Nonmeasurable sets and unions with respect to tree ideals

TL;DR

This paper explores nonmeasurability with respect to various tree ideals, constructs special nonmeasurable sets, introduces $ ext{T}$-Bernstein sets, and examines properties of $ ext{I}$-Luzin sets, revealing new interactions between these concepts.

Contribution

It introduces and studies new classes of nonmeasurable sets related to tree ideals and defines $ ext{T}$-Bernstein sets, expanding understanding of measure and category in descriptive set theory.

Findings

Existence of a set that is nonmeasurable with respect to multiple tree ideals and forms a dominating family.

Introduction of $ ext{T}$-Bernstein sets with specific intersection properties.

Results on the nonexistence of certain $ ext{I}$-Luzin sets and sum properties of Luzin and Sierpiński sets.

Abstract

In this paper we consider a notion of nonmeasurablity with respect to Marczewski and Marczewski-like tree ideals $s_0$, $m_0$, $l_0$, and $cl_0$. We show that there exists a subset $A$ of the Baire space $\omega^\omega$ which is $s$-, $l$-, and $m$-nonmeasurable, that forms dominating m.e.d. family. We introduce and investigate a notion of $\mathbb{T}$-Bernstein sets - sets that intersect but does not containt any body of a tree from a given family of trees $\mathbb{T}$. We also acquire some results on $\mathcal{I}$-Luzin sets, namely we prove that there are no $m_0$-, $l_0$-, and $cl_0$-Luzin sets and that if $\mathfrak{c}$ is a regular cardinal, then the algebraic sum (considered on the real line $\mathbb{R}$) of a generalized Luzin set and a generalized Sierpi\'nski set belongs to $s_0, m_0$, $l_0$ and $cl_0$.