Projective maximal families of orthogonal measures with large continuum

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 ${\Delta }_3^1$-definable well order of the reals, there is a ${\Pi }_2^1$-definable m.o. family, there are no ${\mathbf{\Sigma }}_2^1$-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 ${\Delta }_3^1$-definable well order of the reals, there is a ${\Pi }_2^1$-definable m.o. family, there are no ${\mathbf{\Sigma }}_2^1$-definable m.o. families, $\mathfrak{b}={\omega }_1$ and $\mathfrak{c}={\omega }_2$.

12 pages