Example

Let $A$ be a subset of $T$; for $\varphi_A$ to be locally $\mu$-integrable, it is necessary and sufficient that $A$ be $\mu$-measurable. Assuming the condition to be fulfilled, set $\nu = \varphi_A \cdot \mu$; for every numerical function $f \geq 0$ defined on $T$, we then have

a quantity that is also denoted by $\int_A^\bullet f \, d\mu$ (cf. Ch. IV, §5, No. 6). For a mapping $g$ of $T$ into a topological space $G$ to be $\nu$-measurable, it is necessary and sufficient that the restriction of $g$ to $A$ be $\mu$-measurable. For a mapping $f$ of $T$ into a Banach space $F$ or into $\overline{\mathbf{R}}$ to be essentially $\nu$-integrable, it is necessary and sufficient that $f \varphi_A$ be essentially $\mu$-integrable, in which case

an expression that is also denoted $\int_A f \, d\mu$. Note that if two mappings of $T$ into $G$ (resp. $F, \overline{\mathbf{R}}$) coincide on $A$, then for one of them to be $\nu$-measurable (resp. essentially $\nu$-integrable), it is necessary and sufficient that the other be so. If now $g$ is a mapping into $G$ of an arbitrary subset $B \supset A$ of $T$, one says that $g$ is $\mu$-measurable on $A$ if an arbitrary extension to $T$ of the restriction of $g$ to $A$ is $\nu$-measurable, which amounts to saying that the restriction of $g$ to $A$ is $\mu$-measurable. A mapping $f$ of $B$ into a Banach space $F$, or into $\overline{\mathbf{R}}$, is said to be essentially $\mu$-integrable on $A$ if some extension $\overline{f}$ to $T$ of the restriction of $f$ to $A$ is essentially $\nu$-integrable; one then sets

and one says that $\int_A f \, d\mu$ is the integral of $f$ on $A$ (or extended to $A$). If $f$ is a numerical function $\geqslant 0$ defined on $B \supset A$, one defines similarly $\int_A^* f \, d\mu$ and $\int_A^\bullet f \, d\mu$. Finally, a numerical function $g$ defined on $B \supset A$ is said to be locally $\mu$-integrable on $A$ if some extension $\overline{g}$ to $T$ of the restriction of $g$ to $A$ is locally $\nu$-integrable: this is equivalent to saying that, for every compact subset $K$ of $T$, $\overline{g} \varphi_{K \cap A}$ is $\mu$-integrable.