For every probability $\mu$ on $(\Bbb R,\mathcal{B}(\Bbb R))$ exists at least a real r.v. $X$ s.t. $P^X=\mu$ Given a probability $\mu$ on $(\Bbb R,\mathcal{B}(\Bbb R))$, does exist always some random real-valued variable $X$ (defined on some probability space $(\Omega,\mathcal{A},P)$) such that its distribution $P^X$ be the same probability on $(\Bbb R,\mathcal{B}(\Bbb R))$ as $\mu$?
I think yes, and I think this should be a general result, but I never read it anywhere!
Many thanks!

The answer is: "yes".
Take $\Omega=\mathbb R$, $\mathcal A=\mathcal B(\mathbb R)$ and $P=\mu$. 
For $X$ you can take the identity prescribed by $\omega\mapsto\omega$.
Blandness and elegance hand in hand.
