ZFC proves, among the other things, the existence of a (finitely) additive probability measure $\theta: \mathcal P(\mathbf N) \to \mathbf R$ on the power set of $\mathbf N$ such that $\theta(X) = 0$ for every finite $X \subseteq \mathbf N$, which I will shortly refer to as an ADPM: The proof is actually based on the Hanh-Banach theorem, which, just to recall something that is widely known, is implied by, but not equivalent to, the axiom of choice, see e.g. Section 23.19 in E. Schechter's *Handbook of Analysis and its Foundations* (Academic Press, 1996), so that, in principle, the existence of $\theta$ can even be established in a weaker system than ZFC. On the other hand, ZF proves the following:

> **Proposition.** The existence of an ADPM implies the existence of a subset of $\mathbf R$ without the [Baire property][1].

But it follows from Theorem 1(3) in R. M. Solovay's celebrated paper *A model of set theory in which every set of reals is Lebesgue measurable* (Ann. of Math., 2nd
Ser. **92** (1970), No. 1, l-56) that the existence of an uncountable transitive model of ZFC + I, where I is the statement: ``There exists an inaccessible cardinal'', supplies an uncountable transitive model of ZF in which every subset of $\mathbf R$ *has* the Baire property.

So, putting it all together, we see that the existence of an ADPM is provable in ZFC, but independent of ZF. With this said, my question is:

> Do you know a reference where the proposition above is stated and proved?

To be clear: I am not looking for a proof, and am pretty sure the result is somewhere in Schechter's handbook, but couldn't find it.

  [1]: https://en.wikipedia.org/wiki/Property_of_Baire