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…
Best Answer
$(f_n(x))_n$ 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