Corollary 1

Let $G_i$ ($1 \leq i \leq n$) be $n$ Hausdorff locally convex spaces having the property (GDF), and let $F$ be the space of separately continuous multilinear forms on $\prod_{i=1}^n G_i$, equipped with the topology of pointwise convergence. For every mapping $f$ of $T$ into $F$ that is scalarly essentially $\mu$-integrable, one has $\int f d\mu \in F$.

The space $F$ is in duality with the tensor product $\bigotimes_{i=1}^n G_i$, and the topology of pointwise convergence on $F$ is none other than the topology $\sigma(F, \bigotimes_{i=1}^n G_i)$. The algebraic dual ${F'}^*$ is therefore the space of all multilinear forms on $\prod_{i=1}^n G_i$. Let $z = (z_1, \ldots, z_n)$ be an element of $\prod_{i=1}^n G_i$; for every multilinear form $u \in {F'}^*$, the mapping $x \mapsto u(z_1, \ldots, z_{i-1}, x, z_{i+1}, \ldots, z_n)$ is a linear form on $G_i$, which we shall denote by $\lambda_i(z)(u)$; we thus obtain a linear mapping $\lambda_i(z)$ of ${F'}^*$ into the algebraic dual $G_i^*$ of $G_i$, continuous for the topologies $\sigma({F'}^*, \bigotimes_{i=1}^n G_i)$ and $\sigma(G_i^*, G_i)$. To say that $u \in F$ means that for every index $i$ and every $z \in \prod_{i=1}^n G_i$, one has $\lambda_i(z)(u) \in G_i'$. Now, by Prop. 1 of No. 1, the mapping $\lambda_i(z)$ of $F$ is a scalarly essentially $\mu$-integrable mapping of $T$ into $G_i'$ equipped with the topology $\sigma(G_i', G_i)$, and

By Th. 1, $\int (\lambda_i(z) \circ f) d\mu \in G_i'$ for $1 \leq i \leq n$, therefore $\int f d\mu \in F$.