Sigma algebra/Borel sigma algebra proof problem In my stochastics class we were given the following problem ($\mathcal{B}(\mathbb{R})$ stands for the Borel $\sigma$-algebra on the real line and $\mathbb{R}$ stands for the real numbers):
Let $f : \Omega → \mathbb{R}$ be a function. Let $F = \{ A \subset \Omega : \text{ there exists } B \in \mathcal{B}(\mathbb{R}) \text{ with } A = f^{−1}(B)\}$. Show that $F$ is a $\sigma$-algebra on $\Omega$.
I'm not sure how I should break this down. Apparently the inverse function $f^{-1}$ has some property where its target space is a $\sigma$-algebra when its starting space is a Borel $\sigma$-algebra... or am I going down the wrong path?
I've been going at this for a good while, any help is appreciated.

You have to show that $F$ is (i) non-empty, (ii) closed under complementation, and (iii) closed under countable unions.
