Non-Measurable Set
A Non-Measurable Set is a Set which can not be assigned a measurable size.
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- Counter-Example(s):
- See: Solovay's Model, Mathematics, Mathematical Existence, Length, Area, Volume, Émile Borel, Kolmogorov, Borel Set, Null Set, Robert M. Solovay.
References
2019
- (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Non-measurable_set Retrieved:2019-2-24.
- In mathematics, a non-measurable set is a set which cannot be assigned a meaningful "size". The mathematical existence of such sets is construed to provide information about 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. 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.
- In mathematics, a non-measurable set is a set which cannot be assigned a meaningful "size". The mathematical existence of such sets is construed to provide information about the notions of length, area and volume in formal set theory.