Proving a sufficient and necessary condition for $f:\, X\to\mathbb{R}\cup\{\pm\infty\}$ to be measurable

Solution 1:

I assume that you are topologising $\overline{\mathbb{R}}$ so that the basic open neighbourhoods of $+ \infty$ are of the form $( a , + \infty ] = ( a , + \infty ) \cup \{ + \infty \}$ for $a \in \mathbb{R}$ (and similarly for the basic open neighbourhoods of $- \infty$).

A couple observations that should help you:

• By this previous question it turns out that the Borel subsets of $\overline{\mathbb{R}}$ are precisely those sets of the form $B \cup F$ where $B \subseteq \mathbb{R}$ is Borel, and $F \subseteq \{ - \infty , + \infty \}$.

  It thus follows that $\{ - \infty \} , \{ + \infty \}$ are Borel subsets of $\overline{\mathbb{R}}$, and also every Borel subset of $\mathbb{R}$ is a Borel subset of $\overline{\mathbb{R}}$.

• Given $Y \subseteq X$, let $S^\prime = \{ A \cap Y : A \in S \}$ denote the restriction of the $\sigma$-algebra $S$ to $Y$. If $Y \in S$ it follows that $S^\prime = \{ A \in S : A \subseteq Y \}$.