Definition 3

*Let $m$ be a vectorial measure on $T$ with values in $F$. If $q$ is a lower semi-continuous semi-norm on $F$, $m$ is said to be q-majorizable if there exists a positive measure $\mu$ such that $|z' \circ m| \leq \mu$ for every $z' \in A'_q$; the supremum of the measures $|z' \circ m|$ as $z'$ runs over $A'_q$ (Ch. III, §2, No. 4, Th. 3) is then denoted $q(m)$. One says that $m$ is majorizable if it is q-majorizable for every continuous semi-norm $q$ on $F$.

If $m$ and $m'$ are both q-majorizable, it is immediate that $m + m'$ is also q-majorizable and that

When $F$ is a normed space, with norm denoted $|x|$, to say that $m$ is majorizable therefore means that the measures $|z' \circ m|$, where $|z'| \leq 1$, are bounded above$^1$ by a same positive measure; one then denotes by $|m|$ the supremum of this family of measures.

If $F = \mathbf{R}$, the measure $|m|$ corresponding to the Euclidean norm $|x|$ on $\mathbf{R}$ coincides with the measure denoted by $|m|$ in Ch. III, §1, No. 5.