I'm still not sure what this tells us about the distribution of primes satisfying various algebraic conditions. Perhaps my original question was a red herring.

]]>let X be a set, D any class of subsets of X, and P a real function on D. If P is coherent (i.e., P satisfies de Finetti’s coherence principle) then P admits a coherent extension to the power set of X. This follows easily by Hahn-Banach theorem. Also, if D is an algebra, P is coherent if and only if it is a finitely additive probability. Finaly, if X = {1,2,….}, P is the usual density, and D is the collection of those subsets of X such that the limit exists, then P is coherent. In fact, coherence is preserved under pointwise limits, and P is actually the pointwise limit of the restrictions to D of the coherent mappings

P_n(A) = (1/n) card( A intersection {1,…,n} )

For more details, please, could you give us your email address ?

(My email address is pietro.rigo@unipv.it)

There are already several notions of the density of a set of primes. The most obvious is natural density: d(S) = lim_{x\to\infty} {number in S <= x \over number of primes <= x}. The limit exists for many interesting sets (including the example I gave) and the behaviour under finite disjoint union is obvious, but the collection of sets for which the limit exists is not an algebra. (It is easy to construct A & B with d(A) = d(B) = 1/2, liminf_{x\to\infty} {number in A intersect B <= x \over number of primes <= x} = 0, and limsup_{x\to\infty} {number in A intersect B <= x \over number of primes <= x} = 1/2.) Is there an extension of d to a finitely additive function on an algebra?

Number theorists are particularly interested in sets of primes determined by algebraic equations, for example the set of primes p such that x^3+x+1=0 has a solution modulo p. There is a notion of density which applies to these sets, and I assumed in my previous note that they formed an algebra. The collection is obviously closed under finite union (disjoint or not), but I'm not sure about complements. Contrary to what I said before, it is not obvious to me that any of the usual notions of density is defined on a reasonably interesting algebra.

I'm puzzled by your statement that a measure defined on an algebra automatically extends to the power set. I don't know the extension theorems for finitely additive probabilities, but the countably additive extension of Lebesgue measure to the power set of the real line is non-obvious. A reasonably large subalgebra of the power set of the set of primes would be interesting.

Finally, once such a finitely additive measure is constructed, does it help number theorists in some way? (My experience is limited to groupings of the prime factorisation of n! and other highly divisible numbers. For a Diophantine equation like n! = P(x)^k Q(x), where P & Q are polynomials, Daniel Berend & I considered those primes which could contribute to the factor P(x)^k, and were sometimes able to show that there are at most finitely many solutions.) I know the question is vague, but I am new to this: we all know that set functions can be finitely additive without being countably additive, but it is not clear to me how much study such functions deserve.

]]>we apologize for answering with so big delay, but we are not familiar with normal deviate and we learned of your remarks just now.

For answering your question (or trying to do it) we need some more information.

You are looking for a finitely additive probability P on the power set of the

primes such that

(*) P(p: p is congruent to 1 modulo 6) = 1/2.

But, in addition to (*), what other properties of P are desirable for you ?

And also, you say that such a P already

exists on certain subalgebras. So, clearly, it can be extended to the

power set. What’s the problem ? Perhaps, such

subalgebras do not include certain sets of interest for you ?

Or what else ?

Patrizia Berti Eugenio Regazzini Pietro Rigo

]]>A more interesting application (to me) is the long-run behaviour of primes. (Certainly any particular prime is negligible, so sigma-additivity is out of the question.) It would be nice to say that if p is a randomly selected prime then P(p congruent to 1 modulo 6) = 1/2. Number theorists use various definitions of density which are finitely additive on subalgebras of the power set of the set of primes. (I think this includes all sets determined by solvability of polynomial equations modulo the prime.) Does the theory of finitely additive probabilities have something interesting to say about this?

]]>Here’s another funny one. The dual of $L^\infty(\mu)$ is the Banach space of finitely additive finite signed measures absolutely continuous wrt $\mu$ with total variation norm. On the one hand, this makes it useful to have an understanding of finitely additive measures; on the other hand, this gives yet another reason to avoid the space $L^\infty(\mu)$.

Also, with all due respect, I am not sure I find your negative example very convincing, since it works with a countable partition (i.e., the problem seems adapted to countably additive measures).

Bes twishes

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