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 $\Delta^1_3$-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 $\mathfrak{c}=\omega_2$.