It is a well-known fact in probability theory that if $X$ is a continuous random variable and $F_X$ is a cdf of $X$, then $F_X(X)$ is uniformly distributed over $[0, 1]$.
Is there a two-dimensional version of this statement?
In other words, given a continuous random vector $(X, Y)$,
can we find some function $T: \mathbb{R}^2 \rightarrow [0, 1]^2$ such that $T(X, Y)$ is uniformly distributed over $[0, 1]^2$?
Thank you.