Combinatorial Properties of the Raisonnier Filter
Spyridon Dialiatsis

Abstract:
In 1970, Solovay produced a model of ZF + DC in which all sets of reals are Lebesgue measurable. In order to achieve this, he worked in ZFC with the assumption that an inaccessible cardinal exists (I). In 1984, Shelah proved that this cannot be achieved without assuming the existence of an inaccessible cardinal, by showing that the theories ZFC + I and ZF + DC + "every set of reals is Lebesgue measurable" are equiconsistent. His main theorem states that if every Σ^1_3 set is Lebesgue measurable, then א_1 is an inaccessible cardinal in Gödel’s constructible universe L. In the same year, Raisonnier gave a simpler proof of this fact using a construction now known as the "Raisonnier filter". This filter remained underutilized outside of Raisonnier’s proof.
This thesis is an investigation into the properties of the Raisonnier filter. First, we prove some basic facts about the Raisonnier filter constructed starting from various subsets of 2^ω. Second, we generalize Raisonnier’s proof method and obtain a converse to this generalized statement. This result is significant, as it suggests that the Raisonnier filter is strongly related to Lebesgue measurability and can thus not be used to obtain results about other regularity properties. Third, we obtain a new characterization of Σ^1_2 Lebesgue measurability through the Raisonnier filter. This characterization is also related to the concept of "Laver measurability" introduced by Brendle and Löwe. Finally, we define a new ideal R consisting of the subsets of 2^ω for which the Raisonnier filter is rapid and we establish relationships between its cardinal characteristics and those of the ideal of null sets.