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

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.

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

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.