Automatic time continuity of positive matrix and operator semigroups

We consider a matrix semigroup $T:[0,\mathrm{\infty })\to {\mathbb{R}}^{d\times d}$ without assuming any measurability properties and show that, if $T$ is bounded close to $0$ and $T(t)\ge 0$ entrywise for all $t$, then $T$ is continuous. This complements classical results for the scalar-valued case. We also prove an analogous result if $T$ takes values in the positive operators over a sequence space.