On one paper I saw the function:
$$f_n=n_{\chi[0,1/n]}$$
What is this function?
I read that it's $n$ multiplied by the characteristic function on set $[0,1/n]$. But since $f_n$ in this case is supposed to defined for
$f \in L^1([0,1])$ s.t. $\int_0^1 f(t)dt=1$
then I don't see how $f_n$ necessarily stays bounded inside this.
Wait, there's a misunderstanding here!
Let: $$f_n:=n\chi_{[0,1/n]}$$ and also let: $$f:=\lim_{n\to\infty}f_n$$ Then, it is clear that: $$I_n:=\int_0^1f_nd\lambda=\int_0^{1/n}ndt=1$$ for every $n\in\mathbb{N}$. So, every $f_n$ seperately is bounded.
However, the sequence $\{f_n\}$ is not uniformly bounded!. Moreover, $$f(x)=\left\{\begin{array}{ll} \infty & x=0\\ 0 & x\neq0 \end{array}\right.$$ which is clearly not bounded.
Also note that: $$\int fd\lambda=0$$