- On the McShane integrability
- Kim, Jin-Yee ;
- Communications of the Korean Mathematical Society, volume 11, issue 2, 1996, Pages 377~383
Abstract
For a given separable space X which contains no copy of $C_0$ and a weakly compact T, we show that a Dunford integrable function $f : [a,b] \to X$ is intrinsically-separable valued if and only if f is McShane integrable. Also, for a given separable space X which contains no copy of $C_0$, a weakly compact T and a Dunford integrable function f we show that if there exists a sequence $(f_n)$ of McShane integrable functions from [a,b] to X such that for each $x^* \in X^*, x^*f_n \to x^*f$ a.e., then f is McShane integrable. Finally, let X contain no copy of $C_0$. If $f : [a,b] \to X$ is McShane integrable, then F is a countably additive on $\sum$.