traveling hump sequence
For every positive integer , let . Then every is a subset of (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 , define by , where denotes the characteristic function of the set . The sequence 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 to , 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 .
convergence (http://planetmath.org/L1muConvergence) implies convergence almost everywhere with respect to .
Note that is a sequence of measurable functions that does not converge pointwise (http://planetmath.org/PointwiseConvergence). For every , there exist infinitely many positive integers such that , and there exist infinitely many positive integers such that .
On the other hand, converges in measure to and converges in (http://planetmath.org/ConvergesInL1mu) to .
|Title||traveling hump sequence|
|Date of creation||2013-03-22 16:14:08|
|Last modified on||2013-03-22 16:14:08|
|Last modified by||Wkbj79 (1863)|