On the structure of measurable filters on a countable set

The existence of an uncountable family of nonmeager filter whose intersection is meager is consistent with MA(Suslin)

The goal of this paper is to prove the theorem in the title.

We characterize winning strategies in various infinite games involving filters on the natural numbers in terms of combinatorics or structural properties of the given filter. These generalize several ultrafilter games of Galvin.

We analyze several ``strong meager'' properties for filters on the natural numbers between the classical Baire property and a filter being $F_\sigma$. Two such properties have been studied by Talagrand and a few more combinatorial ones are investigated. In particular, we define the notion of a P$^+$-filter, a generalization of the traditional concept of P-filter, and prove the existence of a non-meager P$^+$-filter. Our motivation lies in understanding the structure of filter...

We study various combinatorial properties, and the implications between them, for filters generated by infinite-dimensional subspaces of a countable vector space. These properties are analogous to selectivity for ultrafilters on the natural numbers and stability for ordered-union ultrafilters on $\mathrm{FIN}$.

We obtain game-theoretic characterizations for meagerness and rareness of filters on a countable set.

Using a new concept of conglomerated filter we demonstrate in a purely combinatorial way that none of Erd\"{o}s-Ulam filters or summable filters can be generated by a single statistical measure and consequently they cannot be represented as intersections of countable families of ulrafilters. Minimal families of ultrafilters and their intersections are studied and several open questions are discussed.

In a recent paper, two multi-representations for the measurable sets in a computable measure space have been introduced, which prove to be topologically complete w.r.t. certain topological properties. In this contribution, we show them recursively complete w.r.t. computability of measure and set-theoretical operations.

In this paper we showed that the existence of a space $X$ possessing a free $\omega_1$-complete open filter which is contained in $<2^{\mathfrak c}$ open ultrafilters implies the existence of a measurable cardinal. This result implies a negative answer to an old question of Liu. Also, we investigate the poset $\mathbf{OF}(X)$ of free open filters on a space $X$. In particular, we characterize spaces for which $\mathbf{OF}(X)$ is a lattice. For each $n\in\mathbb{N}$ we constru...

We give several topological/combinatorial conditions that, for a filter on $\omega$, are equivalent to being a non-meager $\mathsf{P}$-filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager $\mathsf{P}$-filter. Here, we identify a filter with a subspace of $2^\omega$ through characteristic functions. Along the way, we generalize to non-meager $\mathsf{P}$-filters a result of Miller about $\mathsf{P}$-points, and we employ...