Proposition 4 ("continuity of an improper integral with respect to a parameter")

Let $F$ be a compact space, let $I$ be any interval in $\mathbf{R}$, and $f$ a continuous map from $I \times F$ into a complete normed space $E$ over $\mathbf{R}$; if the integral $h(\alpha) = \int_I f(x, \alpha)\,dx$ is uniformly convergent on $F$, it is a continuous function of $\alpha$ on $F$.

In view of prop. 2 of II, p. 69, this proposition also follows from the continuity of a uniform limit of continuous functions (Gen. Top., X, p. 282, th. 2).