Quasi-invariant measure under a locally compact group action

Definition

Let a locally compact group $G$ act measurably on a measurable space $X$ , and let $\mu$ be a $\sigma$ -finite measure on $X$ . The measure $\mu$ is quasi-invariant under the action if, for every $g\in G$ , the pushforward $g_*\mu$ , defined by $g_*\mu(E)=\mu(g^{-1}E)$ , is mutually absolutely continuous with $\mu$ . Equivalently,

\mu(E)=0\quad\Longleftrightarrow\quad\mu(gE)=0

for every measurable $E\subseteq X$ and $g\in G$ . Thus the action preserves the measure class of $\mu$ , though not necessarily its values. Invariance, meaning $g_*\mu=\mu$ for every $g$ , is a stronger condition.

Radon–Nikodym cocycle

By the Radon–Nikodym theorem, quasi-invariance yields derivatives

j(g,x)=\frac{d(g_*\mu)}{d\mu}(x)

defined almost everywhere. With consistent representatives they satisfy a cocycle identity, with the order depending on whether the action and pushforward conventions are written on the left or right. These square-root derivatives correct pullback operators so that actions on $L^2(X,\mu)$ become unitary Folland, §2.6.

Homogeneous spaces

For a closed subgroup $H\leq G$ , the locally compact [[lie-groups/homogeneous-space|homogeneous space]] $G/H$ always carries a natural quasi-invariant measure class under standard locally compact hypotheses, even when it has no $G$ -invariant measure. This measure class is sufficient for quasi-regular and induced-representation constructions Folland, §2.6.

Conventions and near misses

A measure for which $g_*\mu\ll\mu$ but $\mu\not\ll g_*\mu$ is not quasi-invariant; one-sided absolute continuity does not preserve the null-set class.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: §2.6, quasi-invariant measures on homogeneous spaces.