Remark

— 1) If $\theta$ is a complex measure on $T$, one says that a function $f$ is $\theta$-moderated (resp. that $\theta$ is moderated) if $f$ is $|\theta|$-moderated (resp. if $|\theta|$ is moderated). 2) Every bounded measure is moderated; if $T$ is a countable union of compact sets, then every measure on $T$ is moderated. 3) Let $(f_n)$ be a sequence of $\mu$-moderated functions with values in $\overline{\mathbf{R}}$. For each $n$, let $U_n$ be an open set that is a countable union of open sets of finite outer measure, such that $f_n$ is zero outside $U_n$. The function $s = \sum_{n \in \mathbf{N}} |f_n|$ is then zero outside $\bigcup_{n \in \mathbf{N}} U_n$; it is therefore $\mu$-moderated, and the same is true of all functions $f$ such that $|f| \leq s$. This applies in particular to the functions $\liminf_{n \to \infty} f_n$, $\limsup_{n \to \infty} f_n$ and $\sum_{n \in \mathbf{N}} f_n$ (if the sum is defined).

4) A function equal almost everywhere to a moderated function is moderated.