A *non-principal* [*probability*] *measure* on a set X is a function $\mu$ defined on all subsets of $X$, with values in $[0,1]$, which is finitely additive, satisfies $\mu(X)=1$, and vanishes on singletons. 


Can one prove in ZF + DC that the existence of such a measure on $\bf N$ (or on $\omega$), the set of natural integers, implies that of a non-Lebesgue measurable subset of $\bf R$?


Comments: 1) Sierpinski proved this in 1938 in the special case that $\mu$ takes values in $\{0,1\}$ (*non-principal,* or *non-free ultrafilter*). It suffices to take $X=\{A\subset{\bf N}\mid\mu(A)=0\}\subset\{0,1\}^{\bf N}$, equipped with the standard Bernoulli measure $\lambda$ (so that $\{0,1\}^{\bf N},\lambda)\approx([0,1],{\rm Leb}$). If $X$ were $\lambda$-measurable, one would have $\mu(X)={1\over2}$ since $X^c=\{A^c\mid A\in X\}$. But $X$ is a queue event, thus by Kolmogorov, $\mu(X)=0$ or $1$, contradiction.


2) In the 1998 book by Howard and Rubin, *Consequences of the axiom of choice*, it is stated that the existence of a non principal measure on $\bf N$ implies that of a non-Lebesgue measurable set [Form 222 implies Form 93], and that it can be found in articles by Pincus in 1972 and by Foreman-Wehrung in 1991. However, I could not find this implication in these articles.


3) The most famous non-principal measures on countable sets are invariant means on amenable groups, starting with $\bf Z$ [*mean* is another name for a finitely additive probability measure defined on all subsets]. Question: does the existence of a non-principal measure on $\bf N$ (or $\bf Z$) imply (in ZF + DC) that of an invariant mean on $\bf Z$?



Does the existence of a non-principal measure on ω imply that of a non Lebesgue measurable set?
This is stated as an open conjecture in Pincus' 1974 paper [The Strength of the Hahn-Banach Theorem](http://link.springer.com/chapter/10.1007/BFb0066014), which is a pretty good sign that he didn't prove it in a paper published in 1972. That same paper contains the first proof of the analogous question for the Baire Property.


As far as I am aware, this problem is still open, even assuming the stronger assumption of a Banach limit (such a measure that is translation-invariant and whose integral gives the limit for convergent sequences).


