On the structure of measurable filters on a countable set

We show that under some conditions on a family $\mathcal{A}\subset\bbi$ there exists a subfamily $\mathcal{A}_0\subset\mathcal{A}$ such that $\bigcup \mathcal{A}_0$ is nonmeasurable with respect to a fixed ideal $\bbi$ with Borel base of a fixed uncountable Polish space. Our result applies to the classical ideal of null subsets of the real line and to the ideal of first category subsets of the real line.

An ideal on a set $X$ is a collection of subsets of $X$ closed under the operations of taking finite unions and subsets of its elements. Ideals are a very useful notion in topology and set theory and have been studied for a long time. We present a survey of results about ideals on countable sets and include many open questions.

We study the ideal of meager sets and related ideals.

This article is a fundamental study in computable measure theory. We use the framework of TTE, the representation approach, where computability on an abstract set X is defined by representing its elements with concrete "names", possibly countably infinite, over some alphabet {\Sigma}. As a basic computability structure we consider a computable measure on a computable $\sigma$-algebra. We introduce and compare w.r.t. reducibility several natural representations of measurable s...

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.

We introduce a notion of integration defined from filters over families of finite sets. This procedure corresponds to determining the average value of functions whose range lies in any algebraic structure in which finite averages make sense. The average values so determined lie in a proper extension of the range of the original functions. The most relevant scenario involves algebraic structures that extend the field of rational numbers; hence, it is possible to associate to t...

This paper proves the existence of nonmeasurable dense sets with additional properties using combinatorial techniques.

Local versions of measurability have been around for a long time. Roughly, one splits the notion of $\mu $-completeness into pieces, and asks for a uniform ultrafilter over $\mu $ satisfying just some piece of $\mu $-completeness. Analogue local versions of weak compactness are harder to come by, since weak compactness cannot be defined by using a single ultrafilter. We deal with the problem by restricting just to a subset $P$ of all the partitions of $\mu $ into $<\mu $ cl...

Sets satisfying Central sets theorem and other Ramsey theoretic large sets were studied extensively in literature. Hindman and Strauss proved that product of some of these large sets is again large. In this paper we show that if we take two combinatorially large sets along idempotent filters, then their product is also a filter large set. The techniques used here to prove our results are completely elementary in nature.

In this paper, we prove a result on nonmeasurable subgroups in commutative Polish groups with respect to more generalized structures than sigma-finite measures.