Corollary 1

— Let $\Omega$ be a topological space, $t_0$ a point of $\Omega$ admitting a countable fundamental system of neighborhoods, $f$ a mapping of $X \times \Omega$ into $F$ having the following properties: a) for every $t \in \Omega$, the function $x \mapsto f(x, t)$ is integrable; b) for every $x \in X$, the function $t \mapsto f(x, t)$ is continuous at $t_0$;

c) there exist a neighborhood U of t_0 and a numerical function g \geqslant 0 defined on X, such that $\int^* g d|\mu| < +\infty$ and $|\mathbf{f}(x,t)| \leqslant g(x)$ for all $x \in X$ and $t \in U$.

Under these conditions, the mapping $t \mapsto \int \mathbf{f}(x,t) d\mu(x)$ of $\Omega$ into F is continuous at the point $t_0$.