It is a basic fact in probability theory that, for every random variable
$$
X: (\Omega, \mathcal{F}, \mathbb{P}) \to (\mathbb{R}, \mathcal{B}),
$$
we have an associated measure $\mathbb{P}_X$ on the borelians of $\mathbb{R}$ given by
$$
\mathbb{P}_X(B) := \mathbb{P}(X \in B) = \mathbb{P}(X^{-1}(B)), \forall B \in \mathbb{B}.
$$
$\mathbb{P}_X$ is called the measure induced by $X$.
My question: is the opposite also true? That is, given a probability measure on the borelians of $\mathbb{R}$, is it induced by a random variable? More precisely:
Is there a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ such that, given a probability measure $\mu$ in $(\mathbb{R}, \mathcal{B})$, one can find a random variable $X: (\Omega, \mathcal{F}, \mathbb{P}) \to (\mathbb{R}, \mathcal{B})$ such that $\mu$ is induced by $X$?
I aprreciate your concern!
[Edit]
What I really want is a probability space that can be used for global representation of probability measures as measures induced by random variables. I have a guess that the space $\Omega = \{f: \mathbb{R} \to \mathbb{R} | f \:\text{is Borel measurable}\}$ with the $\sigma$-algebra of cilinders and some appropriate measure is the space desired, but I'm not pretty sure on how to prove it.
Best Answer
Yes, trivially: consider the probability space $\Omega = \mathbb{R}$, $\mathcal{F} = \mathcal{B}$, $\mathbb{P} = \mu$, and let $X : \Omega = \mathbb{R} \to \mathbb{R}$ be the identity map $X(\omega) = \omega$.
Alternatively, use inverse transform sampling. Let $F(x) = \mu((-\infty, x])$ be the cumulative distribution function of $\mu$, and let $G(t) = \inf\{x : F(x) \ge t\}$ be its "inverse". (If $F$ is actually 1-1 then $G$ is truly the inverse of $F$.) Now let $U$ be a Uniform(0,1) random variable on any probability space $(\Omega, \mathcal{F}, \mathbb{P})$ (e.g. the identity map on $[0,1]$ with Lebesgue measure) and set $X = G(U)$. It is then easy to see that the measure induced by $X$ is again $\mu$. Proof: we have $\mathbb{P}(X \le x) = \mathbb{P}(G(U) \le x) = \mathbb{P}(U \le F(x)) = F(x)$. So the measure induced by $X$ has $F$ as its cdf, and this uniquely determines it as being $\mu$.