in my homework assignment, I am trying to show the following:
Show that if $f_n \ge 0 $ and $f_n \in L$ and $\int {f_n} \rightarrow 0 $ this does not imply $f_n \rightarrow 0 $ almost everywhere
now I am trying to think what could produce such a sequence of functions, but dont you need the fact that the functions could be negative? else you can only have nonzero vaues countable times. But clearly I am misunderstanding something very crucial.
Any help would be appreciated!
Standard example: define $I_1 = [0,1]$, $I_2 = [0,1/2]$, $I_3 = [1/2, 1]$, $I_4 = [0,1/4]$, $I_5 = [1/4,1/2]$, $I_6 = [1/2,3/4]$, $I_7 = [3/4,1]$, etc. Hopefully the pattern is clear. Let $f_n$ be the characteristic function of $I_n$. Then the integral of $f_1$ is $1$, the integrals of $f_2$ and $f_3$ are $1/2$, the integrals of $f_4$ through $f_7$ are $1/4$, etc. Clearly we have $\int f_n \to 0$. But $(f_n)$ does not converge at any point in $[0,1]$.