Definition 2

— A subset of $T$ is said to be $\mu$*-moderated if it satisfies the equivalent conditions of Proposition 5*. A function defined on $T$, with values in a vector space or in $\overline{\mathbf{R}}$, is said to be $\mu$*-moderated if it is zero on the complement of a* $\mu$*-moderated subset of* $T$. The measure $\mu$ is said to be moderated if $T$ is a $\mu$*-moderated set*.

If $\mu$ is a moderated measure, then every function on $T$ is $\mu$-moderated and every subset of $T$ is $\mu$-moderated.