The Egoroff Theorem for Operator-Valued Measures in Locally Convex Spaces

The Egoroff theorem for measurable $\mathbf{X}$-valued functions and operator-valued measures $\mathbf{m}:\mathrm{\Sigma }\to L(\mathbf{X},\mathbf{Y})$, where $\mathrm{\Sigma }$ is a $\sigma$-algebra of subsets of $T\mathrm{\ne }\mathrm{\varnothing }$ and $\mathbf{X}$, $\mathbf{Y}$ are both locally convex spaces, is proved. The measure is supposed to be atomic and the convergence of functions is net.