This question is related to this question. Let $G$ be a (countably infinite, discrete?) group, and let $\pi : G \to U(H)$ be a unitary representation of $G$ on the Hilbert space $H$. Does anyone know of a proof or reference of the fact that $\pi$ can be written as a direct sum of indecomposable unitary representations, which I guess is related to writing $H = \bigoplus_{i \in I} H_i$ where each $H_i$ is a $G$-invariant Hilbert subspace of $H$. By the way, $\pi$ being indecomposable means that $H$ cannot be written as a direct sum of $G$-invariant subspaces.
I guess I am interested in the irreducible case as well, but I suppose that should be similar. And by irreducible, I mean that $H$ does not contain a non-trivial, proper $G$-invariant subspace.
Let $G=\mathbb{Z},$ $\mathcal{H}=L^2(-\pi,\pi)$ and $$(π(n)f)(x)=e^{inx}f(x)$$ ($π$ is unitarily equivalent to the regular representation of $G).$ Then $π$ is decomposable as $\mathcal{H}=L^2(A)\oplus L^2(B)$ for any decomposition of $(-π,π)=A\overset{.}{\cup} B.$ into sets of positive Lebesgue measure. However $π$ is not a direct sum of indecomposable representations. Indeed, any closed $G$-invariant subspace is of the form $L^2(A).$ Nonetheless it is the direct integral of unitary representations.