Invariant measure
![Squeeze mapping leaves hyperbolic angle invariant as it moves a hyperbolic sector (purple) to one of the same area. Blue and green rectangles also keep the same area](/Images/godic/202501/20/Hyperbolic_sector_squeeze_mapping.svg4832.png")
In mathematics, an invariant measure is a measure that is preserved by some function. 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.