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单词 TravelingHumpSequence
释义

traveling hump sequence


In this entry, denotes the floor function and m denotes Lebesgue measureMathworldPlanetmath.

For every positive integer n, let An=[n-2log2n2log2n,n-2log2n+12log2n]. Then every An 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 fn:[0,1] by fn=χAn, where χS denotes the characteristic functionMathworldPlanetmathPlanetmathPlanetmathPlanetmath of the set S. The sequence {fn} 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 [0,12k] to [2k-12k,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:

  • Convergence in measurePlanetmathPlanetmath implies convergence almost everywhere with respect to m.

  • L1(m) convergence (http://planetmath.org/L1muConvergence) implies convergence almost everywhere with respect to m.

Note that {fn} is a sequence of measurable functionsMathworldPlanetmath that does not converge pointwise (http://planetmath.org/PointwiseConvergence). For every x[0,1], there exist infinitely many positive integers a such that fa(x)=0, and there exist infinitely many positive integers b such that fb(x)=1.

On the other hand, {fn} converges in measure to 0 and converges in L1(m) (http://planetmath.org/ConvergesInL1mu) to 0.

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更新时间:2025/5/4 22:15:09