(1) Are there interesting or natural conditions on the probability space $(\Omega,\mathfrak A, \mathbb P)$ under which the push-forward map \begin{equation} \pi_\mathbb P : L^0(\Omega,\mathbb R) \longrightarrow \mathcal P(\mathbb R), X \mapsto X_\ast \mathbb P \end{equation} is surjective?
(1.1) Motivation: Under which conditions on the probability space in question does every probability distribution on $\mathfrak B(\mathbb R)$ derive from the distribution of some random variable on it?
(1.2) Notations: As usual, $L^0(\Omega,\mathbb R)$ stands for the measurable mappings from $(\Omega,\mathfrak A)$ to $(\mathbb R,\mathfrak B(\mathbb R))$, and $\mathcal P(\mathbb R)$ for the set of probability measures on the latter measurable space.
(1.3) Let be $(\Omega,\mathfrak A) = (\mathbb R,\mathfrak B(\mathbb R))$. Under which conditions upon $\mathbb P$ is $\pi_\mathbb P$ surjective?