Definition 9

The uniform structure on $\mathscr{S}(A, \mu; F)$ of which the $W(V, B, \delta)$ form a fundamental system of entourages is called the uniform structure of convergence in measure in $A$..

The corresponding topology is called the topology of convergence in measure in $A$, and a filter (or a sequence) that converges for this topology is said to be convergent in measure in $A$; the mention of $A$ is often suppressed when $A = X$.

Suppose $F$ is Hausdorff; then, for every $\mu$-integrable set $B \subset A$, the intersection of the entourages $W(V, B, \delta)$, where $V$ runs over a fundamental system of entourages of $F$ and $\delta$ runs over the set of numbers $> 0$, is the set of pairs $(f, g)$ such that $f(x) = g(x)$ almost everywhere in $B$ (with respect to $\mu$). For, the set $M$ of $x \in B$ such that $f(x) \neq g(x)$ is $\mu$-integrable, because it is the inverse image, under the $\mu$-measurable mapping $x \mapsto (f(x), g(x))$, of the complement of the diagonal in $F \times F$, which is open (No. 5, Prop. 7); if $|\mu|(M) = \alpha > 0$, there exists a compact subset $K \subset M$ such that $|\mu|(M - K) < \alpha/2$ and such that the restrictions of $f$ and $g$ to $K$ are continuous; therefore, there exists an entourage $V_0$ of $F$ such that $(f(x), g(x)) \notin V_0$ for all $x \in K$, consequently $(f, g) \notin W(V_0, B, \alpha/2)$.

From this it follows that if $F$ is Hausdorff, then the intersection of all the entourages of $\mathscr{S}(A, \mu; F)$ is the set of pairs $(f, g)$ such that $f(x) = g(x)$ locally almost everywhere in $A$. The Hausdorff uniform space associated with $\mathscr{S}(A, \mu; F)$, which we shall denote $S(A, \mu; F)$ or $S_F(A, \mu)$ (or even $S_F(\mu)$ or $S_F$ when $A = X$), therefore consists of the equivalence classes for the relation « $f(x) = g(x)$ locally almost everywhere in $A$ » in the set $\mathscr{S}(A, \mu; F)$.