Proposition 3

Let $(\mathbf{f}_\alpha )$ be a family of regulated functions on an interval $I$ such that: 1 with respect to the filter $\mathfrak{F}$ the family $(\mathbf{f}_\alpha )$ converges uniformly to a function $f$ (regulated on $I$) on every compact interval contained in $I$; 2 the integral $\int_I \mathbf{f}_\alpha(t)\,dt$ is uniformly convergent for every $\alpha \in A$. Under these hypotheses the integral $\int_I f(t)\,dt$ is convergent, and one has

The hypotheses of prop. 3 are fulfilled when for example $I$ is a bounded interval, the $\mathbf{f}_\alpha$ are uniformly bounded on $I$, and converge uniformly to $f$ on every compact interval contained in $I$; indeed, if $\| \mathbf{f}_\alpha(x) \| \leq h$ for all $x \in I$ and all $\alpha$, and if $J_0$ is such that the difference between the lengths of $I$ and $J_0$ is $\leq \varepsilon / h$, then condition (7) is satisfied for every interval $K \subset I$ with no interior point in common with $J_0$.

As with prop. 1 of II, p. 68, two corollaries to prop. 3 are important in applications: