Measure-preserving semiflows and one-parameter Koopman semigroups

Abstract

For a finite measure space $$\mathrm {X}$$ , we characterize strongly continuous Markov lattice semigroups on $$\mathrm {L}^p(\mathrm {X})$$ by showing that their generator A acts as a derivation on the dense subspace $$D(A)\cap \mathrm {L}^\infty (\mathrm {X})$$ . We then use this to characterize Koopman semigroups on $$\mathrm {L}^p(\mathrm {X})$$ if $$\mathrm {X}$$ is a standard probability space. In addition, we show that every measurable and measure-preserving flow on a standard probability space is isomorphic to a continuous flow on a compact Borel probability space.