Definition 37.4.1 (Spectral Measure).label Let $X$ be a compact Hausdorff space, $H$ be a complex Hilbert space, and $E: \cb_{X} \to B(H)$, then $E$ is a spectral measure relative to $H$ if:

1. (1)

   For each $B \in \cb_{X}$, $E(B)$ is an orthogonal projection.

2. (2)

   $E(\emptyset) = 0$, $E(X) = I_{B(H)}$.

3. (3)

   For each $B, C \in \cb_{X}$, $E(B \cap C) = E(B)E(C)$.

4. (4)

   For each $x, y \in H$, the mapping

   \[E_{x, y}: \cb_{X} \to \complex \quad B \mapsto \dpn{E(B)x, y}{H}\]

   
   

   is a complex Radon measure on $X$.