Remark

Suppose that there exists in $A_q'$ a countable subset D that is dense for $\sigma(F', F)$; then, the function $q \circ f$ is always $\mu$-measurable, because $q(f(t)) = \sup_{z' \in D} |\langle z', f(t) \rangle|$ (Ch. IV, §5, No. 4, Cor. 1 of Th. 2). Then, for every compact subset K of T, $\int_K^*(q \circ f)\, d\mu = \sup_J \int_K g_J\, d\mu$, where J runs over the countable directed set of finite subsets of D (Ch. IV, §1, No. 1, Cor. of Th. 3); thus, one sees that in this case the condition that $\int_K^*(q \circ f)\, d\mu < +\infty$ for every compact subset K of T is necessary and sufficient for m to be $q$-majorizable.