The Hopf Decomposition

Let G be a locally compact second countable group. We present a comprehensive treatment of the classical Hopf Decomposition (also known as the Conservative-Dissipative Decomposition) for general nonsingular G-spaces, and provide several fundamentally different characterizations of it. Subsequently, we establish a complete structure theorem for totally dissipative nonsingular G-spaces through the construction of Krengel G-spaces, extending Krengel's structure theorem on flows.