Remark

Let $f\in \mathscr{L}^p(X)$ and $g\in \mathscr{L}^q(Y)$. Since $p <+\infty$ and $q <+\infty$, the functions $f$ and $g$ are moderate, and the function defined by $(x, y)\mapsto f(x)g(y)$ is $(\mu\otimes \nu$)-moderate (cf. INT, V, p. 92, § 8, n$^o3$, Corollary 2). Consequently, the function $(x, y)\mapsto k(x, y)f(x)g(y)$ is $(\mu\otimes \nu$)-moderate, and in particular the upper integral of its absolute value coincides with the essential upper integral (INT, V, p. 6, § 1, n$^o3$, Proposition 7).

The set $\mathscr{N}^{p,q}(X\times Y)$ is a vector subspace of the space of functions from $X\times Y$ into $\mathbf{C}$ and the mapping defined by $k\mapsto \|k\|_{p,q}$ is a semi-norm on $\mathscr{N}^{p,q}(X\times Y)$.