Definition 2

Given a locally compact space $X$, a measure $\mu$ on $X$, and a Banach space $F$, we denote by $\mathscr{L}_F^p(X, \mu)$ (or simply $\mathscr{L}_F^p(\mu)$, or $\mathscr{L}_F^p$) the closure, in the locally convex space $\mathscr{F}_F^p(X, \mu)$, of the vector space $\mathscr{H}(X; F)$ of continuous mappings of $X$ into $F$ with compact support. We denote by $L_F^p(X, \mu)$ (or $L_F^p(\mu)$, or $L_F^p$) the (normed) Hausdorff space associated with $\mathscr{L}_F^p(X, \mu)$. The functions belonging to $\mathscr{L}_F^p$ are called the $p$*-th power integrable functions* (*).

Obviously $\mathscr{L}_F^p(X, |\mu|) = \mathscr{L}_F^p(X, \mu)$ and $L_F^p(X, |\mu|) = L_F^p(X, \mu)$.

We shall write $\mathscr{L}^p$ and $L^p$ instead of $\mathscr{L}_R^p$ and $L_R^p$ (or of $\mathscr{L}_C^p$ and $L_C^p$ when this causes no confusion). If $F$ is a complex Banach space, $\mathscr{L}_F^p$ and $L_F^p$ are equipped with the structure of a topological vector space over the field $\mathbf{C}$ (No. 3, Remark).

It is clear that every function in $\mathscr{F}_F^p$ that is equivalent to a function in $\mathscr{L}_F^p$, belongs to $\mathscr{L}_F^p$. A function with values in $F$ and defined almost everywhere in $X$ is again said to be $p$-*th power integrable* if it is equivalent to a function in $\mathscr{L}_F^p$; similarly a function with values in $\overline{\mathbf{R}}$, defined and finite almost everywhere in $X$, is said to be $p$-*th power integrable* if it is equivalent to a function in $\mathscr{L}^p$.

The functions in $\mathscr{L}_F^p$ (resp. in $\mathscr{L}^p$) are thus the $p$-*th power integrable functions* that are defined on all of $X$ (resp. defined and finite on all of $X$). In this section and the following one, most of the propositions proved for the functions in $\mathscr{L}_F^p$ (resp. $\mathscr{L}^p$) may immediately be extended to $p$-*th power integrable functions* that are not everywhere defined (resp. that are not everywhere defined and finite); we shall usually leave to the reader the task of formulating and proving these results.

(*) The justification for this terminology will be given in §4, No. 2.