Koopman representation of a nonsingular action

Core idea

Let a countable discrete group act by measurable bijections $T_g$ on a sigma-finite measure space and preserve its measure class: $(T_g)_*\mu$ and $\mu$ have the same null sets. Its nonsingular Koopman representation is

\kappa(g)f(x)= \left(\frac{d(T_g)_*\mu}{d\mu}(x)\right)^{1/2}f(T_{g^{-1}}x).

The square-root Radon–Nikodym weight gives unitarity; the derivative chain rule gives the representation law.

Reduction to measure preservation

For a measure-preserving action, the derivative is one and this reduces to ordinary inverse pullback. In the general nonsingular case constants need not be fixed, so removing constants and using the probability-preserving fixed-vector criterion would not be justified.

References

Jesse Peterson, Lecture notes on ergodic theory, 2011. Definition 2.2.1.