19/05/27 07:45:42.74 gJBhsSTX.net
前 ｽﾚﾘﾝｸ(math板:984番)
>>「塵積もれば山となる」がConglomerabilityで
>>その否定、「０の塵が積もれば、山でなく、やっぱり０となる」が、
>>”non-conglomerability”の辞書的説明だろう
>
>逆だろｗ
（前スレ>>751より）
The reason we do not have countable additivity differs depending on whether the probability of particular ticket winning is zero or infinitesimal.
If the probability is exactly zero, then we lack countable additivity because
1 = P(E1 ∨ E2 ∨・・・) if En is the probability of ticket n being picked (it’s certain that some ticket or other is picked)
whereas P(E1) + P(E2) +・・・ = 0 + 0 + ・・・ = 0.
If on the other hand, P( En ) = α for some (positive) infinitesimalsα,
The standard systems for construction of infinitesimal do not in general define a countable in finite sum of infinitesimals, at least in our case where the summands are the same. Thus, the required equation P(E1 ∨ E2 ∨・・・ ) =P(E1 ) + P(E2)+・・・ does not hold,
since although the left .hand side is defined, the right-hand side is not.
In our infinite fair lottery case, we can intuitively see why we shouldn't be able to have a meaningful sum.
For consider our infinite sum:
α +α+α+α+ ・・・
= (α +α) +(α +α) +・・・
=2α+2α+ ・・・
=2(α+α+・・・ ).
If the value of this sum is x, then x =2x, But if x is not zero, then we can divide both side, by x to yield 1 = 2, and so x must be zero. However, x cannot be zero since it must be at least as big as α, and hence a contradiction follows, from the assumption that the sum has a value.
The lack of countable additivity in the case of an infinite lottery is responsible for a phenomenon known as non-conglomerability.
(引用終り)
”The lack of countable additivity in the case of an infinite lottery is responsible for a phenomenon known as non-conglomerability.”です
以上