Weak convergence on L^p

Question

Let Let $X=[0,1]$ with the Lebesgue measure, find a sequence $\{f_n\}$ of measurable functions $f_n:X \rightarrow{ \mathbb{R} } $ such that:

1. $f_n(x)\rightarrow{0}$ almost everywhere $x∈[0,1]$

2. $f_n$ converge to $0$ in measure.

3. $f_n$ not converge weakly to $0$ in $L^p([0,1])$ for any $p$, $1≤p< \infty$

Answer 1

Let $$ f_n(x)= \begin{cases} n^2&\text{for $0\leq x\leq\frac1n$},\\ 0&\text{for $\frac1n< x\leq1$}. \end{cases} $$ The sequence is unbounded, hence not weakly convergent, but convergent in the sense of 1 and 2.