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

be a measurable space be a measurable function is said to be invariant under

In terms of the pushforward measure

The collection of measures (usually probability measures The collection of ergodic measures Moreover, any convex combination is a convex set

In the case of a dynamical system is a monoid is said to be an invariant measure is invariant if and only if

is an invariant measure for a sequence of random variables (perhaps a Markov chain or the solution to a stochastic differential equation

When the dynamical system can be described by a transfer operator this being the largest eigenvalue as given by the Frobenius–Perron theorem.

Examples

• Consider the real line with its usual Borel σ-algebra Then one-dimensional Lebesgue measure

• -dimensional Euclidean space is an invariant measure for any isometry orthogonal matrix

• Area measure in the Euclidean plane is invariant under the special linear group real matrices of determinant

• Every locally compact group has a Haar measure that is invariant under the group action (translation).

See also

References

• John von Neumann (1999) Invariant measures, American Mathematical Society