All meager filters may be null
by Bartoszynski and Goldstern and Judah and Shelah. [BGJSh:434]
Proc American Math Soc, 1993
We show that it is consistent with ZFC that all filters
which have the Baire property are Lebesgue measurable. We also
show that the existence of a Sierpinski set implies that there
exists a nonmeasurable filter which has the Baire property.
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