Given a measure $\mu$ on some measure space $X$ and a measurable functions $f:X\to Y$ we can define a measure $f_* \mu(A)=\mu(f^{-1}(A))$ on $Y$.
If $f=g$ almost everywhere then $g_*\mu =f_*\mu$.
I could show that the converse holds by placing assumptions on $Y$ (i.e two real valued random variables with the same law are equal almost surely). Is the converse true more generally?
So far I have the supposition $g_*\mu =f_*\mu$ is equivalent to the measure of the symmetric difference of $f^{-1}(A)$ and $g^{-1}(A)$ being $0$. However with no other structure on $Y$ I am not sure whether more can be said.
My question is whether the converse is true. If not, are there any general sufficient measure theoretic conditions for it to be true?
Obviously false. If $\mu \equiv 0$, then for any $f,g$, $f_*\mu = g_*\mu$.
Less trivially, let $\mu$ be the counting measure on a finite set $X$. Let $f:X \to \mathbb{C}$ be a function and $g = f\circ \sigma$ for some permutation $\sigma \in S_X$. Then $f_*\mu = g_*\mu$, but $g$ isn't necessarily $f$.