Banach-valued simple function and integral

Let $(\mathrm{\Omega },\mathcal{A},\mu )$ be a measure space and let $X$ be a real or complex Banach space. An $X$-valued measurable simple function is a function $s:\mathrm{\Omega }\to X$ having a representation

$s={\sum }_{j=1}^m{x}_j{\mathbf{1}}_{A_j},$

where $m\ge 0$, the sets $A_j\in \mathcal{A}$ are pairwise disjoint, and the vectors $x_j\in X$ are distinct and nonzero. The value of $s$ off ${\bigcup }_j{A}_j$ is $0$. The empty representation is therefore the zero function.

The simple function is integrable when $\mu (A_j)<\mathrm{\infty }$ for every nonzero coefficient $x_j$. Its integral over $E\in \mathcal{A}$ is

${\int }_{E}sd\mu :={\sum }_{j=1}^m\mu (E\cap A_j)x_j\mathrm{.}$

In particular, $\int sd\mu$ means ${\int }_{\mathrm{\Omega }}sd\mu$. Requiring finite measure only for the nonzero level sets avoids the undefined product $0\cdot \mathrm{\infty }$. The next lemma proves that the displayed value does not depend on the chosen disjoint representation.

• The canonical representation uses the nonzero fibres $A_j=s^{-1}(\{x_j\})$. Thus repetitions may always be merged and an explicit zero fibre may be discarded.
• When $m=0$, every displayed sum is empty and the integral is $0_X$.