Could it make sense to calculate the "distance correlation" among two probability measures (probability distributions)?

Question

Introduction. I post here the same question that I asked on math.stackexchange, since it might be more relevant to this forum: Could it make sense to calculate the "distance correlation" among two probability measures (probability distributions)?.

Question. Would it make sense to calculate the "distance correlation" introduced by Székely et al.(2007) (also here on Wikipedia) among two probability measures $P$ and $Q$, i.e. $\mathcal{R}(P,Q)$, rather than between two random vectors $X$ and $Y$, as stated on the Székely et al.(2007) paper?

Edited after the Christian Hennig's Answer. @Christian Hennig, thanks! Do you mean something like this?

1. If $X \sim P$ and $Y \sim Q$ and
2. If (from @Artem Mavrin's reply in Joint probability measure - note that I used your notation for the joint probability, i.e. $J$)

the joint distribution of random variables $X$ and $Y$, defined on a common probability space $(\Omega, \mathcal{A}, J)$ and taking values in measurable spaces $(\mathcal{X}, \mathcal{B})$ and $(\mathcal{Y}, \mathcal{C})$, respectively, is the probability measure defined on $(\mathcal{X} \times \mathcal{Y}, \mathcal{B} \otimes\mathcal{C})$ by $$ J_{X, Y}(E) = J((X, Y) \in E) $$ for all $E \in \mathcal{B} \otimes \mathcal{C}$.

3. Then, we can calculate the correlation distance among two probability measures $P$ and $Q$, i.e. $\mathcal{R}(P,Q)=\mathcal{R}\left({J_{X, Y}(E)}\right)$.

But then, I would not know how to connect $\mathcal{R}(P,Q)=\mathcal{R}\left({J_{X, Y}(E)}\right)$ to $\mathcal{R}(X,Y)$, which is (the latter) what stated on the Székely et al.(2007) paper... I think I did not get this connection: $\mathcal{R}(P,Q) = \mathcal{R}(X,Y)$.

Answer 1

If you just specify the distributions $P$ and $Q$ (of the two random variables $X$ and $Y$), you don't specify the dependence structure between them, which is essential for distance correlation. You'd need to define it depending on the joint distribution (let's call it $J$) of $X$ and $Y$, so you can have ${\cal R}(J)$, which would be the same thing as defining it based on $X$ and $Y$ (involving the dependence structure between them).