The simple integral against a signed or complex measure

Let $s={\sum }_{j=1}^m{c}_j{\mathbf{1}}_{E_j}$ be the canonical disjoint representation of a complex simple function on $(X,\mathcal{A})$ using only its nonzero level sets, so every $c_j\mathrm{\ne }0$. Let $\nu$ be a signed measure or a complex measure on $(X,\mathcal{A})$.

Assume that $\mathrm{\mid }\nu \mathrm{\mid }(E_j)<+\mathrm{\infty }$ for every $j$.

Define the simple integral of $s$ against $\nu$ by $\int sd\nu :={\sum }_{j=1}^m{c}_j\nu (E_j)\mathrm{.}$ If $F\in \mathcal{A}$ is measurable and $\mathrm{\mid }\nu \mathrm{\mid }(E_j\cap F)<+\mathrm{\infty }$ for every $j$, define likewise ${\int }_{F}sd\nu :={\sum }_{j=1}^m{c}_j\nu (E_j\cap F)\mathrm{.}$

The finiteness hypotheses make every $\nu (E_j)$ and $\nu (E_j\cap F)$ a finite real or complex number: the one-piece partition of the relevant set contributes at least its single term to the defining supremum for total variation. Because the canonical representation is unique up to deleting empty level sets, the value above is well defined.