Absolutely continuous measure

Summary

A measure $\nu$ is absolutely continuous with respect to a measure $\mu$, written $\nu \ll \mu$, when every $\mu$-null set is also $\nu$-null. By the Radon–Nikodym theorem, for σ-finite measures this holds exactly when $\nu$ has a density with respect to $\mu$. It is the measure-theoretic counterpart of the absolute continuity of functions. A nondecreasing function on a closed bounded interval is absolutely continuous exactly when its Lebesgue–Stieltjes measure is absolutely continuous with respect to Lebesgue measure. In probability, random variables with a density are those whose law is absolutely continuous with respect to Lebesgue measure.