Convergence in probability, To which P are they referring to? I was reading about convergence in probability and read this:

We say that $(X_n)_{n=1}^{\infty}$ converges in probability to $X$, if
for every $\epsilon>0$:
$$ \lim_{n->\infty} P(|X_n-X|>\epsilon)=0 $$

I'm a little bit confused, to what probability function $P$ are they referring to?
The one for the random variable $X$ or the one for $X_n$ Since each random variable can have its own probability function...
 A: Let's look at some basic definitions in probability theory. First we define ourself a probability space $(\Omega, \mathcal{A}, \mathbb{P}) $. We have a sequence of real random variables $ (X_n)_{n \in \mathbb{N}} $ that means for every $ n \in \mathbb{N} $ we have a measurable function $ X_n : \Omega \rightarrow \mathbb{R} $. To be precise each $X_n$ is a function from the probability space $(\Omega, \mathcal{A}, \mathbb{P})$ into the measurable space $(\mathbb{R}, \mathcal{B})$, where $\mathcal{B}$ denotes the Borel-$\sigma$-algebra. Now when we write
$$ \mathbb{P}(|X_n - X| > \epsilon) $$ we apply the probability measure to a subset of $\Omega$ that is define by the following:
$$\{ \omega \in \Omega: |X_n(\omega) - X(\omega)| > \epsilon \} \in \mathcal{A} $$ to conclude we have that
$$ \lim_{n \rightarrow \infty} \mathbb{P}(|X_n - X| > \epsilon) = \lim_{n \rightarrow \infty} \mathbb{P}(\{ \omega \in \Omega: |X_n(\omega) - X(\omega)| > \epsilon \}) $$
So it is always the same probability measure applied to different subsets.
