On \(H_1\)-integrable functions
Integrals defined using Moore-Smith limits are discussed in this interesting paper. A real-valued function defined on \([a,b]\) is said to be \(H_0\)-integrable to \(A\) if there exists a gauge \(\delta\) such that for every \(\varepsilon> 0\), one can find a (McShane) tagged partition \(\pi_0\) of \([a,b]\) such that for every \(\delta\)-fine \(\pi\supseteq \pi_0\), \(|\sum_{(I, x)\in \pi} f(x)|I|- A|< \varepsilon\). \(\pi\supseteq \pi_0\) if and only if for every \((I,x)\in \pi_0\), \(I\) is the union of some intervals from \(\pi\). This definition is a little bit different from, but equivalent to, the definition of the \(H_1\)-integrals, which is given by Garces, Lee and Zhao. The necessary and sufficient conditions for the characteristic function of a set of measure zero to be \(H_0\)-integrable are given in this paper. The \(H_0\)-integral is not absolute. Some interesting problems have also been raised.
scientific article; zbMATH DE number 1848918