  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 $\mathfrak{c}=\omega_2$.

Logic