Corollary 2

Let $A$ be a subset of a topological space $F$, and $f$ a map of $I \times A$ into a complete normed space $E$ over $\mathbf{R}$, such that, for every $\alpha \in A$, the function $x \mapsto f(x, \alpha)$ is regulated on $I$. If, on the one hand, the functions $x \mapsto f(x, \alpha)$ converge uniformly on every compact interval contained in $I$ to a function $x \mapsto f(x)$ as $\alpha$ tends to $\alpha_0 \in \overline{A}$ while remaining in $A$; if, on the other hand, the integral $\int_I f(x, \alpha)\,dx$ is uniformly convergent on $A$, then the integral $\int_I f(x)\,dx$ is convergent, and one has

(10)

In particular: