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| In [[mathematics]], an '''invariant measure''' is a [[measure (mathematics)|measure]] that is preserved by some [[function (mathematics)|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.
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| ==Definition==
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| Let (''X'', Σ) be a [[measurable space]] and let ''f'' be a [[measurable function]] from ''X'' to itself. A measure ''μ'' on (''X'', Σ) is said to be '''invariant under''' ''f'' if, for every measurable set ''A'' in Σ,
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| :<math>\mu \left( f^{-1} (A) \right) = \mu (A).</math>
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| In terms of the [[pushforward measure|push forward]], this states that ''f''<sub>∗</sub>(''μ'') = ''μ''.
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| The collection of measures (usually [[probability measure]]s) on ''X'' that are invariant under ''f'' is sometimes denoted ''M''<sub>''f''</sub>(''X''). The collection of [[ergodic (adjective)|ergodic measures]], ''E''<sub>''f''</sub>(''X''), is a subset of ''M''<sub>''f''</sub>(''X''). Moreover, any [[convex combination]] of two invariant measures is also invariant, so ''M''<sub>''f''</sub>(''X'') is a [[convex set]]; ''E''<sub>''f''</sub>(''X'') consists precisely of the extreme points of ''M''<sub>''f''</sub>(''X'').
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| In the case of a [[Dynamical system (definition)|dynamical system]] (''X'', ''T'', ''φ''), where (''X'', Σ) is a measurable space as before, ''T'' is a [[monoid]] and ''φ'' : ''T'' × ''X'' → ''X'' is the flow map, a measure ''μ'' on (''X'', Σ) is said to be an '''invariant measure''' if it is an invariant measure for each map ''φ''<sub>''t''</sub> : ''X'' → ''X''. Explicitly, ''μ'' is invariant [[if and only if]]
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| :<math>\mu \left( \varphi_{t}^{-1} (A) \right) = \mu (A) \qquad \forall t \in T, A \in \Sigma.</math>
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| Put another way, ''μ'' is an invariant measure for a sequence of [[random variable]]s (''Z''<sub>''t''</sub>)<sub>''t''≥0</sub> (perhaps a [[Markov chain]] or the solution to a [[stochastic differential equation]]) if, whenever the initial condition ''Z''<sub>0</sub> is distributed according to ''μ'', so is ''Z''<sub>''t''</sub> for any later time ''t''.
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| ==Examples==
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| :[[File:Hyperbolic sector squeeze mapping.svg|250px|right|thumb|[[Squeeze mapping]] leaves [[hyperbolic angle]] invariant as it moves a purple [[hyperbolic sector]] to one of the same area. Blue and green rectangles also keep the same area]]
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| * Consider the [[real line]] '''R''' with its usual [[Borel sigma algebra|Borel σ-algebra]]; fix ''a'' ∈ '''R''' and consider the translation map ''T''<sub>''a''</sub> : '''R''' → '''R''' given by:
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| ::<math>T_{a} (x) = x + a.</math>
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| : Then one-dimensional [[Lebesgue measure]] ''λ'' is an invariant measure for ''T''<sub>''a''</sub>.
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| * More generally, on ''n''-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup> with its usual Borel σ-algebra, ''n''-dimensional Lebesgue measure ''λ''<sup>''n''</sup> is an invariant measure for any [[isometry]] of Euclidean space, i.e. a map ''T'' : '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup> that can be written as
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| ::<math>T(x) = A x + b</math>
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| : for some ''n'' × ''n'' [[orthogonal matrix]] ''A'' ∈ O(''n'') and a vector ''b'' ∈ '''R'''<sup>''n''</sup>.
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| * 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 <math> \boldsymbol{\rm S}=\{A,B\}</math> and the identity map <math>T={\rm Id}</math> which leaves each point fixed. Then any probability measure <math>\mu : \boldsymbol{\rm S} \rightarrow \boldsymbol{\rm R} </math> is invariant. Note that '''S''' trivially has a decomposition into ''T''-invariant components ''{A}'' and ''{B}''.
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| * The measure of [[angle|circular angles]] in [[degree (angle)|degree]]s or [[radian]]s is invariant under [[rotation]]. Similarly, the measure of [[hyperbolic angle]] is invariant under [[squeeze mapping]].
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| * [[Area]] measure in the Euclidean plane is invariant under [[2 × 2 real matrices#Equi-areal mapping|2 × 2 real matrices with determinant 1]], also known as the ''special linear group'' [[SL(2,R)]].
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| * Every [[locally compact group]] has a [[Haar measure]] that is invariant under the group action.
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| ==See also== | |
| *[[Quasi-invariant measure]]
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| ==References==
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| *Invariant measures, John Von Neumann, AMS Bookstore, 1999, ISBN 978-0-8218-0912-9
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| {{DEFAULTSORT:Invariant Measure}}
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| [[Category:Dynamical systems]]
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| [[Category:Measures (measure theory)]]
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