Let $X$ be a random variable on the probability space $(\Omega,\mathcal B,P)$, with distribution $P_{X}$. Consider the random variable $\hat X$ on the probability space $(\mathbb R,\mathcal B_{\mathbb R},P_{X})$,defined by $\hat X(x)=x$ . Then $P_{\hat X}=P_{X}$.
$\mathcal B $ is $\sigma$-algebra.
thanks for help.