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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: $f^{-1}([alpha,infty]) = f^{-1}([alpha,infty))cup f^{-1}({infty}).$