Definition 3

Let f be a locally $\mu$-integrable positive function. The measure $f \cdot \mu : K \mapsto f_K \cdot \mu_K$ is called the measure with density f with respect to $\mu$, or the product measure of $\mu$ by the function $f$. Every measure of the form $f \cdot \mu$, where $f$ is positive and locally $\mu$-integrable, is called a measure with base $\mu$.