Nonmeasurable sets and unions with respect to selected ideals especially ideals defined by trees

TL;DR

This paper investigates nonmeasurable sets with respect to ideals defined by trees, demonstrating the existence of sets nonmeasurable in multiple senses and exploring conditions under which unions of small sets are completely nonmeasurable.

Contribution

It introduces new examples of sets nonmeasurable in multiple ideals simultaneously and examines the relationship between nonmeasurability and set unions under various conditions.

Findings

Existence of a set nonmeasurable in s, l, and m ideals simultaneously.

Construction of a m.a.d. family that is also dominating.

Equivalence of certain nonmeasurability conditions to CH.

Abstract

In this paper we consider nonmeasurablity with respect to sigma-ideals defined be trees. First classical example of such ideal is Marczewski ideal s_0. We will consider also ideal l_0 defined by Laver trees and m_0 defined by Miller trees. With the mentioned ideals one can consider s, l and m-measurablility. We have shown that there exists a subset A of the Baire space which is s, l and m nonmeasurable at the same time. Moreover, A forms m.a.d. family which is also dominating. We show some examples of subsets of the Baire space which are measurable in one sense and nonmeasurable in the other meaning. We also examine terms nonmeasurable and completely nonmeasurable (with respect to several ideals with Borel base). There are several papers about finding (completely) nonmeasurable sets which are the union of some family of small sets. In this paper we want to focus on the following problem: "Let P be a family of small sets. Is it possible that for all A which is a subset of P, union of A is nonmeasurable implies that union of A is completely nonmeasurable?" We will consider situations when P is a partition of R, P is point-finite family and P is point-countable family. We give an equivalent statement to CH using terms nonmeasurable and completely nonmeasurable.