For a long time I thought it was established that (given a standard axiomatisation of the probability calculus) countable additivity and countable conglomerability were equivalent. But I’ve lost confidence in my belief. So I’m wondering if anyone can tell me exactly what the answers are to a few questions below.

Just to make sure we’re clear, I’m taking countable additivity to be the principle that if each of the E_{i} in {E_{1}, …, E_{n}, …} are disjoint, then Pr(E_{1} v … v E_{n} v …) = Pr(E_{1}) + … + Pr(E_{n}) + ….

And I’m taking countable conglomerability to be the following. Again, if each of the E_{i} in {E_{1}, …, E_{n}, …} are disjoint, then there is some E_{i} such that Pr(E | E_{i}) <= Pr(E).

*Question One*: Does a failure of countable additivity entail a failure of countable conglomerability?

I’m pretty sure that, as stated, the answer to that is **no**. Consider a standard finitely additive probability function. So there’s some random variable X, and for all natural x, Pr(X=x)=0, while the Pr(X is a natural number)=1. Now insist that Pr is only defined over propositions of the form *X is in S*, where S is a finite or cofinite set of natural numbers. (By a cofinite set, I mean a set whose complement, relative to the naturals, is finite.) I’m reasonably sure that there’s no way to generate a failure of countable conglomerability.

*Question Two*: Assume there is a random variable X such that Pr(X is in S1 | X is in S2) is defined for every S1, S2 that are non-empty subsets of the naturals. And assume that whenever S2 is infinite, and the intersection of S1 with S2 is finite, then Pr(X is in S1 | X is in S2) is 0. (So Pr violates countable additivity.) Does Pr fail to respect countable conglomerability?

I’m even more confident that the answer to this is **yes**. Here’s the proof. Any positive integer *x* can be uniquely represented in the form 2^{n}(2*m*+1), with *n* and *m* non-negative integers. For short, let a statement of the form *n=x* mean that X is one of the numbers such that when represented this way, *n=x*, and similarly for *m*. Then for any non-negative integer, Pr(X is odd | *m=x*) = 0, since for any given *m* there is one way to be odd, and infinitely many ways to be even. By conglomerability, that implies Pr(X is odd) = 0. But an exactly parallel argument can be used to argue that Pr(X+1 is odd) = 0. And this leads to a contradiction.

*Question Three*: Assume there is a random variable X such that for any x, Pr(X=x)=0, while Pr(X is a natural number)=1, and that Pr(X is in S1 | X is in S2) is defined for every S1, S2 that are non-empty subsets of the naturals. Does Pr fail to respect countable conglomerability?

This is what I don’t know the answer to. I think the answer is **yes**, but I can’t see any obvious proof. Nor can I come up with a counterexample. Does anyone know (a) what the answer to this question is, and (b) where I might find a nice proof of the answer?

Much thanks in advance for helpful replies!

In “Statistical Implications of Finitely Additive Probability” Kadane, Schervish and Seidenfeld state that “non-conglomerability in

denumerablepartitions characterizes those which fail to be countably additive, confirming a statement of de Finetti (1972,p. 99)”. This looks like it implies that the answer to your third question is “yes”, because a countable partition where countable additivity fails is basically a random variable where the probability of having a natural number value is 1 but the probability of each particular natural number adds up to less than 1. However, looking at their 1984 paper “The Extent of Non-Conglomerability of Finitely Additive Distributions” (where they say they prove it), I can’t find the actual result, so I can’t tell if they make assumptions about the field of events the probability function must be defined over. But I just skimmed the paper – you can probably find it if you read more closely. I think most of their results assume that it’s the full powerset, which can probably generally be relaxed to the countable and co-countable sets, though they seem to be focusing on countable spaces.Both papers are available on Teddy Seidenfeld’s website, so you don’t have to go digging through statistics journals and collections in honor of de Finetti.

However, the Goel and Zellner volume where “Statistical Implications…” is located also contains a paper by Hill and Lane that might prove the result you want more clearly. I can’t seem to find a copy of it right now though, so I’m not certain.

Thanks Kenny. I’ll try to track down those papers. I had glanced at one of the Kadane et al papers, and I thought they were making slightly stronger assumptions than I had in mind. But I don’t think I’d looked at the first on you cited, so I should read that. And I should find the Hill and Lane paper.

Thanks again.