Definition 1

For every function $f \in \mathscr{F}_+(T)$ one calls essential upper integral of $f$ with respect to $\mu$, and denotes by $\mu^\bullet(f)$, the supremum, finite or not, of the set of numbers $\mu^*(f \varphi_K)$, where $K$ runs over the set of compact subsets of $T$. For every subset $A$ of $T$, one sets $\mu^\bullet(A) = \mu^\bullet(\varphi_A)$.

The notations $\int^\bullet f d\mu$, $\int^\bullet f(t) d\mu(t)$, $\int^\bullet f \mu$ are also used. Since $f \varphi_K \leq f$ for every compact subset $K$ of $T$, one has

It can happen that $\mu^\bullet(f) \neq \mu^*(f)$; for, the condition $\mu^*(f) = 0$ means that $f$ is negligible, whereas the condition $\mu^\bullet(f) = 0$ means that $f$ is locally negligible (Ch. IV, §5, No. 2, Prop. 5), and there may exist locally negligible sets that are not negligible (Ch. IV, §1, Exer. 5).

The mapping $\mu^\bullet$ of $\mathscr{F}_+(T)$ into $\overline{\mathbf{R}}$ coincides with $\mu$ on $\mathscr{K}_+(T)$. It follows that two measures $\mu_1$ and $\mu_2$ such that $\mu_1^\bullet = \mu_2^\bullet$ are equal.