Suppose I've got a probability space $(\Omega, \mathscr F, P)$ and a random variable $X: (\Omega, \mathscr F) \to (\mathbb R, \mathbb B)$ where $\mathbb B$ is the Borel $\sigma$-algebra. Let $\mu$ be a $\sigma$-finite measure on $(\mathbb R, \mathbb B)$ such that $\mu \gg P_X$ where $P_X = P \circ X^{-1}$ so that the Radon-Nikodym derivate $\frac{dP_X}{d\mu}$ exists. Now let $g : (\mathbb R, \mathbb B) \to (\mathbb R, \mathbb B)$ be an invertible function such that $\mu \gg P_X \circ g^{-1}$ as well.
My question:
If we know $\frac{dP_X}{d\mu}$, can we say anything about $\frac{dP_X \circ g^{-1}}{d\mu}$?
For convenience, let $P_g = P_X \circ g^{-1}$, $f_X = \frac{dP_X}{d\mu}$, and $f_g = \frac{dP_g}{d\mu}$.
I've been trying to explore this with a change of variables and the definition of the RND (Radon-Nikodym derivative) as $$ \lambda(A) = \int_A \frac{d\lambda}{d\nu} d\nu $$ but so far I've had no success and at this point I'm quite stumped.