Definition 1

A function $g$, defined locally almost everywhere in $T$ (for the positive measure $\mu$), with values in a Banach space $F$ (resp. in $\overline{\mathbf{R}}$), is said to be locally integrable for $\mu$ (or locally $\mu$*-integrable*) if it satisfies the conditions a), b), c) of Prop. 1. If $\theta$ is a complex measure, a function $g$ defined locally $\theta$*-almost everywhere is said to be locally* $\theta$*-integrable if it is locally integrable for the positive measure* $|\theta|$.

If $g$ is locally $\theta$-integrable, then every function equal to $g$ locally almost everywhere is locally integrable. It is clear that the sum of two locally integrable functions is locally integrable. The functions with values in $F$, everywhere defined and locally integrable for $\theta$, form a vector space denoted $\mathscr{L}_{loc}^1(T, \theta; F)$; when $F = \mathbf{R}$ or $\mathbf{C}$, the mention of $F$ is often omitted if there is no ambiguity. This space will always be equipped (absent express mention to the contrary) with the topology defined by the semi-norms $g \mapsto \int |g \varphi_K| \, d|\theta|$, where $K$ runs over the set of compact subsets of $T$. The associated Hausdorff space, the quotient of $\mathscr{L}_{loc}^1(T, \theta; F)$ by the subspace $\mathscr{N}_F^\infty$ of mappings that are zero locally almost everywhere, is denoted $L_{loc}^1(T, \theta; F)$. The spaces $L_{loc}^1(T, \theta; F)$ and $L_{loc}^1(T, |\theta|; F)$ are identical.

It can be shown that the topological vector spaces just defined are complete (Exer. 31).

Every measurable function $g$, that is essentially bounded on every compact set, is locally integrable. For every number $p$ such that $1 \leq p \leq +\infty$, every function $g \in \mathscr{L}_F^p$ is locally integrable; indeed, for every function $h \in \mathscr{H}(T)$, $h$ belongs to $\mathscr{L}^q$ (where $q$ is the exponent conjugate to $p$), therefore $gh$ is integrable (Ch. IV, §6, No. 4, Cor. 4 of Th. 2).

Let $F, G, H$ be three Banach spaces, and $(u, v) \mapsto \Phi(u, v)$ a continuous bilinear mapping of $F \times G$ into $H$. If $f$ is locally integrable and takes its values in F, and if $g \in \mathscr{L}_G^\infty$, then $\Phi(f, g)$ is locally integrable (Ch. IV, §6, No. 4, Cor. 1 of Th. 2).