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The relation $P\ll \tilde{P}$ is an instance of absolute continuity of measures: it means that every $\tilde{P}$-null event is also $P$-null,By the Radon-Nikodym theorem, this is equivalent to the existence of a nonnegative Radon-Nikodym derivative $dP/d\tilde{P}$ whose $\tilde{P}$-expectation is one.

The map $F$ is a nonsingular transformation with respect to $\omega$ whenEquivalently, the pushforward measure $F_{\ast }\omega (B)=\omega (F^{-1}(B))$ satisfies $F_{\ast }\omega \ll \omega$.

For an essentially bounded observable $g$, define the Koopman operatorNonsingularity makes this well defined on almost-everywhere equivalence classes. By the Radon-Nikodym theorem,Consequently the bounded Koopman operator criterion is

Each component of a finite vector measure is absolutely continuous with respect to its variation measure. The Radon-Nikodym theorem gives a vector density $\sigma$. Taking the variation measure of this representation gives $|\mu |=|\sigma ||\mu |$, hence the unit-length property. For a BV derivative this separates the magnitude of variation from its local direction.

On a finite measure space, a positive functional $\Lambda \in (L^p{)}^{\ast }$ defines the finite measure $\nu (E)=\Lambda (1_E)$. The Radon-Nikodym theorem gives $d\nu =\omega d\mu$ with $\omega \ge 0$. The boundand $L^p$-$L^q$ norm duality imply $\omega \in L^q$, after which density extends the integral representation to every $f\in L^p$.