Remark 1

The definition of $f \cdot \mu$ extends to the case that $f$ is a complex locally integrable function; one then has $|f \cdot \mu| = |f| \cdot \mu$, which implies at once that $f \cdot \mu$ is a measure, not just a premeasure. We retain the expression 'measures with base $\mu$' to designate the complex measures so defined.