Corollary

— *With notations as in Prop. 5, suppose that there exists a bounded positive measure $\mu$ on $T$ such that $|I(f)| \leq \mu(|f|)$ for all $f \in \mathscr{C}^b(T; \mathbf{C})$; then there exists a complex measure $\theta$ on $T$ such that $\theta(f) = I(f)$ for all $f \in \mathscr{C}^b(T; \mathbf{C})$.