Example

— Let L be a compact subset of T, $\lambda$ a measure on L, and $\mu$ the measure on T defined by $\lambda$ (No. 3, Example 2). The formula (3) implies at once that a function $f \in \mathscr{F}_+(T)$ is locally $\mu$-negligible if and only if $f_L$ is $\lambda$-negligible.

It follows immediately from formula (1) that a function $f \in \mathscr{F}_+(T)$ is locally $\mu$-negligible if and only if $f_K$ is $\mu_K$-negligible for every compact subset K of T. Thus the properties of locally negligible sets reduce at once to those of negligible sets in compact spaces, treated in Ch. IV. Here are some results that will be used henceforth without further reference.

— For a function $f \geqslant 0$ to be locally negligible, it is necessary and sufficient that $f(t) = 0$ locally almost everywhere (Ch. IV, §2, No. 3, Th. 1). If $f$ is a function with values in a Banach space, it is therefore equivalent to say that $f = 0$ locally almost everywhere or that $\mu^*(|f|) = 0$; in this case we shall again say that $f$ is locally negligible.

— The sum and upper envelope of a sequence of locally negligible functions $\geqslant 0$ are locally negligible (loc. cit., No. 1, Prop. 2).

— If $f$ and $g$ are two functions $\geqslant 0$ that are equal locally almost everywhere, then $\mu^\bullet(f) = \mu^\bullet(g)$ (loc. cit., No. 3, Prop. 6).