Exercise 10, § 1

Let $G, H$ be two Fréchet spaces, $G_0$ (resp. $H_0$) a dense subset of $G$ (resp. $H$), and $t \mapsto \Phi_t$ a mapping of $T$ into the space $F = \mathscr{B}(G,H)$ of continuous bilinear forms on $G \times H$, equipped with the topology of pointwise convergence. Assume that: $1^\circ$ for every pair $(a,b) \in G_0 \times H_0$, $t \mapsto \Phi_t(a,b)$ is essentially $\mu$-integrable; $2^\circ$ if, for every pair of bounded subsets $A \subset G$, $B \subset H$, one sets $q_{A,B}(\Phi) = \sup_{(x,y) \in A \times B} |\Phi(x,y)|$ for every $\Phi \in \mathscr{B}(G,H)$, then $\int^* q_{A,B}(\Phi_t)\,d\mu(t) < +\infty$. Under these conditions, show that $t \mapsto \Phi_t$ is scalarly essentially $\mu$-integrable and $\int \Phi_t\,d\mu(t) \in F$ (using Lebesgue’s theorem, reduce to applying Cor. 1 of Th. 1). Special case where $H = \mathbf{R}$.