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This question is inspired by this other question which asks for an example of a strictly finitely-additive probability measure. To answer that question, I use the existence of a non-principal ultrafilter (assuming the ultrafilter lemma) on $\mathbb{N}$ to construct a $\{0,1\}$-valued example on $\mathcal{P}(\mathbb{N})$.

I'm no expert on set theory, however I am aware that it is consistent with ZF that all ultrafilters on $\mathbb{N}$ are principal$^{[1]}$. After some digging, I also found out that it is consistent with ZF+DC that there are no non-principal measures (finitely additive probability measures that assign $0$ measure to singletons) on $\mathbb{N}.$$^{[2],[3]}$ This at least shows that I really need something like the ultrafilter lemma to construct an example like the one I give (which is a non-principal measure).

Unfortunately, as far as I can tell, this doesn't quite complete the picture since it is possible that one could find an example which assigns some singleton positive measure. A silly way to do this assuming that a non-principal measure $\mu$ on $\mathbb{N}$ does exist is to consider $\frac12(\mu + \delta_0)$. Obviously here I use again the ultrafilter lemma (and I expect that again, I really need to to get such an example), but the possibility of such examples leads me to ask;

Is it consistent with ZF that all finitely additive probability measures on $\mathbb{N}$ are countably additive?


[1]: This is asserted in this answer by Asaf Karagila.

[2]: See this answer from the same user and the reference therein (my [3])

[3]: David Pincus and Robert M. Solovay, Definability of measures and ultrafilters, J. Symbolic Logic 42 (1977), no. 2, 179--190.

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  • $\begingroup$ What is ZF (other than my initials)? $\endgroup$ Commented Jul 24, 2019 at 20:30
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    $\begingroup$ ZF is the usual abbreviation for Zermelo-Fraenkel set theory. $\endgroup$ Commented Jul 24, 2019 at 20:30

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Let $\mu$ be a finitely additive probability measure on $\mathcal{P}(\mathbb{N})$. Note that for any $A\subseteq\mathbb{N}$, $\mu(A)\geq\sum_{a\in A}\mu(\{a\})$ (since $\mu$ is monotone and the right-hand side is the supremum of $\mu(F)$ for finite subsets $F\subseteq A$). Defining $\mu'(A)=\mu(A)-\sum_{a\in A}\mu(\{a\})$, we see that $\mu'$ is nonnegative and finitely additive. If $\mu'$ is nonzero, we can scale it by a constant to be a non-principal probability measure. So, if no non-principal probability measures exist, $\mu'=0$ and so $\mu$ is countably additive.

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