Remark 4

Let $m$ be a vectorial measure on $X/H$ with values in $E$, and let $q$ be a lower semi-continuous semi-norm on $E$. For $m$ to be $q$-majorizable (Ch. VI, §2, No. 3, Def. 3), it is necessary and sufficient that $m^\#$ be so, in which case $q(m^\#) = q(m)^\#$. This follows at once from the definitions and Cor. 3 a).

On the other hand, let $\mu$ be a positive measure on $X/H$. For $m$ to be scalarly of base $\mu$, it is necessary and sufficient that $m^\#$ be scalarly of base $\mu^\#$: this follows from Cor. 1.

Finally, if $m$ has base $\mu$, with density $f$ with respect to $\mu$ (Ch. VI, §2, No. 4, Def. 4), then $m^\#$ has base $\mu^\#$, with density $f \circ \pi$: this follows from Prop. 6 c).