a Lebesgue measurable but non-Borel set
We give an example of a subset of the real numbers which is Lebesgue measurable, but not Borel measurable.
Let S consist of the set of all irrational real numbers with continued fraction of the form
x=a0+1a1+1a2+1⋱ |
such that there exists an infinite sequence
0<i1<i2<⋯ where each aik divides aik+1.
It can be shown that this set is Lebesgue measurable, but not Borel measurable.
In fact, it can be shown that S is an analytic set (http://planetmath.org/AnalyticSet2), meaning that it is the image of a continuous function
f:X→ℝ for some Polish space
X and, consequently, S is a universally measurable set.
This example is due to Lusin (1927).
Title | a Lebesgue measurable but non-Borel set |
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Canonical name | ALebesgueMeasurableButNonBorelSet |
Date of creation | 2013-03-22 18:37:01 |
Last modified on | 2013-03-22 18:37:01 |
Owner | gel (22282) |
Last modified by | gel (22282) |
Numerical id | 6 |
Author | gel (22282) |
Entry type | Theorem |
Classification | msc 28A05 |
Classification | msc 28A20 |