Let $H$ be a complex [Hilbert space](/page/Hilbert%20Space), let $K$ be a compact [metric space](/page/Metric%20Space), and let $E:\mathcal B(K)\to\mathcal L(H)$ be a projection-valued measure. For $x,y\in H$, define the finite complex measure $\mu_{x,y}$ on $\mathcal B(K)$ by
If $s:K\to\mathbb C$ is a bounded Borel [simple function](/page/Simple%20Function), then the simple spectral integral $\int_K s\,dE\in\mathcal L(H)$ satisfies