Set of Points for which Measurable Function is Real-Valued is Measurable/Corollary

Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $f: X \to \overline \R$ be a $\Sigma$-measurable.

Then:

$\set {x \in X : \size {\map f x} = +\infty}$ is $\Sigma$-measurable.

Proof

We have:

$\set {x \in X : \size {\map f x} = +\infty} = X \setminus \set {x \in X : \map f x \in \R}$

From Set of Points for which Measurable Function is Real-Valued is Measurable, we have:

$\set {x \in X : \map f x \in \R}$ is $\Sigma$-measurable.

Since $\sigma$-algebras are closed under complementation, we have that:

$\set {x \in X : \size {\map f x} = +\infty}$ is $\Sigma$-measurable.

$\blacksquare$