I want to prove the following Lemma:
Let $\mathcal{A}$ be a $\sigma$-algebra in $X$ and let $f:X\rightarrow\mathbb{R}$, then TFAE:
- $f$ is measurable.
- For each Borel set $B\subset\mathbb{R}$ holds $f^{-1}(B)\in\mathcal{A}$.
- For each open set $B\subset\mathbb{R}$ holds $f^{-1}(B)\in\mathcal{A}$.
I don't know how to start. I think one implication is some more work, but the other must follow. I think you have to prove 1-->3-->2-->1.
Some help maybe? Thanks