Non-measurable set
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In
The notion of a non-measurable set has been a source of great controversy since its introduction. Historically, this led
In 1970, Robert M. Solovay constructed the Solovay model, which shows that it is consistent with standard set theory without uncountable choice, that all subsets of the reals are measurable. However, Solovay's result depends on the existence of an inaccessible cardinal, whose existence and consistency cannot be proved within standard set theory.
Historical constructions
The first indication that there might be a problem in defining length for an arbitrary set came from Vitali's theorem.[1] A more recent combinatorial construction which is similar to the construction by Robin Thomas of a non-Lebesgue measurable set with some additional properties appeared in American Mathematical Monthly.[2]
One would expect the measure of the union of two disjoint sets to be the sum of the measure of the two sets. A measure with this natural property is called finitely additive. While a finitely additive measure is sufficient for most intuition of area, and is analogous to
In this respect, the plane is similar to the line; there is a finitely additive measure, extending Lebesgue measure, which is invariant under all
Example
Consider the set of all points in the unit circle, and the
Consistent definitions of measure and probability
The Banach–Tarski paradox shows that there is no way to define volume in three dimensions unless one of the following five concessions is made:[citation needed]
- The volume of a set might change when it is rotated.
- The volume of the union of two disjoint sets might be different from the sum of their volumes.
- Some sets might be tagged "non-measurable", and one would need to check whether a set is "measurable" before talking about its volume.
- The axioms of ZFC (Zermelo–Fraenkel set theory with the axiom of choice) might have to be altered.
- The volume of is or .
Standard measure theory takes the third option.[
In 1970,
The axiom of choice is equivalent to a fundamental result of
See also
- Banach–Tarski paradox – Geometric theorem
- Carathéodory's criterion – necessary and sufficient condition for a measurable set
- Hausdorff paradox
- Measure (mathematics) – Generalization of mass, length, area and volume
- Non-Borel set– Class of mathematical sets
- Outer measure – Mathematical function
- Vitali set – Set of real numbers that is not Lebesgue measurable
References
Notes
Bibliography
- Dewdney, A. K. (1989). "A matter fabricator provides matter for thought". Scientific American (April): 116–119. .