Let $H$ be a [Hilbert space](/page/Hilbert%20Space), let $M\subseteq \mathcal{L}(H)$ be a von Neumann algebra, let $\mathcal{P}(M)$ denote the set of projections in $M$, and let $\phi:M\to\mathbb C$ be a state. Then $\phi$ is normal if and only if, for every index set $I$ and every pairwise orthogonal family $(p_i)_{i\in I}$ in $\mathcal{P}(M)$, if

\begin{align*} p=\bigvee_{i\in I}p_i \end{align*}

in the projection lattice $\mathcal{P}(M)$, then

\begin{align*} \phi(p)=\sup\left\{\sum_{i\in F}\phi(p_i):F\subset I\text{ is finite}\right\}. \end{align*}