Let $K$ be a [topological space](/page/Topological%20Space), let $\mathcal B(K)$ be its Borel $\sigma$-algebra, and let $H$ be a [Hilbert space](/page/Hilbert%20Space). Let
be a projection-valued set function satisfying the finite projection-valued measure axioms: $E(\varnothing)=0$, $E(K)=I_H$, each $E(\Delta)$ is an [orthogonal projection](/theorems/437), for every $\Delta\in\mathcal B(K)$ one has
in $H$ for every $x\in H$. Then $E$ is countably additive in the strong operator topology: for every pairwise disjoint sequence $(\Omega_n)_{n=1}^{\infty}$ in $\mathcal B(K)$, if $\Omega=\bigcup_{n=1}^{\infty}\Omega_n$, then for every $x\in H$,