When I do my work, I come across a problem. Here is it.
Let $(f_n)_{n \in \mathbb{N}}$ be a sequence of measurable functions on a set $E$. Suppose there exists $M>0$ such that for all $n \in \mathbb{N}$, we have
$$\int_E |f_n| \le M.$$
My question is "Does this imply that the sequence $f_n$ is uniformly bounded on $E$, which means there exists some $C>0$ such that $|f_n| \le C$ on $E$"?
Any advice is highly appreciated.
Best Answer
Take $E=(0,1), f_n=n\chi_{(0,\frac 1 n)}, M=1$.