Proposition 4

Let $k\in \mathscr{N}^{p,q}(X\times Y)$ and $f\in \mathscr{L}^p(X)$. The mapping defined on X by $x\mapsto k(x, y)f(x)$ is $\mu$-integrable for every $y$ outside a locally $\nu$-negligible set, and $u_k(f)$ coincides with the class of the function

defined locally $\nu$-almost everywhere.

Let $Y'$ denote the set of $y\in Y$ such that the mapping $x\mapsto k(x, y)f(x)$ is not $\mu$-integrable. Let $y\in Y$ and let C be a compact neighbourhood of $y$, whose characteristic function is denoted by $\varphi$. By condition (2) applied to $f$ and $\varphi$, the function $(x, y)\mapsto k(x, y)f(x)\varphi (y)$ is integrable on $X\times Y$ with respect to the measure $\mu\otimes \nu$. By the Lebesgue-Fubini theorem (INT, V, p. 96, § 8, No.$^o4$, Theorem 1, a)), the function $x\mapsto k(x, y)f(x)\varphi (y)$ is $\mu$-integrable for $y$ outside a $\nu$-negligible set $Y_C$. Since $Y'\cap C\subset Y_C$, it follows that $Y'$ is locally $\nu$-negligible in Y (INT, IV, p. 172, § 5, No.$^o2$, Definition 3).

Let $h$ denote the mapping defined locally $\nu$-almost everywhere on Y by $y\mapsto \int k(x, y)f(x)d\mu$. It is locally $\nu$-integrable (INT, V, loc. cit.). Let $g\in \mathscr{K}(Y)$. We have

(INT, V, loc. cit.). Hence $u_k(f) =h$ locally $\nu$-almost everywhere (cf. INT, V, p. 46, § 5, No.$^o3$, Corollary 2).