example of $(f_n)$ does not converge pointwise, but $f_n \rightarrow 0$ in $L^1[0, 1]$

Question

Define functions $f_n$ on [0, 1] by

$f_n=\begin{cases} 1 & x \in[k2^{-N}, (k+1)2^{-N}], n=2^{N}+k \\ 0 & otherwise \end{cases}$

show that $f_n \rightarrow 0$ in $L^1[0, 1]$, but the sequence $(f_n(x))_n$ does not converge pointwise for any $x \in [0, 1]$

Hint: Every positive integer $n$ can be written as $n=2^N+k$ for a unique choice of integres $k, N$ with $0 \leq k \leq<2^N$

How do I approach this problem?

Answer 1

firstly show the support of $f_n$ has measure $2^{-N}$, and $|f_n| \le 1$, so $||f_n||_1 \to 0$

let $S_N$ denote the set of intervals $[k2^{-N}, (k+1)2^{-N}]$. then you can show that any $x \in [0,1]$ belongs to at least $1$ and at most $2$ of these intervals. draw the appropriate conclusion for the sequence $f_n(x)$