Let $H$ be a complex [Hilbert space](/page/Hilbert%20Space), let $K$ be a compact [metric space](/page/Metric%20Space), and let $\mathcal B(K)$ be the Borel $\sigma$-algebra of $K$. Let
be a normalized projection-valued measure, meaning that $E(\varnothing)=0$, $E(K)=I$, each $E(\Delta)$ is an [orthogonal projection](/theorems/437) on $H$, the identity
holds for all $\Delta,\Omega\in\mathcal B(K)$, and $E$ is countably additive in the strong operator topology on pairwise disjoint Borel sets. Let $B_b(K)$ denote the algebra of bounded Borel functions $f:K\to\mathbb C$, equipped with the supremum norm $\|f\|_\infty$.
as the operator-norm limit of $\int_K s_n\,dE$ for any sequence of bounded Borel simple functions $(s_n)_{n=1}^{\infty}$ with $\|s_n-f\|_\infty\to 0$. Then