Convergence theorems for the Henstock integral involving small Riemann sums. (English) Zbl 1071.26006

A measurable function \(f\) has functionally small Riemann sums or the FSRS property on an interval \(E\) in \(\mathbb{R}^n\) if for every \(\varepsilon> 0\), there exist a nonnegative Lebesgue integrable function \(\delta\) and a positive function \(\delta\) on \(E\) such that for any \(\delta\)-fine partition \(D= \{(I, x)\}\) of \(E\), we have \(|(D)\sum_{|f(x)|> g()}|I|\,|<\varepsilon\), where the sum is taken over \(D\) for which \(|f(x)|> g(x)\).
The following theorem is proved in this note.
If a sequence \(\{f_k\}\) of Henstock integrable functions on an interval \(E\) has the uniform FSRS property and \(f_k\to f\) a.e. in \(E\), then \(\{f_k\}\) is equi-Henstock integrable on \(E\). Conversely, if \(\{f_k\}\) is equi-Henstock integrable on \(E\), \(f_k\to f\) a.e. in \(E\) and the primitive \(\{F_k\}\) of \(\{f_k\}\) satisfies the strong Lusin condition, then \(\{f_k\}\) has the uniform FSRS property.


26A39 Denjoy and Perron integrals, other special integrals
Full Text: DOI