Which random variables are representable via the transformation of a $[0,1]$-valued uniform random variable?

Let me be more specific about what I'm looking for, reformulating the problem in the language of measure theory.

Suppose $(\mathcal{X},\mathcal{F})$ is a measurable space. Let $\mathcal{B}$ be the family of Borel subsets of $[0,1]$ and let $\mu_L \colon \mathcal{B} \to [0,1]$ be the Lebesgue measure.

Are there any fairly general sufficient conditions on the measurable space $(\mathcal{X},\mathcal{F})$ which guarantee that for every probability measure $\mu \colon \mathcal{F} \to [0,1]$ there exists a measurable function $\varphi$ from $\big([0,1],\mathcal{B}\big)$ to $(\mathcal{X},\mathcal{F})$ for which $$\forall F \in \mathcal{F}\;, \qquad \mu[F] = \mu_L\big[\varphi^{-1}(F)\big] \;?$$

(here, $\varphi^{-1}(F):= \{ u \in [0,1] \mid \varphi(u)\in F\}$).

The idea is then that, if $(\Omega,\mathcal G,\mathbb{P})$ is a probability space, then whatever random variable $X\colon \Omega \to \mathcal{X}$ we may choose, the distribution $\mathbb{P}_X$ of $X$ is the same as the distribution $\mathbb{P}_{\varphi(U)}$ of $\varphi(U)$ for some measurable map $\varphi \colon [0,1] \to \mathcal{X}$, where $U\colon \Omega \to [0,1]$ is any uniform $[0,1]$-valued random variable.

I'm aware that the result is true for $\mathcal{X} = \mathbb{R}$ and $\mathcal{F} = \mathcal{B}_\mathbb{R}$, where $\mathcal{B}_\mathbb{R}$ is the family of Borel subsets of $\mathbb{R}$, but I'm wondering if there are general enough conditions holding in basically every practical case (maybe something like being $(\mathcal{X},\mathcal{F})$ be the Borel measurable space generated by a nicely behaved metric, for example) under which the result holds.

References are very welcome.


1 Answer 1


The property you seek is called a "standard probability space." The wiki article [1] is a good starting point. Quoting from that source:

"A (complete) probability space is standard, if it is isomorphic mod sets of measure zero to an interval with Lebesgue measure, a finite or countable set of atoms, or a combination (disjoint union) of both."

"Every probability distribution on a polish space turns it into a standard probability space. (Here, a probability distribution means a probability measure defined initially on the Borel sigma-algebra and completed.) See (Rokhlin 1952, Sect. 2.7 (p. 24)), (Haezendonck 1973, Example 1 (p. 248)), (de la Rue 1993, Theorem 2-3), and (Itô 1984, Theorem 2.4.1)."

[1] https://en.wikipedia.org/wiki/Standard_probability_space

  • $\begingroup$ Just a comment to point to a precise reference in the literature for those who need it: a sufficient condition, which is a simple consequence of Theorem A.11 in Theory of Operator Algebras I by Takesaki Masamichi, is that $(\mathcal{X}, \mathcal{F})$ is the Borel space associated with a complete and separable metric space $(\mathcal{X}, d)$. $\endgroup$
    – Bob
    Commented Dec 23, 2022 at 15:17

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