Let $H$ be a [Hilbert space](/page/Hilbert%20Space), let $M\subseteq \mathcal{L}(H)$ be a von Neumann algebra, and let $(T_i)_{i\in I}$ be a net in $M$ indexed by a nonempty directed set $I$. Suppose that each $T_i$ is self-adjoint, that $T_i\le T_j$ whenever $i\le j$, and that there exists $T\in M$ such that $T_i\to T$ in the strong operator topology. If $\varphi:M\to\mathbb C$ is a positive normal linear functional, then the scalar net $(\varphi(T_i))_{i\in I}$ is increasing and

\begin{align*} \sup_{i\in I}\varphi(T_i)=\lim_{i\in I}\varphi(T_i)=\varphi(T). \end{align*}

Equivalently,

\begin{align*} \varphi(T_i)\uparrow\varphi(T). \end{align*}