can you tell me how to construct a nonmeasurable set from a free (non-principal) ultrafilter on ω?

How is it possible to reconcile the following... In 1970, Solovay constructed Solovay's model, which shows that it is consistent with standard set theory, excluding uncountable choice, that all subsets of the reals are measurable. A not too well known application of the Boolean prime ide...

It is well-known that any non-principal ultrafilter on $\omega$ is non-measurable regarded as a subset of $2^{\omega}$. My question is "how well-known" is this fact? Here is the only proof I know: Let $\mathcal{U}$ be a non-principal ultrafilter on $\omega$. If $\mathcal{U}$ were measurable, the...

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 $\omega$), th...

...as soon as you have a non-standard number, you get a non-measurable set. Every nonstandard natural number $N$ gives rise to a nonprincipal ultrafilter $U$ on $\mathbb{N}$, by saying that a set $X\subset\mathbb{N}$ is in $U$ if and only if $N\in X^*$, the nonstandard analogue of $X$. In ot...

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" stan...