Definition 2

Let $\theta$ be a complex measure, and let $u$ be a complex function (or a function with values in $\overline{\mathbf{R}}$) defined locally $\theta$-almost everywhere and locally $\theta$-integrable. The complex measure $f \mapsto \int fu\,d\theta$ on T is said to be the product of the measure $\theta$ by the function $u$, or the measure with density $u$ with respect to $\theta$, and is denoted $u \cdot \theta$.

Every complex measure that is the product of a positive measure $\mu$ by a locally $\mu$-integrable function is called a measure with base $\mu$.

The relation $\eta = u \cdot \theta$ is again, by convention, written

When $u$ is everywhere defined and continuous, one recovers the definition given in Ch. III, §1, No. 4. It is clear that if $u_1$ and $u_2$ are locally $\theta$-integrable, then $(u_1 + u_2) \cdot \theta = u_1 \cdot \theta + u_2 \cdot \theta$. Similarly, if $\theta_1$ and $\theta_2$ are two measures on $T$, and if $u$ is a function locally integrable for $\theta_1$ and $\theta_2$, then $u$ is locally integrable for $\theta_1 + \theta_2$ and one has $u \cdot (\theta_1 + \theta_2) = u \cdot \theta_1 + u \cdot \theta_2$.

We shall henceforth restrict ourselves to the case of functions defined everywhere; the extension to functions defined locally almost everywhere, which is always obvious, is left to the reader.

The following proposition permits, for the most part, reducing the case of complex measures to that of positive measures: