>>414-415
(引用開始)
Alexander Pruss本人のBlogが見つかった
https://alexanderpru...onglomerability.html
Alex Pruss’s Blog 2024/9/11 ”Independence conglomerability”
冒頭に定義が書いてある
Conglomerability says that if you have an event E and a partition {Ri : i ∈ I} of the probability space, then if P(E∣Ri) ≥ λ for all i, we likewise have P(E) ≥ λ.
Conglomerabilityとは、ある事象Eと確率空間の分割{Ri:i∈I} があるとき、
すべてのi に対してP(E∣Ri) ≥λならば、同様にP(E) ≥λ が成り立つというものである。
(引用終り)

更にPruss氏引用
・Absence of conglomerability leads to a variety of paradoxes, but in various infinitary contexts, it is necessary to abandon conglomerability.
・(e.g., because P(J=i) = 0 and we don’t have a way of conditionalizing on zero probability events)
・Example: I am going to uniformly randomly choose a positive integer (using a countably infinite fair lottery, assuming for the sake of argument such is possible). For each positive integer n, you have a game available to you: the game is one you win if n is no less than the number I am going to pick. You despair: there is no way for you to have any chance to win, because whatever positive integer n you choose, I am infinitely more likely to get a number bigger than n than a number less than or equal to n, so the chance of you winning is zero or infinitesimal regardless which game you pick. But then you have a brilliant idea. If instead of you choosing a specific number, you independently uniformly choose a positive integer n, the probability of you winning will be at least 1/2 by symmetry. Thus a situation with two independent countably infinite fair lotteries and a symmetry constraint that probabilities don’t change when you swap the lotteries with each other violates independence conglomerability.

要点
・Conglomerability or Independence conglomerability
　どちらも、infinite(無限集合) Eにおける 確率空間の分割{Ri:i∈I} を問題にしている
　つまり、finite(有限集合)のEにおける 確率空間の分割{Ri:i∈I} ならば、conglomerability 成立
・infinite(無限集合)のEでは、例示 ”using a countably infinite fair lottery”において、Conglomerability or Independence conglomerability が問題となる
　即ち、二人のゲームで 相手の選んだn よりも大きな数を 選ぶ人が勝ち。先に選んだ方が負けてします。なぜならば、正整数Nは可算無限だから
　そして、お互い独立に正整数Nを選べば 確率1/2だが、これは”independence conglomerability”違反 だと Pruss氏

要するに、infinite(無限集合）Eを扱うとき
Conglomerability or Independence conglomerability を満たさないとき ”paradoxes”が導かれる