Theorem 1.14

If $f_{n}:X\to\overline{\mathbb{R}}$ is measurable, for $n=1,2,3,\dots$ and $$g=\sup_{n\ge 1}f_{n},\quad h=\limsup_{ n \to \infty }f_{n}$$(respectively infimum) then $g$ and $h$ are measurable.

Corollary

1. If $(f_{n})_{n\ge 1}$ is a sequence of $\mathbb{R}$-valued measurable functions, converging pointwise to $f$, then $f$ is measurable.
2. If $f,g$ are measurable (with range $[-\infty,\infty]$) then so are $f\vee g$, $f\wedge g$, $f^{+}$,$f^{-}$.