Proposition 10

Let $\mathfrak{K}$ be a $\mu$-dense set of compact subsets of $T$. a) If $f$ is a numerical function $\geq 0$, then

b) If $f$ is an essentially integrable function with values in a Banach space $F$, then the limit being taken with respect to the directed (for $\subset$) set $\mathfrak{K}$.

To establish a), it suffices to show that for every compact subset $L$ of $T$,

where $K$ runs over the set of subsets of $L$ belonging to $\mathfrak{K}$. Since $L$ is the union of a negligible set and an increasing sequence $(K_n)$ of elements of $\mathfrak{K}$ (Ch. IV, §5, No. 8, Prop. 12), this follows from the theorem on passage to the limit in upper integrals (Ch. IV, §1, No. 3, Th. 3).

Suppose now that $f$ belongs to $\mathscr{L}_F^1$; let $\varepsilon$ be a number $> 0$, and let $K$ be an element of $\mathfrak{K}$ such that

(such a K exists by a)). Then, for every compact set H containing K,

Extension to complex Banach spaces and measures. Let F be a complex Banach space; by an abuse of notation, the real Banach space underlying F will also be denoted by F. The Banach space $\overline{\mathscr{L}}_F^p(T, \mu)$ may then be equipped with a natural complex Banach space structure, and it is necessary to be specific as to whether one is using the real or the complex structure of this space. In this chapter, and absent express mention to the contrary, it will always be understood to be the real structure.

Let $\theta$ be a complex measure; we set $\overline{\mathscr{L}}_F^p(T, \theta) = \overline{\mathscr{L}}_F^p(T, |\theta|)$; if F is a complex Banach space, one can make the same remarks as above. In particular, a function f with values in F will be called essentially integrable for $\theta$ if it is essentially integrable for $|\theta|$. Assertion b) of Prop. 10 then extends at once to complex measures.