Proposition 11

Let f be a lower semi-continuous function $\geqslant 0$ defined on $X = S \times T$.

a) The function $f_s : t \mapsto f(s, t)$ is lower semi-continuous on T for every $s \in S$.

b) The function $I_f : s \mapsto \int^\bullet f(s, t) d\mu(t)$ is lower semi-continuous on S, and

The property a) is obvious, since the mapping $t \mapsto f(s, t)$ of T into $\overline{\mathbf{R}}$ is the composition of $f$ with the continuous mapping $t \mapsto (s, t)$ of T into X. To establish b), we will make use of a lemma: