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.

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.

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 (where "D" stands for "diffuse", a term used e.g. by E. K. van Douwen in Finitely additive measures on $\mathbb{N}$, Topology Appl. 47 (1992), No. 3, 223-268): 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, for the record and those who care, 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.

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, 1-56) 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 simply:

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.

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.

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. I am pretty sure the result is mentioned in Schechter's handbook, but I couldn't find it.

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.

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.