Projective maximal families of orthogonal measures with large continuum

TL;DR

The paper constructs models of set theory where certain definable maximal orthogonal families of measures exist with large continuum, highlighting the interplay between definability, measure families, and continuum size.

Contribution

It introduces models with definable maximal orthogonal measure families and specific continuum and bounding number configurations, advancing understanding of measure family definability.

Findings

Existence of $ ext{Pi}^1_2$-definable m.o. families in certain models.

Models with large continuum and definable well orders of reals.

No $ ext{Sigma}^1_2$-definable m.o. families in these models.

Abstract

We study maximal orthogonal families of Borel probability measures on $2^\omega$ (abbreviated m.o. families) and show that there are generic extensions of the constructible universe $L$ in which each of the following holds: (1) There is a $\Delta^1_3$-definable well order of the reals, there is a $\Pi^1_2$-definable m.o. family, there are no $\mathbf{\Sigma}^1_2$-definable m.o. families and $\mathfrak{b}=\mathfrak{c}=\omega_3$ (in fact any reasonable value of $\mathfrak{c}$ will do). (2) There is a $\Delta^1_3$-definable well order of the reals, there is a $\Pi^1_2$-definable m.o. family, there are no $\mathbf{\Sigma}^1_2$-definable m.o. families, $\mathfrak{b}=\omega_1$ and $\mathfrak{c}=\omega_2$.