In mathematics, a non-measurable set is a set whose structure is so complicated that it cannot be assigned any meaningful measure. The mathematical existence of such sets is construed to shed light on the notions of length, area and volume in formal set theory.
The notion of a non-measurable set has been a source of great controversy since its introduction. Intuition suggests to many people that any subset S of the unit disk (or unit line) should have a measure, because one can throw darts at the disk (see Freiling's axiom of symmetry), and the probability of landing in S is the measure of the set.
Historically, this led Borel and Kolmogorov to formulate probability theory on sets which are constrained to be measurable. The measurable sets on the line are iterated countable unions and intersections of intervals (called Borel sets) plus-minus null sets. These sets are rich enough to include every conceivable definition of a set that arises in standard mathematics, but they require a lot of formalism to prove that sets are measurable.
In 1970, Solovay constructed Solovay's model, which shows that it is consistent with standard set theory, excluding uncountable choice, that all subsets of the reals are measurable.
Read more about Non-measurable Set: Historical Constructions, Example, Consistent Definitions of Measure and Probability
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