If $\mu$ is a Borel measure on $\mathbb{R}$, then $\mu$ is absolutely continuous with respect to Lebesgue measure if and only if $\mu\left((-\infty,x]\right)$ is an absolutely continuous function from $\mathbb{R}$ to $\mathbb{R}$.
I'm wondering if this can be generalized to two dimensions. Let $\mu$ be a Borel measure on $\mathbb{R}^2$. My question is, is there some notion of absolute continuity for multivariable functions such that we can say that $\mu$ is absolutely continuous with respect to Lebesgue measure if and only if $\mu\left((-\infty,x]\times(-\infty,y]\right)$ is an absolutely continuous function from $\mathbb{R}^2$ to $\mathbb{R}$?
This paper, this paper, and this paper discuss generalizations of absolute continuity to functions of multiple variables. Do any of these generalizations yield the notion I want?
This is all an attempt to better understand jointly continuous random variables.
Let $\mu$ be absolutely continuous w.r.t. Lebesgue measure. For a rectangle $R=(a,b]\times (c,d]$ define $F(R)=F(b,d)-F(a,d)-F(b,c)+F(a,c)$. Then, given $\epsilon >0$ three exists $\delta >0$ such that for any finite disjoint collection of $(R_i)$ of rectangles with total area less than $\delta$ we have $\sum F(R_i) <\epsilon$. The converse is also true.