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I am trying to build intuition over the following matter: let $X,Y$ be two random variables with corresponding probability measures $P_X,P_Y$. Assume also there exists a joint measure $P_{XY}$ such that for every measurable set $E$ $P_{XY}(\mathcal{X}\times E) = P_Y(E)$ and for every $F$ $P_{XY}(F\times \mathcal{Y}) = P_X(F)$.

My question is: is there a characterisation for when the joint is guaranteed to be absolutely continuous wrt to $P_XP_Y$? All the counterexamples I could come up with are somehow dimension-related or "pathological", like: let $P_XP_Y$ be the Lebesgue measure over the unit square and $P_{XY}$ the joint induced by $X=Y$.

If we exclude this type of settings, is the constraint of $P_X,P_Y$ being the marginals enough to guarantee absolute continuity? Does anyone have intuition on this problem?

user1868607
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1 Answers1

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One sufficient condition for this to hold (a coupling $P_{XY}$ is absolutely continuous w.r.t. the product measure of marginals $P_{X} \times P_{Y}$) is when the coupling $P_{XY}$ is already absolutely continuous with respect to some reference measure on $\mathcal{X} \times \mathcal{Y}$ (e.g., Lebesgue measure on $\mathbb{R}^2$ in the 2D case), see this example: Marginally continuous measures. This example excludes your "pathological" counterexample when $P_{XY}$ is supported on a Lebesgue measure zero subset, but the product measure is ``larger''.

Yibo Yang
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  • The reference measure on $X\times Y$ seems to be required to be of the product type, i.e. there are $\sigma$-fimite measures $m_X$ and $m_Y$ on $\mathcal{X}$ and $\mathcal{Y}$ such that $P_{X,Y}\ll m_X\otimes m_Y$. – Mittens Feb 16 '23 at 20:56