Relationships between AC, Ultrafilter Lemma/BPIT, Non-measurable sets

Free ultrafilters on $\mathbb N$ translate to non-measurable sets of real numbers.

Recall that the real numbers and the power set of the natural numbers, and the Cantor space. The Cantor space can be endowed with the product measure such that the full measure is $1$, and the measure is translation invariant (in the sense of finite modifications). Now it turns out that this is actually the completion of the Borel measure of the Cantor set and it is isomorphic, as a measure space, to the real numbers and the Lebesgue measure.

So what is an ultrafilter? It is a partition of $\mathcal P(\mathbb N)$ into two parts, and by translating a set of integers to its characteristics function we have a partition of the Cantor space into two parts. The full measure is one, and the complement map $A\mapsto\mathbb N\setminus A$, which is a bijection between the ultrafilter and its complement, is measure preserving. Therefore the Cantor set is the union of two disjoint sets of equal measure, which has to be $\frac12$. However by Kolmogorov's zero-one law we have that a free ultrafilter has to have either measure zero or measure one if it is measurable. Since $\frac12$ is neither zero nor one, we have to have that it is a non-measurable set.

So free ultrafilters make non-measurable sets. In a model where all sets are measurable, there are no free ultrafilters and so BPIT fails.

In Herrlich's The Axiom of Choice, Diagram 5.10 p. 122 has some nice principles which imply the existence of non-measurable sets. This is not a question of consistency, but rather actual implications. I believe that Gregory Moore's Zermelo's Axiom of Choice should have a similar diagram at the end of the book, but I cannot recall for sure (I will check and update tomorrow).

I will also add that the same argument shows that free ultrafilters cannot have Baire property (and therefore in models where every set of real numbers have the Baire property there are no free ultrafilters as well).

There are two steps in the ultrafiler proof of the Tychonoff theorem that require AC. You take an ultrafilter on the product and this ultrafilter induces along the projections an ultrafilter on every component space. You use the fact that these ultrafilters converge by compactness. Being able to do this step is equivalent to the BPI.

Now, convergence in the product topology is equivalent to coordinate-wise convergence, so you want to find a point in the product that the original ultrafilter converges to. You do that by picking for each coordinate a point the induced ultrafilter converges to. It is here that you need the full strenght of AC. For Hausdorff spaces, you can skip this step since in a Hausdorff space, a filter can not converge to more than one point. So you do not have to make arbitrary choices at that step.