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Let $F:\mathbb{R} \rightarrow \mathbb{R}$ be a distribution function, i.e., it is monotonically increasing and $F(-\infty)=0,F(+\infty)=1$ and let $P$ be a probability measure on some space $\Omega$. Then is there some random variable $X$ on $\Omega$ such that $F$ is the distribution function associated with $X$, i.e., $F(\lambda) = P(X \le \lambda)$.

Remarks. I know that if $P$ is the Lebesgue measure on $(0,1)$, then we could just define $X(\omega) = \sup \{\lambda\in \mathbb{R}):F(\lambda) < \omega\}$, but does it also hold for general spaces $\Omega$?

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Certainly not. If $\Omega$ has single point then any random variable on it is a constant. You need at least a nice space $\Omega$ and a non-atomic probability measure $P$ to be able to find such random variables.

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