General version of Skorokhod representation of random variables

Question

Let $F: \mathbb{R} \to [0,1]$ be cumulative distribution function (cdf). The standard way to build a random variable $\tau$ on $([0,1],\mathcal{B},\text{Leb})$ with $F$ as its cdf is using the Skorokhod representation: For $x \in [0,1]$, let $$ \tau(x) := \inf \{ z : F(z) > x \} $$ be the generalized inverse of $F$.

I'm looking for a more general version of this. Let $X = \mathbb{R}^n$ (to begin with), and let $\mu$ be a probability measure on it. How does one build a random variable $\tau$ on $([0,1],\mathcal{B},\text{Leb})$ that takes values in $X$, so that the pushforward measure $\text{Leb}_* (\tau)$ is $\mu$?

Can this be done if $X$ is a Polish space and $\mu$ is a Borel measure? My feeling that this is generally true, and the result is probably due to von Neumann. I can't remember the right reference though.

Can anyone help? Note that a statement like "an isomorphism between these measure spaces exists" is not what I'm looking for. I want an explicit way to construct $\tau$.

Answer 1

The statement is true for probability measures on a complete separable metric space. I made a slight variation of the Skorokhod representation theorem in my PhD thesis (way long time ago!) so I am more or less cutting and pasting, sorry for the errors. I got the original proof from Ikeda, Watanabe, Stochastic differential equations and diffusion processes, North Holland Mathematical Library 24, North Holland/Kodansha 1989.

Let $(S,d)$ be a complete separable metric space, and $\mu$ a probability measure on $S$ (on the Borel $\sigma$-algebra of $S$). Set $\Omega=[0,1)$, $\mathcal{F}$ the Borel $\sigma$ algebra of $[0,1)$, and $\mathcal{L}$ the Lebesgue measure on $[0,1)$.

Step 1: construction of suitable families of subsets of $S$. For each $k\geq0$ let $(B_m^k)_{m\geq1}$ be balls with radius less than $2^{-(k+1)}$ which cover $S$ and such that $\mu(\partial B_m^k)=0$. This is possible since $\mu$ is a finite measure, hence in each point only a countable set of radii gives balls with ``fat'' boundaries. Set $$ D_1^{k} = B_1^k, \qquad D_2^k = B_2^k\setminus B_1^k, \quad \ldots, \quad D_n^k = B_n^k\setminus\bigcup_{i=1}^{n-1}B_i^k, \quad \ldots $$ and $$ S_{i_1\ldots i_k}=D_{i_1}^1\cap D_{i_2}^2\cap\ldots\cap D_{i_k}^k. $$ It can be easily seen that

• if $(i_1,\ldots, i_k)$ and $(i_1',\ldots, i_k')$ are different, then $S_{i_1\ldots i_k}$ and $S_{i_1'\ldots i_k'}$ are disjoint,
• $\bigcup_i S_i=S$ and $\bigcup_i S_{i_1\ldots i_ki}=S_{i_1\ldots i_k}$,
• the diameter of $S_{i_1\ldots i_k}$ is $\leq 2^{-k}$,
• $\mu(\partial S_{i_1\ldots i_k})=0$.

Order the $k$-tuples $S_{i_1\ldots i_k}$ by lexicographic order.

Step 2: Define a family of sub--intervals $\Delta_{i_1\ldots i_k}$ (of the type $[a,b)$ in $[0,1)$), such that $$ \mathcal{L}(\Delta_{i_1\ldots i_k}) = \mu(S_{i_1\ldots i_k}), $$ such that $\Delta_{i_1\ldots i_k}$ is on the left of $\Delta_{i_1'\ldots i_k'}$ if $(i_1,\ldots, i_k)<(i_1',\ldots,i_k')$, and such that together they cover $[0,1)$.

Step 3: Define the random variable. For each $(i_1,\ldots, i_k)$ such that $S_{i_1\ldots i_k}$ is non-empty, choose a point $x_{i_1\ldots i_k}$ in the interior of $S_{i_1\ldots i_k}$. For each $\omega\in[0,1)$ set $X_k(\omega)=x_{i_1\ldots i_k}$ if $\omega\in\Delta_{i_1\ldots i_k}$. The random variable is well defined since, if $\Delta_{i_1\ldots i_k}$ is non-empty, then the interior of $S_{i_1\ldots i_k}$ is non-empty. Observe that $d(X_k(\omega),X_{k+p}(\omega))\leq\frac1{2^k}$, hence by the completeness, the limit $X(\omega)=\lim_k X_k(\omega)$ always exists.

Step 4: $X$ has law $\mu$. Indeed $$ \mathbb{P}[X_{k+p}\in\bar S_{i_1\ldots i_k}] = \mathbb{P}[\omega:\omega\in\Delta_{i_1\ldots i_k}] = \mu(S_{i_1\ldots i_k}), $$ and, since each open set $A$ in $S$ is a disjoint countable union of sets $S_{i_1\ldots i_k}$, by Fatou, $\mu(A)\leq\liminf_k\mathbb{P}[X_k\in A]$, that is $X$ has law $\mu$.