Convergence of the characteristic function Check that $$A_n \rightarrow A \iff 1_{A_n} \rightarrow 1_A$$ pointwise.
Is my proof correct?



Proof so far:
$A_n \rightarrow A \iff \limsup_{n \rightarrow \infty} A_n=\liminf_{n \rightarrow \infty}A_n=A$. 
$\iff \limsup_{n \rightarrow \infty} 1_{A_n}=1_{\limsup_{n \rightarrow \infty} A_n}=1_A= 1_{\liminf_{n \rightarrow \infty} A_n}=\liminf
_{n \rightarrow \infty} 1_{A_n} \iff \lim_{n \rightarrow \infty}1_{A_n}=1_A \iff 1_{A_n} \rightarrow 1_A$ .
I'm not sure the problem makes sensea. It seems like you are asked to prove:


  Given a probability space $(\Omega, \mathscr F, \mathbb P)$, let $A_1, A_2, ...$ be events.
  
  
  If $\limsup A_n = \liminf A_n := A$ and $\limsup 1_{A_n} = \liminf 1_{A_n}$, then $\lim 1_{A_n} = 1_A$
  If $\lim 1_{A_n} = 1_A$, then $\limsup A_n = \liminf A_n := A$
  


If we change all your $\iff$ to $\rightarrow$, you've proven #1, but...

What does #2 even mean? By definition $\limsup A_n = \liminf A_n := A$.

I think #2 should be restated:


  If $1_{A_n} \to 1_{\limsup A_n}$ and $1_{A_n} \to 1_{\liminf A_n}$,
  then $\limsup A_n = \liminf A_n$


which I'm not even sure is true:

We have $\lim 1_{A_n} = 1_{\limsup A_n} = 1_{\liminf A_n}$

$\to 1_{\limsup A_n} = 1_{\liminf A_n}$

$\to \limsup A_n = \liminf A_n?$

I think we merely have $\limsup A_n = \liminf A_n$ a.s.
I'm not sure the problem makes sensea. It seems like you are asked to prove:


  Given a probability space $(\Omega, \mathscr F, \mathbb P)$, let $A_1, A_2, ...$ be events.
  
  
  If $\limsup A_n = \liminf A_n := A$ and $\limsup 1_{A_n} = \liminf 1_{A_n}$, then $\lim 1_{A_n} = 1_A$
  If $\lim 1_{A_n} = 1_A$, then $\limsup A_n = \liminf A_n := A$
  


If we change all your $\iff$ to $\rightarrow$, you've proven #1, but...

What does #2 even mean? By definition $\limsup A_n = \liminf A_n := A$.

I think #2 should be restated:


  If $1_{A_n} \to 1_{\limsup A_n}$ and $1_{A_n} \to 1_{\liminf A_n}$,
  then $\limsup A_n = \liminf A_n$


which I'm not even sure is true:

We have $\lim 1_{A_n} = 1_{\limsup A_n} = 1_{\liminf A_n}$

$\to 1_{\limsup A_n} = 1_{\liminf A_n}$

$\to \limsup A_n = \liminf A_n?$

I think we merely have $\limsup A_n = \liminf A_n$ a.s.