Corollary 1

— Let $\lambda, \lambda'$ be two positive measures on $X/H$. For $\lambda'$ to have base $\lambda$, it is necessary and sufficient that ${\lambda'}^\#$ have base $\lambda^\#$. For $\lambda$ and $\lambda'$ to be equivalent, it is necessary and sufficient that $\lambda^\#$ and ${\lambda'}^\#$ be equivalent.

The first assertion follows from Prop. 6, a) and c). The second follows from the first.