On the summability of certain \(\mu\)-integrable functions
In Arch. Math. 49, 245-256 (1987; Zbl 0612.28010) we introduced an analogue to Daniell's extension process which works without any continuity conditions on the non-negative functional I. Here we show that integration with respect to finitely additive set function \(\mu\) is mod null-functions a special case: If \(\Omega\) is a ring of sets from X and \(\mu:\quad \Omega \to [0,\infty [\) is additive on \(\Omega\), by \textit{N. Dunford} and \textit{J. T. Schwartz} [Linear operators. I: General theory (1958; Zbl 0084.104)], extended in \textit{H. Günzler} [Integration (1985; Zbl 0568.28009)], one can define the set \(R_ 1(\mu,I)\) of ``Riemann- \(\mu\)-integrable functions'' f: \(X\to {\mathbb{R}}\) and \(\int f d\mu \); if B \(=\) function vector lattice, \(B_{\Omega}=real\)-valued step function over \(\Omega\) and \(I=I_{\mu}=\int \cdot d\mu,\) then any \(f\in R_ 1(\mu,{\mathbb{R}})\) can be written \(f=g+h\) with \(g\in (B_{\Omega})_ 0\cap R_ 1(\mu,{\mathbb{R}}),\) \(h\in R_ 1(\mu,{\mathbb{R}})\) with \(\int | h| d\mu =0,\int f d\mu =I_{\mu}(g),\) where \(B_ 0=I\)-summable functions of the first reference quoted above. The results hold also for \(R_ 1(\mu,\overline{{\mathbb{R}}})\).
scientific article; zbMATH DE number 4103394