traveling hump sequence

In this entry, $\lfloor \cdot \rfloor$ denotes the floor function and $m$ denotes Lebesgue measure.

For every positive integer $n$ , let $\displaystyle A_n=\left[ \frac{n-2^{\left\lfloor \log_2 n \right\rfloor}}{2^{\left\lfloor \log_2 n \right\rfloor}} , \frac{n-2^{\left\lfloor \log_2 n \right\rfloor}+1}{2^{\left\lfloor \log_2 n \right\rfloor}} \right]$ . Then every $A_n$ is a subset of $[0,1]$ (click here to see a proof) and is Lebesgue measurable (clear from the fact that each of them is closed).

For every positive integer $n$ , define $f_n \colon [0,1] \to \mathbb{R}$ by $f_n=\chi_{A_n}$ , where $\chi_S$ denotes the characteristic function of the set $S$ . The sequence $\{f_n\}$ is called the traveling hump sequence. This colorful name arises from the sequence of the graphs of these functions: A ``hump'' seems to travel from $\displaystyle \left[ 0, \frac{1}{2^k} \right]$ to $\displaystyle \left[ \frac{2^k-1}{2^k}, 1 \right]$ , then shrinks by half and starts from the very left again.

The traveling hump sequence is an important sequence for at least two reasons. It provides a counterexample for the following two statements:

• Convergence in measure implies convergence almost everywhere with respect to $m$ .
• $L^1(m)$ convergence implies convergence almost everywhere with respect to $m$ .

Note that $\{f_n\}$ is a sequence of measurable functions that does not converge pointwise. For every $x \in [0,1]$ , there exist infinitely many positive integers $a$ such that $f_a(x)=0$ , and there exist infinitely many positive integers $b$ such that $f_b(x)=1$ .

On the other hand, $\{f_n\}$ converges in measure to $0$ and converges in $L^1(m)$ to $0$ .