Suppose $f$ and $g$ are defined on $\mathbb{R}^n$ and are $\mathcal{B}(\Bbb{R}^n)$-$\mathcal{B}(\Bbb{R})$ measurable.
Is it true that the $\sigma$-algebra generated by $f$ belong to the $\sigma$-algebra generated by $g$ iff $g(x)=g(y)$ implies $f(x)=f(y)$?
I think this is true because I can’t find any counterexample. Could anyone help?
I found this on the book statistical inference by casella, in the chapter about minimal sufficient statistics. the original statement is: $f(x)=r(g(x))$ for some $r(x)$ from $\Bbb{R}$ to $\Bbb{R}$ iff $g(x)=g(y)$ implies $f(x)=f(y)$. But the book by casella doesn’t mention anything about measurability of $r(x)$, so I wonder if we can say that $r(x)$ is actually measurable.