Definition 2

For every mapping $f$ of $X$ into a Banach space $F$, one sets $N_\infty(f) = M_\infty(|f|)$; $f$ is said to be bounded in measure (for the measure $\mu$) if $N_\infty(f)$ is finite. The set of mappings of $X$ into $F$ that are measurable and bounded in measure is denoted $\mathscr{L}_F^\infty(X, \mu)$ (or $\mathscr{L}_F^\infty(\mu)$, or simply $\mathscr{L}_F^\infty$).

A function $f$ in $\mathscr{L}_F^\infty$ may thus be characterized by the fact that there exists a bounded measurable function equal locally almost everywhere to $f$.

It follows immediately from (1) that

on the other hand, $N_\infty(\alpha f) = |\alpha| N_\infty(f)$ for every scalar $\alpha$. The set $\mathscr{L}_F^\infty$ is therefore a linear subspace of the space of all mappings of $X$ into $F$, and $N_\infty(f)$ is a semi-norm on this vector space. Let $(f_n)$ be a sequence of functions in $\mathscr{L}_F^\infty$ that converges to $f \in \mathscr{L}_F^\infty$ for the topology defined by the semi-norm $N_\infty(f)$; for every integer $m$, there exist a locally negligible set $H_m$ and an integer $n_0$ such that $|f(x) - f_n(x)| \leq 1/m$ for every integer $n \geq n_0$ and every $x \notin H_m$ (every countable union of locally negligible sets being locally negligible); the union $H$ of the $H_m$ is locally negligible, and one sees that $f_n(x)$ tends uniformly to $f(x)$ on the complement of the locally negligible set $H$; the converse is immediate.

It is clear that every function equal locally almost everywhere to a function in $\mathscr{L}_F^\infty$ belongs to $\mathscr{L}_F^\infty$. In particular, the locally negligible functions defined on $X$ with values in $F$ form a linear subspace $\mathscr{N}_F^\infty$ of $\mathscr{L}_F^\infty$, characterized by the relation $N_\infty(f) = 0$ (the closure of 0 for the topology defined by $N_\infty(f)$). The Hausdorff space associated with $\mathscr{L}_F^\infty$, that is, the quotient space $\mathscr{L}_F^\infty / \mathscr{N}_F^\infty$, is denoted $L_F^\infty(X, \mu)$ (or $L_F^\infty(\mu)$ or $L_F^\infty$); its topology is defined by the norm deduced from $N_\infty$ by passage to the quotient; the norm of a class $\dot{f} \in L_F^\infty$ is denoted $N_\infty(\dot{f})$, or also $\| \dot{f} \|_\infty$. When $F = \mathbf{R}$ (resp. $\mathbf{C}$), we write $\mathscr{L}^\infty$ and $L^\infty$ in place of $\mathscr{L}_\mathbf{R}^\infty$ and $L_\mathbf{R}^\infty$ (resp. $\mathscr{L}_\mathbf{C}^\infty$ and $L_\mathbf{C}^\infty$) if this can cause no confusion.