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On the second page of Perfect-Set Properties in $L(\mathbb{R})[U]$ by Di Prisco and Todocevic, there is the following sentence:

By a well known classical argument of W. Sierpinski, nonprincipal ultrafilfers on $\mathbb{N}$ can naturally be identified with sets of reals which are neither Lebesgue measurable nor have the propertly of Baire.

Do the authors mean there is a natural bijection

$f:\textrm{nonprincipal ultrafilters}\leftrightarrow\textrm{sets that are neither Lebesgue nor Baire measurable}$

or are they merely saying a nonprincipal ultrafilter is nonmeasurable?

new account
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  • The following, in the accepted answer, gives a reference to the Sierpinski result. In a comment to that answer, it is mentioned that Sierpinski is vague on some points, and the comment gives another stack exchange article which may fill in more of the required arguments [it's out of my expertise] https://math.stackexchange.com/questions/1130615/non-measurability-of-ultrafilter-on-omega – coffeemath Jan 28 '24 at 03:58
  • @coffeemath I know ultrafilters are nonmeasurable. But the authors seem to be suggesting a stronger result. Or am I just too bad at reading... – new account Jan 28 '24 at 04:02

1 Answers1

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This is a bit ambiguously written. There is indeed a correspondence being referred to, but it is the much simpler one between sets of sets of natural numbers and sets of reals. I think it would have been more clearly phrased as follows:

Sets of sets of natural numbers can be naturally identified with sets of reals. Under this identification, ultrafilters are converted to sets of reals which are nonmeasurable and lack the property of Baire.

Of course, this raises the question of whether - in $L(\mathbb{R})[\mathcal{U}]$, at least - all "pathological" sets of reals can be viewed as "an ultafilter (perhaps not $\mathcal{U}$ itself) in disguise" in some sense. One way to make this rigorous is the following:

In $L(\mathbb{R})[\mathcal{U}]$, is every non-measurable non-Baire set of reals $X$ Wadge-equivalent to (the set of reals corresponding as usual to) some nonprincipal ultrafilter on the naturals?

While this is not what was meant in the cited article (simply because Sierpinski proved no such result!), I do not know the answer to this question.

Noah Schweber
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