Equivalence of a lim sup of functions with a lim inf of sets.

Question

$\{\omega\in\Omega : \limsup_{n\to\infty}\lvert Y_n(\omega)-Y(\omega)\rvert=0\}$ =$\liminf_{n\to\infty}\{{\omega\in\Omega:\lvert Y_n(\omega)-Y(\omega)\rvert\lt \epsilon\}}\forall\epsilon>0$.

How do we turn the LHS which is a lim sup on functions to the RHS which is a lim inf on sets.

Answer 1

Remember $\limsup_{n\to\infty} \lvert x_n-x\rvert=0$ for $x_n,x\in\mathbb{R}$ means for all $\epsilon>0$, $\lvert x_n-x\rvert<\epsilon$ eventually.

So \begin{align*} &\{\omega\in\Omega:\limsup_{n\to\infty}\lvert Y_n(\omega)-Y(\omega)\rvert=0\}\\ &=\{\omega\in\Omega:(\forall\epsilon>0)(\lvert Y_n(\omega)-Y(\omega)\rvert<\epsilon\text{ eventually})\}\\ &=\bigcap_{\epsilon>0}\{\omega\in\Omega:\lvert Y_n(\omega)-Y(\omega)\rvert<\epsilon\text{ eventually}\}\\ &=\bigcap_{\epsilon>0}\liminf_{n\to\infty}\{\omega\in\Omega:\lvert Y_n(\omega)-Y(\omega)\rvert<\epsilon\}. \end{align*}