For two measures $\mu, \nu$ on the same space say that $\mu$ is *absolutely continuous* with respect to $\nu$ ($\mu \ll \nu$) whenever $\nu(A)=0$ implies that $\mu(A)=0$ too.


Let $(\Omega, \mathsf P$) be a probability space and let $S$ be a separable metric space. For an $S$-valued random variable $W$ on $\Omega$, denote by $\mathsf P\_W$ the distribution of $W$ on $S$ given by $\mathsf P\_W(A)=\mathsf P(W^{-1}(A))$.


Consider four random variables $X, X\_1, Y, Y\_1\colon \Omega\to S$ such that


1. $X\_1, Y\_1$ are independent,
2. $\mathsf{P}\_X \ll \mathsf{P}\_{X\_1}$,
3. $\mathsf{P}\_Y \ll \mathsf{P}\_{Y\_1}$.


I am looking for **a reference** to the statement that
$$\mathsf{P}\_{(X,Y)}\ll \mathsf{P}\_{(X\_1,Y\_1)}.$$
The case where $S=\mathbb R$ would be fine.


Not sure where to start looking. Thank you for your time.



Absolute continuity of measures - reference sought
This statement is false. Example: $Y=X=X\_1=U$ and $Y\_1=V$, where (say) $U$ and $V$ are independent standard normal r.v.'s. 


