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My understanding is that Solovay (1970)'s relative consistency shows that if ZFC+I has a model then ZF+DC has a model in which every subset of the reals is Lebesgue measurable (and hence $\sigma$-additive).

I was wondering if there a relative consistency result that shows that ZF + {some weaker than AC/DC condition} proves the existence of finitely additive atomless probability measure $\mu$ on $\mathcal{P}(\mathbb{R})$ satisfying following conditions? Is ZF alone sufficient to prove the existence such measure? Thanks!

  1. $\mu(\emptyset)=0$ and $\mu(\mathbb{R})=1$
  2. If $X\subseteq Y$ then $\mu(X)\le \mu(Y)$
  3. If $X$ and $Y$ are disjoint, then $\mu(X\cup Y)=\mu(X)+\mu(Y)$
  4. For every $x\in \mathbb{R}$, $\mu(\{x\})=0$
  • Solovay, R. M. (1970). A model of set-theory in which every set of reals is Lebesgue measurable. The Annals of Mathematics, 92(1):1–56.
Martin Sleziak
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Logica
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  • Also, do you want to impose some conditions on this? Or are trivial measures sufficient for your purposes? – Asaf Karagila Jun 12 '15 at 15:45
  • I am mostly concerned with its being a probability measure. – Logica Jun 12 '15 at 15:48
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    $$m(A)=\begin{cases}0 & 1\notin A\ 1 & 1\in A\end{cases}$$ – Asaf Karagila Jun 12 '15 at 15:50
  • hmm, perhaps an atomless measure? I guess what I want to understand is to what extend Choice or some weaker version of it is needed in order to prove the existence of a finitely additive probability measure. – Logica Jun 12 '15 at 15:58
  • Yes, you probably want it to be without atoms. It is of course up to you to ask whether or not it sits with the Lebesgue measure, or if it is at all translation invariant. – Asaf Karagila Jun 12 '15 at 15:59
  • ok, I revised my question to make it more specific. so no requirements for translation invariant nor $\sigma$-additivity – Logica Jun 12 '15 at 16:05

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The issue is that $\sf DC$ is not the right tool for constructing measures or ultrafilters. Both, when constructed by transfinite recursion, require recursion much longer than a countable length, which really all that $\sf DC$ can give you.

The tool for obtaining such objects is instead the Boolean Prime Ideal theorem, or weaker theorems like the Hahn-Banach theorem. Those work "magically" by combining finitely coherent objects into a larger, maximal, object. Although these too have limits, since they are weaker than the axiom of choice (and in fact the Boolean Prime Ideal theorem is consistent with the failure of countable choice).

If you look at the 4th section of the following paper, you will see that the proof given by Blass that it is consistent that there are no free ultrafilters (at all) can be transformed into a proof that there are no non-principal measures either; all the while preserving $\sf DC$.

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

Asaf Karagila
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