Definition 11

For every function $f \in \mathscr{F}_+(T)$, one defines the upper integral of $f$ (with respect to the measure $\mu$) to be the finite or infinite positive number

where $g$ runs over the set of lower semi-continuous functions that are $\geqslant f$.

The notations $\int^* f(t) d\mu(t)$ and $\int^* f d\mu$ are also used. When $T$ is locally compact, this definition coincides with the usual definition (Ch. V, §1, No. 1, Prop. 4). Clearly $\mu^\bullet(f) \leqslant \mu^*(f)$, with equality when $f$ is lower semi-continuous. If $A$ is a subset of $T$, one writes $\mu^*(A)$ instead of $\mu^*(\varphi_A)$, and this number is called the outer measure of $A$. The measurable sets with finite outer measure are called integrable sets, as in the case of locally compact spaces.

A function $f$ with values in a Banach space or in $\overline{\mathbf{R}}$ such that $\mu^*(|f|) = 0$ is said to be negligible; a set $A \subset T$ is said to be negligible if $\varphi_A$ is negligible, that is, if $\mu^*(A) = 0$. The expression almost everywhere is introduced as in Ch. IV, §2, No. 3.