Learn about measure preservation and ergodicity in this mathematics lecture from the Simons Semester on Dynamics series. Explore the Birkhoff ergodic theorem, including its definition, proof, and applications to continuous compactly supported functions. Examine the ergodicity of irrational circle rotations and flows on the 2-torus through various mathematical arguments, including approaches by Ratner/Masur, Codene, and Dani. Master concepts of hyperbolic total automorphisms and their ergodicity and mixing properties using the Hopf argument, while developing a deep understanding of equidistribution theory and dynamical systems.