I'm thinking about the basic types of convergence for sequences of functions: convergence in measure, almost uniform convergence, convergence in Lp and point wise almost everywhere convergence. I'm looking for examples of sequences of functions that converge in one or more of these ways, but fail for others. I keep seeing the same examples over and over and I'd like to think about some new ones. Here are the examples I've seen:
$f_n=\chi_{[n, n+1]}$
$f_n=\chi_{A_{n}}$ where $A_1 = [0,1]$, $A_2 = [0,1/2]$, $A_3 = [1/2,1]$, $A_4 = [0,1/4]$, $A_5 = [1/4,1/2]$, $A_6 = [1/2,3/4]$, $A_7 = [3/4,1]$, $A_8 = [0,1/8]$ …
$f_n= n\chi_{[1/n,2/n]}$
These are a great set of examples since they let you give a counterexample for the relations between the types of convergence when needed, but I would like to know of some more.
(EDIT: I added a bit to the 2nd example to make it more clear.)
Best Answer
The sequence $(f_n)$ given by $f_n=n^{-1}\chi_{[-n,n]}$ converges uniformly, but not in $L^1$.