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 $0$ .

单词	TravelingHumpSequence
释义	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 $0$ .
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