Projective maximal families of orthogonal measures with large continuum

Abstract

We study maximal orthogonal families of Borel probability measures on $2^ω$ (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 $Δ^1_3$-definable well order of the reals, there is a $Π^1_2$-definable m.o. family, there are no $\mathbfΣ^1_2$-definable m.o. families and $\mathfrak{b}=\mathfrak{c}=ω_3$ (in fact any reasonable value of $\mathfrak{c}$ will do). (2) There is a $Δ^1_3$-definable well order of the reals, there is a $Π^1_2$-definable m.o. family, there are no $\mathbfΣ^1_2$-definable m.o. families, $\mathfrak{b}=ω_1$ and $\mathfrak{c}=ω_2$.