Ergodic Theory is the mathematical theory of well-behaved randomness. The random variables can occur discretely or continuously, but the total probability over the time period is always 1 so that the end of the time period, the random event will surely have occurred. The probability distribution takes extreme values over the class of shift invariant processes.
The first serious study of ergodic theory was performed by Ludwig Boltzmann in the 1930s, who also coined the term. His thesis was to justify the use of equilibria to study mechanics in a probabilistic model. John von Neumann and George David Birkhoff are other seminal contributors to the field.
A controversial application of the probability-based ergodic theory is to number theory, which traditionally has been one of the most rigorous fields of mathematics.