TITLE: Spectral theorem for $n$-tuples of selfadjoint operators
QUESTION [4 upvotes]: I need a 'good' reference to the following version of the Spectral Theorem:
Given $n$ commuting selfadjoint operators on an infinite-dimensional Hilbert space, there exist a Borel measure $\mu$ on $\mathbb R^n$ and auxiliary Hilbert spaces $h(x)$ such that the construction is unitarily equivalent to $\int\oplus h(x)\,d\mu(x)$ with selfadjoint operators of multiplication by $x_k$, $k=1, \dots, n$. 
It is important that the construction be based on a measure in $\mathbb R^n$. Thanks in advance.

REPLY [2 votes]: I don't think you can find such theorem precisely in that form. Instead you can find the following 
Spectral theorem. For $n$ bounded commuting self-adjoint operators $X_1,\dots,X_n$ on a Hilbert space $H$ there exists a unique projection valued measure $E$ on $\mathbb R^n$ such that $X_k=\int x_k dE(x_1,\dots,x_n)$ for all $k=1,\dots,n.$ 
Then using the following two (easy) Lemmas you get your version of the spectral theorem.
Lemma 1. $H$ can be decomposed into direct sum of common invariant subspaces $H_i,\ i\in I$ such that every $H_i$ is cyclic for the set $\{X_1,\dots,X_n\}$ with a cyclic vector $\xi_i.$
Lemma 2. Let $H_i,\xi_i$ be as in Lemma 1. Define $\mu_i(\cdot)=\langle E(\cdot)\xi_i,\xi_i\rangle.$ Then every $H_i$ is isomorphic to $L^2(\mathbb R^n,\mu_i)$ and $X_k$ are unitarily equivalent (under that isomorphism) to multiplication operators by $x_k.$
That spectral theorem you can find in the books:
Birman, Solomyak "Spectral theory of self-adjoint operators in Hilbert space,"
Yu. Samoilenko "Spectral theory of families of self-adjoint operators",
Yu. Berezanskii "Self-adjoint operators in spaces of functions of infinitely many variables" (in russian)
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