Integration in vector spaces

Abstract

We define an integral of a vector-valued function $f:\mathrm{\Omega }⟶X$ with respect to a bounded countably additive vector-valued measure $\nu :\mathrm{\Sigma }⟶Y$ 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 $\Vert f\Vert \in L_1(\nu )$. In this case, the indefinite integral of $f$ is of bounded variation if and only if $\Vert f\Vert \in L_1(|\nu |)$. We also define the integral of a weakly $\nu$-measurable function and show that such a function $f$ satisfies $x^{\ast }f\in L_1(\nu )$ for all $x^{\ast }\in X^{\ast }$ and is $|y^{\ast }\nu |$-Pettis integrable for all $y^{\ast }\in Y^{\ast }$.