Definition 4

Let w be a positive premeasure on T. We shall set, for every function $f \in \mathscr{F}_+(T)$,

where K runs over the set of compact subsets of T.

For each compact set K, let $p^K$ be the image encumbrance of the encumbrance $(w_K)^\bullet$ under the canonical injection of K into T; $w^\bullet$ is the upper envelope of the encumbrances $p^K$, hence is an encumbrance (No. 1, Prop. 1). One says that $w^\bullet$ is the essential upper integral associated with the positive premeasure w. One often writes $\int^\bullet f\, dw$ or $\int^\bullet f(t)\, dw(t)$ instead of $w^\bullet(f)$.

Remark 1). — If v and w are two positive premeasures, then $(v + w)^\bullet = v^\bullet + w^\bullet$ (Ch. V, §1, No. 1, Prop. 3). If v and w are two complex premeasures, then $|v + w|^\bullet \leq |v|^\bullet + |w|^\bullet$.