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I have some questions related to a the concept of absolute continuity and hope someone can help me.

The context is the following. Suppose we have two random variables X and Y on $(\Omega, \mathcal{F}, P)$.

1)The question states that there always exists a measure $\mu$ on $ (R, \mathrm{borel}) $ such that $P^{x}$ and $P^{y}$, the marginal laws of X and Y resp., are both absolutely continuous with respect to $\mu$. Give an example of such $\mu$.

If the variables have a density then clearly we can take the lebesgue measure, but this is not always the case. So, is there a $\mu$ for which this is always true (like $P$ itself since $\sigma(X) \subset \mathcal{F}$)?

2)Later on it states that $P^{x,y}$, the joint law is not always absolutely continuous w.r.t. the product measure $\mu \times \mu$. What would be a counterexample with $\mu$ the lebesgue measure?

3) This seems to imply that in the univariate case there is always a reference measure with respect to which the marginal laws are absolutely continuous but not always in the multivariate case?

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1) Just take $\mu=P^x+P^y$.

2) The uniform distribution on the diagonal $\{(x,x):x\in [0,1]\}$.

3) I don't get the question.

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