Supremum & Infimum Preserve Measurability

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Supremum & Infimum Preserve Measurability

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.

Cor

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

Linked from

A Summary of MATH 891