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Why do some people prefer the following axiom (e.g. deFinetti)

If $A,B \in \mathcal{B}$ where $\mathcal{B}$ is a $\sigma$-algebra of sets and $A,B$ are disjoint then $P(A \cup B) = P(A)+P(B)$

over the following

Suppose $A_1, A_2 ,\dots \in \mathcal{B}$ and are pairwise disjoint. Then $$P \left(\bigcup_{i=1}^{\infty} A_i \right) = \sum_{i=1}^{\infty} P(A_i)$$

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Are you missing a $P$ in (2)? I guess so. And I'd say (1) is a special case of (2) if $A$ and $B$ make up the algebra and (1) implyies (2) via induction. Or is your problem the $\infty$ vs. only two elements? – NikolajK Dec 17 '11 at 15:57
You need more than induction to prove 2 from 1. – Mariano Suárez-Alvarez Dec 17 '11 at 16:02
One reason would be to talk about the probability that two natural numbers are co-prime. But I doubt that's what Bruno de Finetti had in mind. – Michael Hardy Dec 17 '11 at 19:00

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