We are on $\mathbb{R}^d$ with Borel $\sigma$-algebra. Let $\mu_1, ..., \mu_d$ be probability measures on $\mathbb{R}$ and $\Pi(\mu_1, \mu_2, ..., \mu_d)$ be the set of probability measures on $\mathbb{R}^d$ with first marginal $\mu_1$, second marginal $\mu_2$, etc.

Given $\pi^{(1)}, \pi^{(2)} \in \Pi(\mu_1, ..., \mu_d)$ I am interested in conditions for the absolute continuity $\pi^{(1)} \ll \pi^{(2)}$. More specifically, I am looking for sufficient conditions that do not depend on the marginals $\mu_1, ..., \mu_d$, but solely on the dependence structures of $\pi^{(1)}$ and $\pi^{(2)}$.

I am first of all very interested whether these types of conditions or related questions have already been studied.

More specifically, I have the following idea for such a condition, which I could so far neither prove nor disprove:

Let $C_1, C_2 : [0, 1]^d \rightarrow [0,1]$ be two copulas. I.e., $C_1, C_2$ are CDFs corresponding to $\nu^{(1)}, \nu^{(2)}$, which are probability measures on $[0, 1]^d$ with uniform marginals.

Let $\pi^{(1)}, \pi^{(2)} \in \Pi(\mu_1, ..., \mu_d)$ be given by the dependence structures corresponding to $C_1, C_2$, respectively.

**Question:** If $\nu^{(1)} \ll \nu^{(2)}$, does then also hold $\pi^{(1)} \ll \pi^{(2)}$?