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

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.

• 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 $(\mathcal{X}, d)$.