Definition 4

Let $\mu$ be a positive measure on T. A vectorial measure $m$ on T, with values in F, is said to be a measure with base $\mu$ if there exists a mapping $f$ of T into F, scalarly locally $\mu$-integrable, such that $m(g) = \int gf\,d\mu$ for every function $g \in \mathscr{K}(T)$. One then says that $f$ is a density of $m$ with respect to $\mu$, and one writes $m = f \cdot \mu$.

It is immediate that if $f_1$ and $f_2$ are two densities of $m$ with respect to $\mu$, then $f_1 - f_2$ is scalarly locally $\mu$-negligible (Ch. V, §5, No. 3, Cor. 2 of Prop. 3); recall that in general this does not imply that $f_1 - f_2$ is zero locally almost everywhere (cf. §1, Exer. 12 and No. 1, Remark 2).

Let $m$ be a measure with base $\mu$, of density $f$. For a numerical function $g$ to be essentially integrable for $m$, it is necessary and sufficient that $gf$ be scalarly essentially $\mu$-integrable (Ch. V, §5, No. 3, Th. 1).