[過去ﾛｸﾞ] 現代数学の系譜11 ガロア理論を読む13 [転載禁止]©2ch.net (654ﾚｽ)
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228(2): 現代数学の系譜11 ガロア理論を読む 2015/04/29(水)14:37 ID:6XYDeD+q(15/25) AAS
>>227
つづき
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 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. 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.
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229(1): 現代数学の系譜11 ガロア理論を読む 2015/04/29(水)14:39 ID:6XYDeD+q(16/25) AAS
>>228
Historical constructions
The first indication that there might be a problem in defining length for an arbitrary set came from Vitali's theorem.[1]
When you form the union of two disjoint sets, one would expect the measure of the result 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 Riemann integration,
it is considered insufficient for probability, because conventional modern treatments of sequences of events or random variables demand countable additivity.
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233: 現代数学の系譜11 ガロア理論を読む 2015/04/29(水)15:19 ID:6XYDeD+q(20/25) AAS
R. Solovay, Ann. of Math., 92 (1970), 1?56 >>220
In 1970, Solovay constructed Solovay's model, >>228
Robert M. Solovay
外部ﾘﾝｸ:en.wikipedia.org
Robert Martin Solovay (born December 15, 1938) is an American mathematician specializing in set theory.
Solovay earned his Ph.D. from the University of Chicago in 1964 under the direction of Saunders Mac Lane, with a dissertation on A Functorial Form of the Differentiable Riemann?Roch theorem.
Solovay has spent his career at the University of California at Berkeley, where his Ph.D. students include W. Hugh Woodin and Matthew Foreman.
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