When do we have $\sigma(X)= \sigma (f(X))$?
Solution 1:
Any injective measurable function will do. It is a well-known fact from descriptive set theory that an injective Borel measurable function $f:\mathbb{R}\to\mathbb{R}$ sends Borel sets to Borel sets. So for every Borel set $B\subseteq \mathbb{R}$, we have that $f(B)$ is a Borel set and $(f\circ X)^{-1}(f(B))=X^{-1}\circ f^{-1}(f(B))=X^{-1}(B)$.