Measure with density

Summary

Given a measure space $(X, \Sigma, \mu)$ and a measurable function $f \ge 0$, the measure with density $f$ gives each set $A$ the mass $\int_A f \, d\mu$. It is written $f \mu$ or $f \, d\mu$. It is the basic way to build new measures from old ones. The Radon–Nikodym theorem says that, for σ-finite measures, every measure absolutely continuous with respect to $\mu$ is of this form.