A well known fact in probability is that a uniform random variable on $[0,1]$ can be used to simulate any other probability distribution on $\mathbb{R}$.

A standard way of doing this is to define, given $\mu$ a probability on $\mathbb{R}$, the random variable $F(u,\mu) = \min\lbrace x \in \mathbb{R}: \mu((-\infty,x]) \ge u\rbrace$. The random variable $F(u,\mu)$ has distribution $\mu$.

Suppose now that $(X,d)$ is a compact metric space and let $\mathcal{P}(X)$ be the space of Borel probability measures on $X$ endowed with the topology of weak convergence.

I'm looking for a reference for the following statement:

There exists a Borel function $F:[0,1]\times \mathcal{P}(X) \to X$ such that if $u$ is a random variable whose distribution is uniform on $[0,1]$ then for each $\mu$ the random variable $F(u,\mu)$ has distribution $\mu$.

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    $\begingroup$ I haven't seen this written down; we might consider it "folklore". But you have basically already written down the proof on $\mathbb{R}$ (just take the inf over rationals to make it clear that $F$ is Borel), and for any other compact metric space $X$, there is a Borel isomorphism from $X$ to $\mathbb{R}$, so the general case follows at once. $\endgroup$ Aug 27, 2016 at 16:10

1 Answer 1


This is a result of Blackwell and Dubins, "An extension of Skorohod's almost sure representation theorem". In fact, your function F can be constructed to be almost surely continuous in the measure argument, and X can be any Polish space.

  • $\begingroup$ According to the comment guidelines I'm supposed to avoid saying thanks. But thanks! $\endgroup$ Aug 27, 2016 at 17:05

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