Pontwise convergence of indicator functions is the same as the convergence of their sets.

Question

I'm studying probability theory, and I'm trying to proove the following proposition:

Limits od indicator functions

$A_n \rightarrow A \Leftrightarrow I_{A_n}(w) \rightarrow I_{A}(w)$ , for all $w$.

Thus the convergence of sets is the same as pointwise convergence of their indicator functions

For this side $\Rightarrow $ I was able to do it, by proving that:

$ \limsup I_{A_n} = I_{\limsup A_n}$

But I haven't been able to prove the other way ($ \Leftarrow$ ).

Answer 1

Use the fact that $$ 1_{\liminf A_n}=\liminf_{n\to\infty} 1_{A_n}=1_A=\limsup_{n\to\infty}1_{A_n}=1_{\limsup A_m}. $$