Definition 3

Let $\mu$ be a positive measure on $X$; for every numerical function $f \geq 0$ (finite or not) defined on $X$, the positive number

(*finite or equal to $+\infty$) is called the upper integral of $f$ (with respect to $\mu$).

When $f \in \mathscr{I}_+$, the number $\mu^*(f)$ thus defined is equal to the upper integral defined in Def. 1, since $\mu^*$ is increasing in $\mathscr{I}_+$.

In place of the notation $\mu^*(f)$, we shall also employ the notations $\int^* f d\mu$, $\int^* f(x) d\mu(x)$, $\int^* f \mu$ and $\int^* f(x) \mu(x)$.