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.
In terms of the pushforward measure, this states that
The collection of measures (usually probability measures) on that are invariant under is sometimes denoted The collection of ergodic measures, is a subset of Moreover, any convex combination of two invariant measures is also invariant, so is a convex set; consists precisely of the extreme points of
In the case of a dynamical system where is a measurable space as before, is a monoid and is the flow map, a measure on is said to be an invariant measure if it is an invariant measure for each map Explicitly, is invariant if and only if
Put another way, is an invariant measure for a sequence of random variables (perhaps a Markov chain or the solution to a stochastic differential equation) if, whenever the initial condition is distributed according to so is for any later time
When the dynamical system can be described by a transfer operator, then the invariant measure is an eigenvector of the operator, corresponding to an eigenvalue of this being the largest eigenvalue as given by the Frobenius–Perron theorem.
- Consider the real line with its usual Borel σ-algebra; fix and consider the translation map given by: Then one-dimensional Lebesgue measure is an invariant measure for
- More generally, on -dimensional Euclidean space with its usual Borel σ-algebra, -dimensional Lebesgue measure is an invariant measure for any isometry of Euclidean space, that is, a map that can be written as for some orthogonal matrix and a vector
- The invariant measure in the first example is unique up to trivial renormalization with a constant factor. This does not have to be necessarily the case: Consider a set consisting of just two points and the identity map which leaves each point fixed. Then any probability measure is invariant. Note that trivially has a decomposition into -invariant components and
- Area measure in the Euclidean plane is invariant under the special linear group of the real matrices of determinant
- Every locally compact group has a Haar measure that is invariant under the group action.
- Geometry/Unified Angles at Wikibooks