Ref. request: Additive probability measure on $\mathcal P({\bf N})$ supplies subset of $\mathbf R$ without Baire property

Question

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.

Answer 1

A very good reference for various forms of AC is the book Howard, Rubin: Consequences of the Axiom of Choice, AMS, 1998. (See AMS website or Google Books.) This book contains a large database of various consequence of choice and tables showing relations between them.

There is also a website where you can search for relations between various forms mentioned in this book.

In this book we can find:

FORM 142. $\neg$PB: There is a set of reals without the property of Baire. Jech [1973b] p 7 and note 28.

FORM 222. There is a non-principal measure on $\mathcal P(\omega)$. Pincus/Solovay [1977] and note 147.

Then in the table which shows implications between various forms we find:

222 142 (1) Pincus [1972c]
142 222 (3) Pincus [1972c]

(1) = "The implication is provable."
(3) = "The implication is not provable in ZF".

The reference given there is:

[1972c] D. Pincus: The strength of the Hahn-Banach theorem, Proc. of the Victoria Symp. on Nonstandard Analysis, Lecture Notes in Math. 369, Springer, Heidelberg, 1973, 203-248. DOI: 10.1007/BFb0066014.

(The correct year for this reference should probably be 1974 not 1973, see the comment below. I left it here in the same way it is written in the book I quoted from.)

D. Pincus writes there that this implication "is due to Solovay. It is stated without proof in [20]. Since it may interest readers of this paper we include our own proof at the end of §I." Where [20] is R. M. Solovay, A model of set theory in which every set of reals is Lebesgue measurable, Ann. of Math. 92 (1970), pp. 1-58; DOI: 10.2307/1970696.

A proof given there seems to be similar to the proof sketched (very briefly) in this comment.

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EDIT: Since the OP also mentioned Schechter's Handbook of Analysis and Its Foundations, I will add that in this book we can find the result in Chapter 29. In 29.37 the following three principles are listed and shown to be equivalent:

• $(\ell_\infty)^* \supsetneqq \ell_1$; there is a bounded functional on $\ell_\infty$ that cannot be represented by a member of $\ell_1$.
• There exists a measurable space $(\Omega,\mathcal S)$ and a bounded scalar-valued charge on $\mathcal S$ that is not a measure.
• There exists a probability charge on $(\mathbb N,\mathcal P(\mathbb N))$ that vanishes on finite sets.

In 29.38 it is shown that they imply: (NBP) There exists a subset of $\{0,1\}^{\mathbb N}$ that lacks Baire property.

As far as the references are concerned, the author says that:

This implication was first stated without proof in Solovay [1970]; the first published proof apparently is that of Pincus [1974]. The slightly shorter proof below is essentially that of Taylor; it was published in Wagon [1985].

The first two references have already been mentioned. The third one is S. Wagon, The Banach-Tarski Paradox, Encyclopedia Math. Appl. 24, Cambridge Univ. Press, Cambridge, 1985. The corresponding result is indeed given in as Theorem 13.5 in this book.