Integration in vector spaces

Abstract

We define an integral of a vector-valued function $f:\Omega\longrightarrow X$ with respect to a bounded countably additive vector-valued measure $\nu:\Sigma\longrightarrow Y$ and investigate its properties. The integral is an element of $X\check{\otimes}Y$, and when $f$ is $\nu$-measurable we show that $f$ is integrable if and only if $\|f\|\in L_{1}(\nu)$. In this case, the indefinite integral of $f$ is of bounded variation if and only if $\|f\|\in L_{1}(|\nu|)$. We also define the integral of a weakly $\nu$-measurable function and show that such a function $f$ satisfies $x^{*}f\in L_{1}(\nu)$ for all $x^{*}\in X^{*}$ and is $|y^{*}\nu|$-Pettis integrable for all $y^{*}\in Y^{*}$.