Proposition 7

— Let $f$ be an element of $\mathscr{F}_+(T)$. 1) If the function $f$ is not $\mu$-moderated, then $\mu^*(f) = +\infty$. 2) If the function $f$ is $\mu$-moderated, then $\mu^*(f) = \mu^*(f)$. 3) If $\mu^*(f) < +\infty$ then there exists a $\mu$-moderated subset $A$, the union of a sequence of compact subsets of $T$, such that $f = f \varphi_A$ locally almost everywhere.

The first assertion follows immediately from Lemma 1 of Ch. IV, §5, No. 6. To establish the second, denote by $A$ a moderated subset such that f is zero outside A; A is the union of a negligible set $A_0$ and a sequence $(A_n)_{n \geq 1}$ of compact sets, which we may suppose to be increasing. The function $f$ is then almost everywhere equal to the upper envelope of the functions $f \varphi_{A_n}$ ($n \geq 1$), therefore (Ch. IV, §1, No. 3, Th. 3 and §2, No. 3, Prop. 6)

whence the equality $\mu^*(f) = \mu^\bullet(f)$ by virtue of the formula (1). Finally, suppose that $\mu^\bullet(f) < +\infty$; there exists an increasing sequence $(A_n)$ of compact sets such that

Set $A = \bigcup_n A_n$; the second member is equal to $\mu^*(f \varphi_A)$ (Ch. IV, §1, No. 3, Th. 3), that is, to $\mu^\bullet(f \varphi_A)$ (by Prop. 1, or by 2) above). Since $\mu^\bullet(f) = \mu^\bullet(f \varphi_A) + \mu^\bullet(f \varphi_{\mathbf{C}_A})$ (Prop. 2), we have $\mu^\bullet(f \varphi_{\mathbf{C}_A}) = 0$, from which 3) follows.