Corollary 2

Let $(F_n)$ be a sequence of topological spaces; for every index n, let $f_n$ be a measurable mapping of X into $F_n$ and let $f(x) = (f_n(x)) \in F = \prod_n F_n$; finally, let u be a continuous mapping of $f(X)$ into a Banach space G. For the function $g(x) = u(f(x))$ to be integrable, it is necessary and sufficient that $N_1(g) < +\infty$.

For, g is measurable (No. 3, Th. 1).