A Riesz-type definition of the Henstock integral in Euclidean space (Q2778266)

If \(E\) is a compact interval in \({\mathbb R}^m\) then a function \(f : E\to\mathbb R\) is Henstock integrable over \(E\), \(\int_Ef=A\in\mathbb R\), if for all \(\varepsilon>0\) there is a gauge function \(\delta : E\to(0,1)\) such that if \(D=\{(x_n,I_n)\}_{n=1}^N\) is a \(\delta\)-fine tagged division of \(E\) then \(\bigl|\sum_{n=1}^Nf(x_n)|I_n|-A\bigr|<\varepsilon\). Here the \(I_n\) are nonoverlapping intervals with union \(E\), \(x_n\in E\) and \(\delta\)-fine means \(I_n\subset B(x_n,\delta(x_n))\) for all \(n\). NEWLINENEWLINENEWLINEThe authors introduce the following notion of a weakly control-convergent sequence of functions. For a sequence of Henstock integrable functions \(\{f_n\}\), let the primitives be \(F_n=\int_Ef_n\) and suppose \(f_n\to f\) almost everywhere on \(E\). There is a sequence of closed sets \(X_i\nearrow E\) such that \(\{F_n\}\) is weakly \(\text{UAC}^{**}_\delta(X_i)\) for each \(i\), each function \(f_{n,i}:=f_n\) on \(X_i\) and \(0\) elsewhere is Henstock integrable with primitive \(F_{n,i}\) and \(\{F_{n,i}\}\) is \(\text{UAC}^{*}_\delta(X_i)\) for each \(i\). The generalised types of uniform absolute continuity, \(\text{UAC}^{**}_\delta\) and \(\text{UAC}^{*}_\delta\), are defined in the paper. Under these hypotheses, \(f\) is Henstock integrable and \(\int_Ef_n\to\int_Ef\). NEWLINENEWLINENEWLINEThis convergence theorem is compared with the usual controlled convergence theorem and used to prove the equivalence of the Henstock and Riesz-type definitions of the integral. The Riesz definition is: there is a weakly control-convergent sequence of step functions \(\{\psi_n\}\) with \(\psi_n\to f\) almost everywhere and \(\int_Ef:=\lim\int_E\psi_n\).