Equi-integrability and controlled convergence for the Henstock integral (Q1322633)

The main result of this note is a relationship between the controlled convergence and the equi-integrability of a sequence of Kurzweil-Henstock integrable functions. A typical result (the proof is based on the Henstock's lemma) is the following: ``Let \(f_ n\), \(n=1,2,\dots\), be Henstock integrable on \([a,b]\) with primitive \(F_ n\), and \(f_ n(x)\to f(x)\) everywhere in \([a,b]\). If \(\{F_ n\}\) is \(\text{UACG}^*\) on \([a,b]\), then there is a subsequence of \(\{f_ n\}\) which is equi-integrable on \([a,b]\)'' (Theorem 3). It is shown that, in the presence of the so-called uniformly strong Lusin condition, ``everywhere'' can be weakened to ``almost everywhere'' in Theorem 3 and the other theorems.