Invariant measure

In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, and a difference of slopes is invariant under shear mapping.^[1]

Ergodic theory is the study of invariant measures in dynamical systems. The Krylov–Bogolyubov theorem proves the existence of invariant measures under certain conditions on the function and space under consideration.