The wide Denjoy integral as the limit of a sequence of stepfunctions in a suitable convergence. (English) Zbl 0943.26020

The author shows that if \(\mathcal A[a,b]\) is an arbitrary dense linear subspace of \(L_1[a,b]\) and \(f:[a,b]\to \overline {\mathbb{R}},\) then \(f\) is \(\mathcal D\)-integrable (\(\mathcal D^*\)-integrable) on \([a,b]\) if and only if there exists a sequence \(\{f_n\}\subset \mathcal A[a,b]\) which is \(\mathcal D\)-controlled convergent (\(\mathcal D^*\)-controlled convergent) to \(f\) on \([a,b]\) and that in such a case it is \[ \lim_{n\to \infty} (\mathcal L)\int_a^b f_n(t)\operatorname{d}t = (\mathcal D)\int_a^b f(t)\operatorname{d}t \quad \Big (\text{respectively } =(\mathcal D^*) \int_a^b f(t)\operatorname{d}t\Big). \] Furthermore, he shows that also Ridder’s \(\alpha\)- and \(\beta\)- integrals can be defined as limits of some controlled convergent sequences of stepfunctions.
Reviewer: M.TvrdĂ˝ (Praha)


26A39 Denjoy and Perron integrals, other special integrals
26A46 Absolutely continuous real functions in one variable