral does not cover the wide Denjoy integral.…”

“…So for any finite subfamily of {(I i , z i )} i the inequality (8) should hold. At the same time by (16) and by the choice of n we have…”

“…Hence µ(P ) ≤ j h k j ≤ µ(G) < 2µ(P ). We can apply (16) to each of these intervals and to the corresponding set…”

On approximately continuous integrals (a survey)

“…In [61], we used the following result due to Ene ([13], Theorem 3): A measurable F : [a, b] → R is VBG if and only if it is so on each nullset. The measurability assumption in this theorem is essential (we showed in[53] that there is a function F : [0, 1] → [0, 1] which is not VBG, but it is so on each null subset of [0, 1]). To obtain the required measurability in this context we proved in[53] that if V ap F is σ -finite on each nullset then F : R → R is measurable.…”