Lemma 11

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

Théories spectrales, chapter IV, § 1. Opérateurs compacts sur un espace hilbertien

If the endomorphism $u_N$ of $L^2(X)$ is positive, then $N(x, x)\geqslant 0$ for every $x$ in the support of $\mu$ .

Since $u_N$ is positive, the families $(f_i)$ and $(g_i)$ may be chosen in such a way that $f_i=g_i$ for all $i\in I$ (remark 3 of IV, p. 151). For all $x\in \widetilde{X}$ , we then have

N(x, x) =\sum_{i\in I}\alpha_i|f_i(x)|^2\geqslant 0

Since N is continuous, and $\mu(X-\widetilde{X}) = 0$ , the function N is positive on the support of $\mu$ .

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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