John von Neumann and the birth of Ergodic Theory

Modern ergodic theory (as we know it today) began around 1930 with groundbreaking work by Birkhoff, Koopman and von Neumann, to list only some of the main contributors in alphabetical order. The most celebrated results from this period were, of course, the ergodic theorems by Birkhoff and von Neumann, but many other concepts and results with significant impact on this new discipline also originated around that time, with major contributions by John von Neumann.
Prominent examples of these are ergodic decomposition (the decomposition of measure-preserving, or non-singular, dynamical systems into ergodic components), and the isomorphism problems (the  problem of classifying the equivalence classes of dynamical systems with respect to various natural notions of isomorphism – including the weakest of these notions, orbit equivalence).