Definition 12

Let $f$ be a function defined on $T$, with values in a Banach space or in $\overline{\mathbf{R}}$. One says that $f$ is moderated for the measure $\mu$, or $\mu$-moderated, if $f$ is zero on the complement of a countable union of integrable open sets. A subset $A$ of $T$ is said to be moderated if the function $\varphi_A$ is moderated. The measure $\mu$ is said to be moderated if the function $1$ is $\mu$-moderated.

For example, since the encumbrance $\mu^\bullet$ is locally bounded, every compact subset $K$ of $T$ is contained in an open set $V$ such that $\mu^\bullet(V) < +\infty$; a function that is zero outside a compact set is therefore moderated. A negligible function is moderated. The remarks following Def. 2 of Ch. V, §1, No. 2 can immediately be extended to the present context. In particular, the sum of a sequence of moderated positive functions is moderated.