Definition 3

The elements of $\overline{\mathscr{L}}_F^1$ are called essentially $\mu$-integrable functions with values in F. On composing the mapping $\tilde{f} \mapsto \mu(f)$ of $L_F^1$ into F with the canonical mapping of $\overline{\mathscr{L}}_F^1$ onto $L_F^1$, one obtains a continuous linear mapping of $\overline{\mathscr{L}}_F^1$ into F that extends the mapping $f \mapsto \int f d\mu$ of $\mathscr{L}_F^1$ into F. One again denotes by $\int f d\mu$ or $\mu(f)$ the value of this mapping for $f \in \overline{\mathscr{L}}_F^1$, and this element is called the integral of f with respect to $\mu$.

Two essentially integrable functions that are equal locally almost everywhere have the same integral. For every function $f \geq 0$ that is finite and essentially integrable, $\int^\bullet f d\mu = \int f d\mu$. If A is a set whose characteristic function is essentially integrable, then A is said to be an essentially $\mu$-integrable set; $\int \varphi_A d\mu$ is also denoted $\mu(A)$ and is again called the measure of A.

If a function $f$, with values in $F$, is defined locally almost everywhere in $T$, we again say that $f$ is essentially integrable if it is equal, locally almost everywhere, to a function $f_1$ that is everywhere defined and integrable; we then set

and this definition is independent of the integrable function $f_1$ everywhere defined and equal locally almost everywhere to $f$ (Lemma). One defines similarly the notion of essentially integrable function for functions with values in $\overline{\mathbf{R}}$ that are defined and finite locally almost everywhere.

The reader will have no difficulty in extending, to essentially integrable functions, the results of Ch. IV, §4 for integrable functions, on replacing 'almost everywhere' in the statements by 'locally almost everywhere.' We note for example the inequality

valid for every essentially integrable function $f$ with values in a Banach space.