For some interval $[a, b]$ of the real line, consider the family of probability distributions $p: [a,b]\times [a,b] \rightarrow \mathbb{R}^+$ such that $p\geq 0$ and $\int dx dy p(x, y) = 1$. So $p(x, y)$ can be interpreted as the probability (density) for sampling a pair $(x, y) \in [a,b]^2$.
Is there any well-defined notion of a maximally correlated distribution (call it $p^*$) that assigns a higher probability density to the outcome $(x, x)$ than any other possible distribution $p$? Naively, the following seems like a possible choice:
$$ p^* (x, y) = q(x) \delta(x - y) \tag{1} $$
where $q(x)$ is appropriately normalized and $\delta$ is the Dirac delta.
What is wrong with the distribution $(1)$, and what kind of better choice/reasoning could be used to describe a "maximally correlated" probability density in the above sense?