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Hometraveling hump sequence
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traveling hump sequence
In this entry, $\lfloor\cdot\rfloor$ denotes the floor function and $m$ denotes Lebesgue measure.
For every positive integer $n$, let $\displaystyle A_{n}=\left[\frac{n2^{{\left\lfloor\log_{2}n\right\rfloor}}}{2^% {{\left\lfloor\log_{2}n\right\rfloor}}},\frac{n2^{{\left\lfloor\log_{2}n% \right\rfloor}}+1}{2^{{\left\lfloor\log_{2}n\right\rfloor}}}\right]$. 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 $0$.
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