On the Rudin-Blass Ordering of Measures

TL;DR

The paper extends Rudin-Blass and Rudin-Keisler orderings from ultrafilters to finite additive measures on $\\omega$, introducing $Q$-measures and selective measures as natural analogues of $Q$-points and selective ultrafilters. It develops multiple construction schemes (density extensions, Sikorski transfinite embeddings, generic and Solovay measures, and Mokobodzki-type transfinite methods) to produce non-atomic measures and to study their minimality in RB/RK orderings, uncovering both symmetries and fundamental differences. A key finding is that RB-minimality does not coincide with $Q$-measures: under CH one can have RB-minimal, shift-invariant measures that are not $Q^+$-measures, illustrating a separation between minimality notions. The work also provides forcing and consistency results, plus a concrete CH-based construction of a shift-invariant RB-minimal measure, highlighting rich independence phenomena in the measure-order landscape on $\\omega$.

Abstract

We study the Rudin-Blass (and the Rudin-Keisler) ordering on the finite additive measures on $ω$. We propose a generalization of the notion of Q-point and selective ultrafilter to measures: Q-measures and selective measures. We show some symmetries between Q-points and Q-measures but also we show where those symmetries break up. In particular we present an example of a measure which is minimal in the sense of Rudin-Blass but which is not a Q-measure.

Theorems & Definitions (85)

• Proposition 1.1
• proof
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3
• Proposition 2.4
• Proposition 2.5
• Lemma 2.6