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Concept
Invariant measure
Summary
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.
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.
Definition
Let (X, \Sigma) be a measurable space and let f : X \to X be a measurable function from X to itself. A measure \mu on (X, \Sigma) is said to be invariant under f if, for every measurable set A in \Sigma,
\mu\left(f^{-1}(A)\right) = \mu(A).
In terms of the pushforward measure, this states that f_*(\mu) = \mu.
Official source
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