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 (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 $f_{n}\colon[0,1]\to\mathbb{R}$ 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 (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	2013-03-22 16:14:08
Last modified on	2013-03-22 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