rhoaction

Rhoaction is a term used in mathematical contexts, notably in ergodic theory and harmonic analysis, to describe a group or semigroup action on a measure space that carries a Radon–Nikodym derivative, or cocycle, describing how the measure changes under the action. The name reflects the role of the function rho in tracking the quasi-invariance of the measure.

Formally, let (X, Σ, μ) be a measure space and let G act on X by measurable transformations T_g: X → X with T_e = identity and T_{gh} = T_g ∘ T_h. The action is called a rhoaction when, for each g ∈ G, there is a nonnegative measurable function ρ(g, x) such that, for all A ∈ Σ,

μ(T_g^{-1}(A)) = ∫_A ρ(g, x) dμ(x).

The family {ρ(g, ·)} forms a Radon–Nikodym cocycle satisfying ρ(e, x) = 1 and ρ(gh, x) = ρ(g, h x) ρ(h, x) almost everywhere. If ρ(g, x) = 1 for all g and x, the action preserves μ and is a standard measure-preserving action.

Examples include translations on the real line with Lebesgue measure, where ρ ≡ 1, and more generally nonuniformly changing measures under affine or nonlinear transformations. Rhoactions also arise in the construction of crossed-product von Neumann algebras and in the analysis of unitary representations associated with quasi-invariant measures.

Related concepts include group actions, quasi-invariant measures, Radon–Nikodym derivatives, cocycles, and modular theory. The rhoaction framework helps describe dynamical systems where the underlying measure is not preserved but transforms in a controlled, computable way.