Proposition 6

— Let $f$ be a positive numerical function defined on $T$ that is $\mu$-measurable and $\mu$-moderated. Then there exists a sequence $(h_n)_{n \in \mathbf{N}}$ of elements of $\mathscr{F}_+(T)$, with sum equal to $f$, having the following properties:

1) The function $h_0$ is $\mu$-negligible. 2) For every $n \geq 1$, there exists a compact set $K_n$ such that $h_n$ is zero outside $K_n$, and such that the restriction of $h_n$ to $K_n$ is finite and continuous.

Suppose that $f$ is the sum of a sequence $(f_n)$ of positive measurable functions, each of which has the property in the statement; it is clear that $f$ also has it. Set

for every $n \in \mathbf{N}$; since $f$ is equal to the sum of the sequence $(f_n)$, it will thus suffice to establish the proposition assuming $f$ to be moderated and bounded. Denote then by $A$ the set of $t \in T$ such that $f(t) > 0$; $A$ is measurable and moderated, therefore there exists a sequence $(A_n)$ of pairwise disjoint integrable sets such that $A = \bigcup_n A_n$. We are reduced to proving the statement for the functions $f \varphi_{A_n}$; in other words, we may suppose $f$ to be bounded and to be zero outside an integrable set $I$. But $I$ is the union of a negligible set $N$ and a sequence $(L_n)$ of pairwise disjoint compact sets (Ch. IV, §4, No. 6, Cor. 2 of Th. 4). We are thus reduced to treating the case that $f$ is bounded and is zero outside a compact set $L$.

Let $\mathscr{K}$ be the set of compact subsets $K$ of $T$ such that $f|K$ is continuous; since $\mathscr{K}$ is $\mu$-dense (Ch. IV, §5, No. 10, Prop. 15), $L$ is the union of a negligible set $N$ and a sequence $(K_n)_{n \geq 1}$ of pairwise disjoint elements of $\mathscr{K}$ (Ch. IV, §5, No. 8, Def. 6). The functions $h_0 = f \varphi_N$, $h_n = f \varphi_{K_n}$ for $n \geq 1$ then satisfy the conditions of the statement.

The following proposition makes it possible to reduce the study of the upper integral to that of the essential upper integral.