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I know (to some extent) how integration over $\mathbb{R}$ of a Borel-measurable function against a projection-valued measure works. Recently while reading a paper I came across calculations in which iterated integrals of functions of several variables against projection-valued measures are manipulated. The particular problem will be described below, but I wonder if there is a well established theory for such things in general--a theory that defines what such integrals mean and has some version of Fubini's theorem. One more problem with Fubini here is that once we start to calculate the iterated integrals, the integrand become operator-valued, and we are then integrating operated-valued functions against operator-valued measures.

Any references would be genuinely deeply appreciated.

The context of the particular calculations is quantum mechanics. The projection-valued measure $E$ is associated (via the spectral theorem) with the Laplacian (the free Hamiltonian) on $\mathbb{R}^n$: $$\Delta=H_0=\int_{\mathbb{R}}\lambda dE(\lambda).$$ The calculations dealt with integrals like this:$$\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{\mathbb{R}} f(\lambda_1,\lambda_2,\lambda_3)dE(\lambda_1)VdE(\lambda_2)\tilde{V}dE(\lambda_3)$$Here $f$ is a scalar-valued function and $V$ and $\tilde{V}$ are multiplier operators on $L^2(\mathbb{R}^d)$ (potentials). I tried to make sense of it by interpreting this as integral over $\mathbb{R}^3$ of $f$ against an "operator-valued measure" $dE(\lambda_1)VdE(\lambda_2)\tilde{V}dE(\lambda_3)$, which takes the value $E(A_1)\circ V \circ E(A_2)\circ \tilde{V}\circ E(A_3)$ on a cylinder set $A_1\times A_2\times A_3\subset \mathbb{R}^3$. And say we integrate out $\lambda_2$, I guess then the integral becomes $$\int_{\mathbb{R}}\int_{\mathbb{R}} dE(\lambda_1)Vf(\lambda_1,\Delta,\lambda_3)\tilde{V}dE(\lambda_3),$$ whatever this means.

While I could live with it, since $dE(\lambda)$ has an integral kernel so everything can be spelt out that way, it would be more elegant if there is already a theory for integrating operator-valued functions against operator-valued measures (or even $C^{\ast}$-algebra-valued, or even Banach algebra-valued ones), or just any theory for such functional calculus in quantum mechanics to be rigorous.

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    $\begingroup$ I guess you should take a look at multiple operator inegrals, e.g. Peller. $\endgroup$ – Mateusz Wasilewski Apr 12 at 9:48

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