Affiliation:
1. Departamento de Matemáticas , 33184 Universidad Metropolitana , Caracas , Venezuela
Abstract
Abstract
In this note we consider a family
𝖪𝖡𝖭
{\sf KBN}
of filters on ω which are null, with respect to
the standard product measure on
2
ω
{2^{\omega}}
in some
strong sense and we prove in
𝖹𝖥𝖢
{{\sf ZFC}}
that this class
includes all universally measurable filters.
Also, we construct
2
𝔠
{2^{\mathfrak{c}}}
-many
𝖪𝖡𝖭
{\sf KBN}
filters which are not universally measurable
and show that
the Filter Dichotomy Principle (
𝖥𝖣𝖯
{{\sf FDP}}
)
of A. Blass and C. Laflamme implies that
every member of
𝖪𝖡𝖭
{\sf KBN}
is meager.
In addition, we prove that if the identity
min
{
𝖺𝖽𝖽
(
𝒩
)
,
𝔱
}
=
𝔡
{\min\{{\sf add}(\mathcal{N}),\mathfrak{t}\}=\mathfrak{d}}
holds,
there are non-meager filters in
𝖪𝖡𝖭
{\sf KBN}
.
Thus, the existence of non-meager
𝖪𝖡𝖭
{\sf KBN}
filters
is independent of the axioms of
𝖹𝖥𝖢
{{\sf ZFC}}
.