Corollary 1

— Let $(f_n)$ be a sequence of numerical functions (finite or not). If the $f_n$ are measurable, then the functions $\sup_n f_n$, $\inf_n f_n$, $\limsup_{n \to \infty} f_n$ and $\liminf_{n \to \infty} f_n$ are measurable.

For, the extended real line $\overline{\mathbf{R}}$, being homeomorphic to a compact interval of $\mathbf{R}$, is metrizable. The function $\sup_n f_n$ is the pointwise limit of the increasing sequence of functions $g_n = \sup(f_1, f_2, \ldots, f_n)$, which are measurable (No. 3, Cor. 1 of Th. 1); similarly, $\limsup_{n \to \infty} f_n$ is the pointwise limit of the decreasing sequence of functions $h_n = \sup_{p \geq 0} f_{n+p}$, each of which is measurable by the foregoing. Finally, since $\inf_n f_n = -\sup(-f_n)$ and $\liminf_{n \to \infty} f_n = -\limsup_{n \to \infty} (-f_n)$, these functions are measurable.