I am reading the paper http://www.ams.org/mathscinet-getitem?mr=3246935 and there some notation that I have found a bit confusing on page 1503 between Lemma 4.2 and 4.3. I'll give as much context as I can.
Let $\mathcal{H}$ be a infinite dimensional separable Hilbert space and $(X, \nu)$ a metric space equipped with a positive locally finite Borel measure. Let $T$ be a bounded linear operator on $L^2(X, \nu) \otimes \mathcal{H}$. Let $\varphi \in L^2(X, \nu) \otimes \mathcal{H}$ and define the finite measure $$ d \mu = \|T \varphi\|_{\mathcal{H}}^2 d \nu $$ My questions are: 1) What does this notation mean? 2) What is the measure $\mu$? 3) How did $\mu$ become a finite measure when $\nu$ is locally finite?
This paper it has given more hints what this notation means. For a Borel subset $\Omega \subset X$ let $P_{\Omega}$ be the orthogonal projection $P_{\Omega} \colon L^2(X, \nu) \otimes \mathcal{H} \to L^2(\Omega, \nu) \otimes \mathcal{H}$. Then $\| P_{\Omega} T \varphi \|^2 = \mu(\Omega)$. This makes some sense but I am confused why there is no reference to $\nu$.