Background to the question is that I have to show the equivalance of the following three statements: $(X, \mathcal{A})$ a measure space and $f: X \to \overline{\mathbb R}$
Show the equivalence of the three following statements:
$i)$ $f$ is $\mathcal{A}-\mathcal{B}(\overline{\mathbb R})-$measureable
$ii)$There exists a dense subset $D \subseteq \mathbb R$, so that $\{f \geq \alpha\}\in \mathcal{A}$, $\forall \alpha \in D$.
$iii)$ $f^{-1}(\{+\infty\}) \in \mathcal{A}$ and $f^{-1}(\mathcal{B}(\mathbb R)) \subseteq \mathcal{A}$
I thought I took the "easy route" when I showed $ii) \Rightarrow i)$ and then $i) \Rightarrow iii)$. Now I am having trouble proving that $iii) \Rightarrow ii)$, which leads me to think that it may not be possible directly and so I'll have to redo my other implications. Is there a way to directly solve:
If $f^{-1}(\{+\infty\}) \in \mathcal{A}$ and $f^{-1}(\mathcal{B}(\mathbb R)) \subseteq \mathcal{A} \Rightarrow$ I can find $D \subseteq \mathbb R$ such that $\{f \geq \alpha\} \in \mathcal{A}$, $\forall \alpha \in D$.
Hint: $f^{-1}([\alpha,\infty]) = f^{-1}([\alpha,\infty))\cup f^{-1}(\{\infty\}).$