Exercise 1, § 3

Let $\mu$ be Lebesgue measure on the interval $X = [0, 1[$ of $\mathbf{R}$. For every integer $n = 2^h + k$ ($h \geqslant 0, 0 \leqslant k < 2^h$) let $f_n$ be the function equal to 1 on the interval $[k/2^h, (k+1)/2^h[$ and to 0 elsewhere in $X$. Show that the sequence $(f_n)$ converges in mean of order $p$ to 0 for every $p > 0$, but that the sequence $(f_n(x))$ does not converge for any point $x \in X$.