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We know that, with Axiom of Choice (AC), it can be shown that there exists a finitely additive uniform distribution defined for all subsets of the integers (see, e.g., Hrbacek and Jech 1999, Ch. 11).

I was wondering, WITHOUT AC, whether or not there exists any finitely additive measure defined over $\mathcal{P}(\mathbb{N})$? Is there a result showing that ZF + {nonexistence of finitely additive measure over $\mathcal{P}(\mathbb{N})$} is consistent? Thanks!

Logica
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    The counting measure is finitely additive. Do you mean a probability measure? Do you mean a measure that assigns 0 to each finite set? – Goldstern Jun 09 '15 at 19:46
  • Yes, finitely additive probability measure. – Logica Jun 09 '15 at 19:51
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    $\mu(A) = \sum_{n\in A } \frac1{2^n}$ for all $A\subseteq {1,2,3,4,\ldots}$. – Goldstern Jun 09 '15 at 19:54
  • Thanks, this is very nice. The next question is this. WITHOUT AC, whether or not it can be shown that there does not exists any nontrivial FINITELY additive measure defined over all subsets of the reals? Is there a result showing that ZF + {nonexistence of finitely additive measure over all subsets of reals} is consistent? – Logica Jun 09 '15 at 20:04
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    You probably meant to ask about the existence of a finitely additive measure that is NOT $\sigma$-additive. – Christian Remling Jun 09 '15 at 20:04
  • The context in which I ask this question is de Finetti's subjective probability theory, where he takes that personal probabilities should be finitely additive and there is no need to impose the stronger condition of countable additivity. I was trying to understand how much set theory he needs at the foundational level for his conviction. – Logica Jun 09 '15 at 20:11
  • Yes, I believe this is consistent. It's consistent with ZF+DC that every set of reals has the Baire property (Solovay / Shelah), and as I recall, under these axioms, you can prove that $\ell^1(\mathbb{N})$ is reflexive. A (strictly) finitely additive measure on $\mathbb{N}$ would give you a continuous linear functional in $(\ell^\infty)^* \setminus \ell^1$, contradicting reflexivity. Unless someone else beats me to it, I will post an answer with details when I have a chance. – Nate Eldredge Jun 09 '15 at 20:30
  • A finitely additive probability measure on the reals (still not good enough for an answer): $\mu(A)= \sum_{n\in A\cap {1,2,\ldots }} \frac 1{2^n}$. – Goldstern Jun 09 '15 at 20:36
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    @AsafKaragila: Yes, thanks for finding that. Clinton's comment on Stefan's answer sketches (without needing functional analysis) why the existence of such a measure is inconsistent with BP, and as I mentioned, we know from Shelah that ZF+DC+BP is consistent. So that resolves the question at hand. – Nate Eldredge Jun 09 '15 at 22:15

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