Sierpiński 1938: no free ultrafilter on $\mathbb{N}$ is measurable or has the Baire property

Identify a subset of $\mathbb{N}$ with its characteristic function, so that $\mathcal{P}(\mathbb{N})$ becomes the Cantor space $\{0,1\}^{\mathbb{N}}$ with its product topology and its uniform product measure; equivalently, transport it to $[0,1]$ by binary expansions and use Lebesgue measure. Then:

no free ultrafilter on $\mathbb{N}$, read as a subset of $\{0,1\}^{\mathbb{N}}$, is measurable, and none has the Baire property.

The non-measurability is Sierpiński (1938), where it appears as the observation that a free ultrafilter yields a non-measurable set and hence a non-measurable additive function. The category form, that no such set has the Baire property, is the standard analogue and is recorded in the same place in the literature on weak choice principles.

The consequence usually wanted is negative: a free ultrafilter can never be exhibited by a construction that produces only measurable sets, or only sets with the Baire property. It is a precise sense in which such an object cannot be written down.

• Not proved in this library. The statement needs the product measure on $\{0,1\}^{\mathbb N}$, or Lebesgue measure on $[0,1]$, and a specialised topological zero-one law on a Polish space. The library now has the general Baire/category background, but not that zero-one-law argument or the measure and integration track.

• What would prove it. A free ultrafilter $\mathcal{U}$ is unchanged by altering finitely many coordinates, since it contains every cofinite set, so it is a tail event: the Kolmogorov zero-one law would force its measure to be $0$ or $1$, and the topological zero-one law would force it to be meagre or comeagre. Complementation $A \mapsto \mathbb{N} \setminus A$ is a measure-preserving homeomorphism of $\{0,1\}^{\mathbb{N}}$ that carries $\mathcal{U}$ exactly onto its own complement, because an ultrafilter contains exactly one of $A$ and $\mathbb{N} \setminus A$ (Characterisation of ultrafilters: every set or its complement ↗). A measurable $\mathcal{U}$ would therefore have measure $\tfrac{1}{2}$, and a $\mathcal{U}$ with the Baire property would be neither meagre nor comeagre; both contradict the zero-one laws. The two zero-one laws are the missing machinery, and they belong to the measure-theory and Baire-category tracks.

• Why it matters elsewhere. The ultrafilter lemma, from the Axiom of Choice: every filter extends to an ultrafilter ↗ produces an ultrafilter from Zorn's lemma with no description of it, and FALSE, once the ultrafilter lemma is available: every ultrafilter is principal ↗ uses that to produce a free ultrafilter on $\mathbb{N}$ and can say nothing further about it. This item is the sharp reason why nothing further can be said by the usual means: every free instance of Ultrafilter ↗ on $\mathbb{N}$ lies outside the measurable sets and outside the sets with the Baire property.

• No consistency hypothesis is needed. Unlike the independence results this library also records, this is an outright theorem: it says of any free ultrafilter that exists that it is non-measurable, and is vacuously true in a model with none.