Let $(\Omega,\mathscr A,P)$ be an arbitrary probability space,
and let $X:\Omega\to\mathbb R$ be a random variable.
Then,
one can generate a random variable $Y$ from the probability space $\big([0,1],\mathscr B,\lambda\big)$ (where $\mathscr B$ denotes the Borel algebra and $\lambda$ denotes the Lebesgue measure) to $\mathbb R$ whose probability distribution $\mu\_Y$ is equal to that of $X$ by means of the quantile function by letting
\begin{align\*}
Y(x):=\sup\big\{y\in\mathbb R:x\leq F\_X(y)\big\},&&x\in(0,1)
\end{align\*}
where $F\_X$ is the cumulative distribution function of $X$.


This has many important theoretical and practical implications.
For instance,
it shows that any cumulative distribution function can be realized by a random variable (which allows one to make statements such as let $X$ be a random variable with distribution $F$);
it can make some technical proofs easier,
as $\big([0,1],\mathscr B,\lambda\big)$ has many *nice* properties (separability, regularity, $[0,1]$ is compact),
and it gives rise to the so-called [inverse transform sampling](http://en.wikipedia.org/wiki/Inverse_transform_sampling),
which is very useful in simulations.


**Question.** Does there exist similar results for random variables $X$ taking values in spaces other than $\mathbb R$ (such as $\mathbb R^n$)?
For example,
if one has a random variable $X:\Omega\to\mathbb C$,
is it always possible to construct a random variable $Y$ defined on the unit square $[0,1]\times[0,1]\subset\mathbb C$ such that $X$ and $Y$ have the same probability distribution ($\mu\_X=\mu\_Y$)?



Is it possible to construct any random variable on the Euclidean Probability space?
Your question is a little bit unprecise, because of the fuzziness of the word "construct". Since all Borel $\sigma$-algebras of complete separable metric spaces are equivalent, your desired $Y$ exists (and in fact, you can even take $Y$ from $[0,1]$ to $\mathbb{R}^n$, for that matter). I guess you would like to have special properties like in the one-dimensional case.


Then I guess that you have to make at least some arbitrary choices, because you do not have a very good linear order on $\mathbb{R}^n$ anymore. You could use desintegration and the Knothe-Rosenblatt rearrangment: taking as you did $n=2$ to simplify, let $X\_1$ be the first coordinate of $X$ and construct $Y\_1$ from $[0,1]$ to $\mathbb{R}$ having the same law as $X\_1$, in the monotonic way you describe. Then, let $(\mu\_a)$ be the defined-almost-everywhere family of laws of the second coordinate of $X$, conditional to $X\_1=a$. Construct, again monotonically, independent random variables $Y\_a$ from $[0,1]$ having law $\mu\_a$. Then take $Y$ to have first coordinate $Y\_1$, and second coordinate $Y\_a$ conditionally to $Y\_1=a$. Then by construction the law of $Y$ is equal to the law of $X$, and you keep some kind of monotonicity.


However, this construction gives a prominent role to the coordinates and their order, so even an isometric modification of $X$ (i.e. compose $X$ with a displacement) would yield a completely different $Y$ (different from the original $Y$ composed with the same displacement).


