There must exist a random variable with certain given law?

Question

Let $(\Omega,\mathcal F,\mathbb P)$ and $(E,\mathcal G,\mu)$ be two probability spaces. My question is the following:

Measure-theoretically, is there exist a measurable mapping $X:(\Omega,\mathcal F)\to(E,\mathcal G)$, such that $\mu$ is just the push-forward measure of $\mathbb P$ w.r.t $X$, i.e., $\mu(A)=\mathbb P(X^{-1}(A))$ for $\forall A\in\mathcal G$.

Or equivalently in probability, is there exist a random variable $X:(\Omega,\mathcal F)\to(E,\mathcal G)$, such that $\mu$ is just the law of $X$.

For stochastic processes, there is the well-known Kolmogorov extension theorem to guarantee the existence of stochastic processes for given finite-dimensional distributions. But for random variables, is there some theorem to guarantee the existence for given law?

Any comments or references will be appreciated.

Answer 1

In general no, but if they are both Polish spaces and the cardinality (either finite, countable or continuous) of $\Omega$ is greater than that of $E$, each endowed with their Borel $\sigma$-algebra and a probability measure, then yes (Kuratowski's theorem)

Answer 2

Consider cases where $\Omega$ is finite and $\mu$ takes too many values...