I have read in some book that in case of almost sure convergence of a sequence of random variable it possible that $|X_{n}(\omega) - X(\omega)|$ can be extremely large for $\omega$ in a small probability set. How is that possible ? Here the sequence {$X_{n}$} of random variables converges to $X$ in almost sure sense.
By almost sure convergence we mean that $P(\omega : X_{n}(\omega)=X(\omega)$ as $n$->infinity)=1 i.e.$P(\omega$ : for any $\epsilon > 0 $there exist an $N$ such that for all $n>=N$ $|X_{n}(\omega)-X(\omega)| < \epsilon)$ = 1 . Then how can $|X_{n}(\omega) - X(\omega)|$ be extremely large for $\omega$ in a small probability set where $|X_{n}(\omega) - X(\omega)| < \epsilon$ with probability 1 ? confused.