Combinatorial Properties of the Raisonnier Filter

TL;DR

This paper explores the combinatorial aspects of the Raisonnier Filter, a tool in descriptive set theory, providing new characterizations of measurability and analyzing related ideal properties.

Contribution

It introduces a generalized Raisonnier filter, offers a partial converse to Raisonnier's theorem, and studies the associated ideal's cardinal characteristics.

Findings

Partial converse to Raisonnier's theorem established

New characterization of $oldsymbol{ abla}^1_2$ set measurability

Connections made between the filter's ideal and Cichoń's Diagram

Abstract

The Raisonnier Filter is a combinatorial object isolated by Jean Raisonnier in order to simplify Shelah's proof that if all $\boldsymbol{\Sigma}^1_3$ sets are Lebesgue-measurable then there is an inner model with an inaccessible cardinal. In this paper, we study the combinatorics of a general version of the Raisonnier filter, with an eye to potential applications in descriptive set theory. Among the most interesting of our results is a partial converse to Raisonnier's theorem, which can be used to provide a new characterisation of the statement "all $\boldsymbol{\Sigma}^1_2$ sets are measurable". We also introduce an ideal on the Cantor Space induced by the Raisonnier filter and study its cardinal characteristics, connecting them to the well-known characteristics in Cicho\'n's Diagram.