Set of points at which sequence of measurable functions converge (another approach) Question is to prove that : 
Set of all points at which a sequence of measurable functions converge is a measurable set..
What i have tried is as follows :
We are looking at the following set : 
$$\{x\in X : (f_n(x)) \text{converges}\}$$
Which is same as 
$$\{x\in X : (f_n(x)) \text{is cauchy}\}$$
Which is same as 
$$\{x\in X : \text{given $\epsilon>0$ there exists $N\in \mathbb{N}$ such that $|f_n(x)-f_m(x)|<\epsilon$ for all $m,n\geq N$}\}$$
I want to write this as unions and intersections of measurable sets and then conlcude this is measurable.. Something like :
$$\bigcap_{p\in \mathbb{R}??}\bigcup_{m,n\in \mathbb{N}??}\{x:|(f_n-f_m)(x)|<p\}$$
Now, As $f_n,f_m$ are measurable so is $f_n-f_m$ and so is $|f_n-f_m|$ and  $\{x:|(f_n-f_m)(x)|<p\}$ being inverse image of open set is measurable..
I am not so sure how to write that as unions and intersections...
 A: $(f_n(x))_n$ is cauchy means that for any positive integer $p$, there exists one integer $k$, such that for all $m > k$ and $n > k$ we have $|f_n(x) - f_m(x)| < \frac{1}{p}$.
Since it's for any positive integer p, we should have something like $\cap_{p=1}^{\infty}$.
And there exists a $k$ such that blabla means $\cup_{k=1}^{\infty}$ blabla..., i.e. for one k in $\{1,2, \cdots\}$, blabla is ok
For all $m,n$ greater than $k$ is $\cap_{m > k, n>k}$.
So finally $(f_n(x))_n$ is cauchy means that $x$ is in the set
$\cap_{p=1}^{\infty} \cup_{k=1}^{\infty}\cap_{m > k, n>k}\{x: |f_m(x) - f_n(x)| < \frac{1}{p}\}$
To resume, when there is "for any, for all", use intersection; when there is "exists", use union
A: The idea is good. But use a different definition of continuity which is in fact equivalent: Namely
$$\{x\in X : \text{given $\mathbb N\ni k>0$ there exists $N\in \mathbb{N}$ such that $|f_n(x)-f_m(x)|<\frac{1}{k}$ for all $m,n\geq N$}\}$$
Then write this as
$$\bigcap_{k\in \mathbb{N}}\bigcup_{N\in\mathbb N}\bigcap_{m,n\geqslant N}\left\{ x:\left|(f_n-f_m)(x)\right|<\frac{1}{k}\right\}$$
As you said the set  $\left\{ x:\left|(f_n-f_m)(x)\right|<\frac{1}{k}\right\}$ is measurable and hence you're done, since you have a countable union respectively intersection of measurable sets. This is why you use the different definition of continuity.
