Let $X$ and $Y$ be indepdendent under the probability measure $P$. Define two equivalent probability measures $Q$ and $R$ by $$\frac{dQ}{dP}=g(X), \frac{dR}{dP}=g(X)h(Y)$$ where $g, h : \mathbb{R} \to (0, \infty)$. I want to show that $X$ has the same probability distribution under $Q$ and $R$, and determine if $X$ nd $Y$ are independent under these measures. I don't think that this is meant to be too difficult, but I'm just stuck.
All I have is that if $f_Q$ and $f_R$ are the probability distributions of $X$ under $Q$ and $R$ respectively, then for any borel set $A$: $$Q(X \in A) = \int_Af_Q(x)g(X)dP = E_P[g(X)1(X \in A)]$$ $$R(X \in A) = \int_Af_R(x)g(X)h(Y)dP = E_P[g(X)Y(X)1(X \in A)]$$ I'm not really sure what to do from here, is there a better way of showing two probability distributions are equivalent? Thanks!
Ps. I'm having trouble finding probability distributions under changes of measure in general, so if there are any good resources please let me know.