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I have a question which is probably well known but I do not find any written reference. Let us consider a probability measure $\mu$ on $\mathbb{R}^2$. I would like to know if one can find a random vector $X:[0,1]\rightarrow \mathbb{R}^2$ whose law is $\mu$ : $$ \forall A \subset \mathbb{R}^2 \quad \mbox{Meas}\{\omega \in [0,1], X(\omega)\in A \}=\mu(A). $$ Here, $A$ is a Borel subset of $[0,1]$ and $\mbox{Meas}$ the Lebesgue measure on $[0,1]$. I find lots of such results if $X$ is of the form $(X_1,X_2)$ where $X_1$ and $X_2$ are independant or if $X$ admits a radial invariance, but never in the former general case. My question is well known in dimension $1$, but I do not know if it is true for any dimension.

Thanks,

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This is true in all dimensions. One source is Royden's Real Analysis, section 15.5 (I have 3rd edition). Some results:

Proposition 12. Every Borel measure on an uncountable complete separable metric space is isomorphic to a Borel measure on $[0,1]$.

An isomorphism means having a bijective map that is measurable both ways and pushes one measure forward to the other.

Since every Borel measure on $[0,1]$ is a pushforward of the Lebesgue measure under some increasing function $f:[0,1]\to [0,1]$, the conclusion follows.

Theorem 16 in the same section makes the above more precise. If the measure in Prop. 12 has no atoms, it is isomorphic to the Lebesgue measure on $[0,1]$. Otherwise, it is isomorphic to the sum of the Lebesgue measure on $[0,b]$ with $0\le b<1$, and point masses $c_k \delta_k$ at integers $k=1,2,\dots$. Naturally, $\sum c_k = 1-b$.

The above result leads to the concept of a standard probability space, and I thought of referring to the Wikipedia article on the subject... but I can't read it myself.

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