traveling hump sequence
In this entry, $\lfloor \cdot \rfloor $ denotes the floor function and $m$ denotes Lebesgue measure^{}.
For every positive integer $n$, let ${A}_{n}=[{\displaystyle \frac{n{2}^{\lfloor {\mathrm{log}}_{2}n\rfloor}}{{2}^{\lfloor {\mathrm{log}}_{2}n\rfloor}}},{\displaystyle \frac{n{2}^{\lfloor {\mathrm{log}}_{2}n\rfloor}+1}{{2}^{\lfloor {\mathrm{log}}_{2}n\rfloor}}}]$. Then every ${A}_{n}$ is a subset of $[0,1]$ (click here (http://planetmath.org/RegardingTheSetsA_nFromTheTravelingHumpSequence) to see a proof) and is Lebesgue measurable (clear from the fact that each of them is closed (http://planetmath.org/Closed)).
For every positive integer $n$, define ${f}_{n}:[0,1]\to \mathbb{R}$ 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 $[0,{\displaystyle \frac{1}{{2}^{k}}}]$ to $[{\displaystyle \frac{{2}^{k}1}{{2}^{k}}},1]$, 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:

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Convergence in measure^{} implies convergence almost everywhere with respect to $m$.

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${L}^{1}(m)$ convergence (http://planetmath.org/L1muConvergence) implies convergence almost everywhere with respect to $m$.
Note that $\{{f}_{n}\}$ is a sequence of measurable functions^{} that does not converge pointwise (http://planetmath.org/PointwiseConvergence). 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 ${L}^{1}(m)$ (http://planetmath.org/ConvergesInL1mu) to $0$.
Title  traveling hump sequence 

Canonical name  TravelingHumpSequence 
Date of creation  20130322 16:14:08 
Last modified on  20130322 16:14:08 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  14 
Author  Wkbj79 (1863) 
Entry type  Definition 
Classification  msc 28A20 
Related topic  ModesOfConvergenceOfSequencesOfMeasurableFunctions 