Lemma 2

With notations as in Lemma 1, for every function $f \in \mathscr{K}(X \times Y, K \times L; \mathbf{C})$ the function

belongs to $\mathscr{K}(Y, L; \mathbf{C})$.

For every function $u \in \mathscr{K}(X; \mathscr{K}(Y, L; \mathbf{C}))$, the integral $\int u(x) d\lambda(x)$ belongs to $\mathscr{K}(Y, L; \mathbf{C})$ since the latter is a Banach space (§ 3, No. 3, Cor. 1 of Prop. 7); but for $u = \omega(f)$ and for every $y \in Y$,

whence the lemma.

Consider, then, for every function $f \in \mathscr{K}(X \times Y; \mathbf{C})$, the number $\nu(f) = \mu(\int f(x, y) d\lambda(x))$ (which we shall also write $\int d\mu(y) \int f(x, y) d\lambda(x)$ by an abuse of notation); if K (resp. L) is a compact subset of X (resp. Y), then there exists a number $a_K$ (resp. $b_L$) such that, for every function $g \in \mathscr{K}(X, K; \mathbf{C})$ (resp. $h \in \mathscr{K}(Y, L; \mathbf{C})$), we have $|\lambda(g)| \leq a_K \|g\|$ (resp. $|\mu(h)| \leq b_L \|h\|$). It follows that for every function $f \in \mathscr{K}(X \times Y, K \times L; \mathbf{C})$,

for every $y \in Y$, whence $|\nu(f)| \leq a_K b_L \|f\|$. The linear form $\nu$ on $\mathscr{K}(X \times Y; \mathbf{C})$ is thus a measure on $X \times Y$ that obviously satisfies (1), which completes the proof of Th. 1.