**Problem**

Given a Hilbert space $\mathcal{H}$.

Let the Lebesgue measure be $\lambda$. 

Consider a Borel spectral measure:
$$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$

Denote its probability measures by:
$$\nu\_\varphi(A):=\|E(A)\varphi\|^2$$

Introduce the spectral space:
$$\mathcal{H}\_\parallel:=\{\varphi:\nu\_\varphi\ll\lambda\}$$
$$\mathcal{H}\_\perp:=\{\varphi:\nu\_\varphi\perp\lambda\}$$

Then they decompose:
$$\mathcal{H}=\mathcal{H}\_\parallel\oplus\mathcal{H}\_\perp$$

How to prove this?

**Attention**

This thread has been split: [Spectral Spaces (II)](https://math.stackexchange.com/q/1036647/79762)