Lemma 1

Tag 02QQ, permanent. Cite it as [Bourbaki, Tag 02QQ].

Théories spectrales, chapter III, § 2. Exemples d’applications linéaires compactes

Let $k\in \mathscr{N}^{p,q}(X\times Y)$ . The function $k$ is locally $(\mu\otimes \nu )$ -integrable.

The mapping $k$ is $(\mu\otimes \nu$ )-measurable by hypothesis. Let A be a compact subset of $X\times Y$ . There exist compact subsets $B\subset X$ and $C\subset Y$ such that $A\subset B\times C$ . Applying (2) to the characteristic functions $\varphi_B$ of B and $\varphi_C$ of C, one obtains

\int_{X\times Y}^*|k(x, y)|\varphi_A(x, y)d(\mu\otimes \nu )(x, y)

\leqslant \int_{X\times Y}^*|k(x, y)|\varphi_B(x)\varphi_C(y)d(\mu\otimes \nu )(x, y)<+\infty

whence the result (INT, V, p. 41, §5, n $^o1$ , Proposition 1 and Definition 1).

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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