[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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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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230(1): 現代数学の系譜11 ガロア理論を読む 2015/04/29(水)14:45 ID:6XYDeD+q(17/25) AAS
>>229
Example
Consider the unit circle S, and the action on S by a group G consisting of all rational rotations.
Namely, these are rotations by angles which are rational multiples of π.
Here G is countable (more specifically, G is isomorphic to Q/Z) while S is uncountable.
Hence S breaks up into uncountably many orbits under G.
Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset X ⊂ S with the property that all of its translates by G are disjoint from X and from each other.
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