In the book "continuous martingales and brownian motion" of Yor-Revuz, page 16 theorem 1.2 says :
Given a probability measure $\mu$ on $\mathbb R$, there exist a probability space $(\Omega ,\mathcal F,\mathbb P)$ and a sequence of independent r.v. $X_n$ defined on $\Omega $ s.t. $X_n(\mathbb P)=\mu$.
1) Could someone explain me what $X_n(\mathbb P)$ mean ?
2) Could someone explain me what this result/theorem mean ?
The distribution of $X_n$ under $\mathbb P$. Remember $X_n$ will be a function from $\Omega$ to $\mathbb R$, so it ``pushes'' the measure $\mathbb P$ forward to the measure $\nu = X_n(\mathbb P)$ defined by $\nu(A) = \mathbb P( \{\omega\in\Omega: X_n(\omega)\in A\})$; what the book claims is that $X_n$ can be picked so this image measure $\nu$ is equal to the specified $\mu$, for all $n$. This is described in the Wikipedia article with a slightly different notation with subscripted stars, as if Y&R had written ${(X_n)}_*(\mathbb P)$ or something.