I am studying for measure theory. Now there is this one question which I am trying to proof, and I think I am on to something. However, there is one part which I can't get my head over, which would help me a lot with my proof.
So we have 3 functions; $g$, $f$ and $h$. We know that $g, f: \Omega \to \overline{\mathbb{R}}$, which are both $(\sigma(f), \overline{\mathcal{B}})$-measurable.
Here $$\sigma(f) = \bigcap_{\substack{\mathcal{A}: \mathcal{A} \text{ is a $\sigma$-algebra}\\\text{of sets of $\Omega$ and $f$ is}\\ \text{$(\mathcal{A},\overline{\mathcal{B}})$-measurable}}}\mathcal{A}.$$
Now I have $h = g \circ f^{-1}$. How do I prove that $h$ is $(\overline{\mathcal{B}}, \overline{\mathcal{B}})$-measurable?
I have already tried several things, but none which seem to work out. Help would be highly appreciated!