Definition 1

One says that the integral $\int_I \mathbf{f}_\alpha(t)\,dt$ is uniformly convergent for $\alpha \in A$ (or uniformly convergent over $A$) if, for every $\varepsilon > 0$ there exists a compact interval $J_0 \subset J$ such that, for every compact interval $K \subset I$ with no interior point in common with $J_0$, and every $\alpha \in A$, one has

This definition is equivalent to saying that the family of maps $\alpha \mapsto \int_I \mathbf{f}_\alpha(t)\,dt$ is uniformly convergent on $A$ (towards the map $\alpha \mapsto \int_I \mathbf{f}_\alpha(t)\,dt$) with respect to the filter of sections $\Phi$ of $\mathscr{K}(I)$; each of the integrals $\int_I \mathbf{f}_\alpha(t)\,dt$ is a fortiori convergent (the converse being false). Further, from what we have just seen (or from Gen. Top., X, p. 281, cor. 2):