My question is regarding the following Proposition:
Let $(X,\mathcal{A})$ be a measurable space and let $f: X \rightarrow \mathbb{R}$ be an $\mathcal{A}$-measurable function. If the set $C$ is in the Borel $\sigma$-algebra on $\mathbb{R}$, then $f^{-1}(C) \in \mathcal{A}$.
My question is about the interpretation of this proposition. If I did understand that Proposition correctly, then this would be very useful when trying to show that for example a function is measurable. Here is an example of how I think this should work:
Consider $f(x)=|sin(x)|$. Since $|.|$ and $\sin$ are continuous, they are Borel measurable. Consider $f^{-1}((a,\infty))$ for $a \in \mathbb{R}$,then $f^{-1}((a,\infty)) \in \mathcal{A}$ since $(a,\infty) \in \mathcal{B}$.
Is my assumption about the use of the Proposition correct? And, does this work like in the example I provided?