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.[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.

Definition

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Let   be a measurable space and let   be a measurable function from   to itself. A measure   on   is said to be invariant under   if, for every measurable set   in    

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.

Examples

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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
  • 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 (translation).

See also

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References

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  1. ^   Geometry/Unified Angles at Wikibooks
  • John von Neumann (1999) Invariant measures, American Mathematical Society ISBN 978-0-8218-0912-9