View source for Non-measurable set

A set that is not a [[Measurable set|measurable set]]. In more detail: A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n0672001.png" /> belonging to a hereditary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n0672002.png" />-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n0672003.png" /> is non-measurable if

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here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n0672005.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n0672006.png" />-ring on which the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n0672007.png" /> is given, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n0672008.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n0672009.png" /> are the exterior and interior measures, respectively (see [[Measure|Measure]]).

For an intuitive grasp of the concept of a non-measurable set, the following  "effective constructions"  are useful.

Example 1. let

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be the unit square and define a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720011.png" /> on the sets

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720013.png" /> runs through the Lebesgue-measurable sets of measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720014.png" />, by setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720015.png" />. Then the set

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is non-measurable, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720018.png" />.

The oldest and simplest construction of a non-measurable set is due to G. Vitali (1905).

Example 2. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720019.png" /> be the set of all rational numbers. Then a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720020.png" /> (called a Vitali set) having in accordance with the [[Axiom of choice|axiom of choice]] exactly one element in common with every set of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720022.png" /> is any real number, is non-measurable. No Vitali set has the [[Baire property|Baire property]].

Example 3. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720023.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720024.png" />) be the set of numbers of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720026.png" /> is an irrational number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720028.png" /> are integers with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720029.png" /> even (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720030.png" /> odd), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720031.png" /> be a set obtained by means of the axiom of choice from the equivalence classes of the set of real numbers under the relation:

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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720033.png" />. Then for every measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720034.png" />:

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Yet another construction of a non-measurable set is based on the possibility of introducing a total order in a set having cardinality of the continuum.

Example 4. There exist a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720036.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067200/n06720038.png" /> intersect every uncountable closed set. Any such set (a Bernstein set) is non-measurable (and does not have the Baire property). In particular, any set of positive exterior measure contains a non-measurable set.

Apart from invariance under a shift (Example 2) and topological properties (Example 3) there are also reasons of a set-theoretical character why it is impossible to define a non-trivial measure for all subsets of a given set; such is, for example, Ulam's theorem (see [[#References|[2]]]) for sets of bounded cardinality.

No specific example is known of a Lebesgue non-measurable set that can be constructed without the use of the axiom of choice.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.R. Halmos,   "Measure theory" , v. Nostrand  (1950)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.C. Oxtoby,   "Measure and category" , Springer  (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.R. Gelbaum,   J.M.H. Olmsted,   "Counterexamples in analysis" , Holden-Day  (1964)</TD></TR></table>



====Comments====
See also [[Measure|Measure]]; [[Gödel constructive set|Gödel constructive set]]; [[Descriptive set theory|Descriptive set theory]].

In certain models of ZF (not ZFC) every set of real numbers is Lebesgue measurable (a result of Solovay), so the axiom of choice is needed to construct a Lebesgue non-measurable set.

For Ulam's theorem see [[Cardinal number|Cardinal number]].