Corollary 2

— Let $f$ be a function $\geq 0$ defined on $X$, $\nu$-measurable and $\nu$-moderated. The set of $t \in T$ such that $f$ is not $\lambda_t$-moderated is then locally $\mu$-negligible (and even $\mu$-negligible if $\Lambda$ is vaguely continuous).

For, $f$ is the sum of a sequence of functions $f_n \geq 0$ such that $f_n$ is zero outside a compact set $K_n$ for $n \geq 1$, and $f_0$ is $\nu$-negligible (§1, No. 2, Prop. 6); $f_0$ is then $\lambda_t$-negligible except for $t$ forming a set that is locally $\mu$-negligible (and even $\mu$-negligible, if $\Lambda$ is vaguely continuous) by Cor. 1, and the statement then follows at once.