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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.

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rdfs:label	Invariant measure (en) 不変測度 (ja) Miara niezmiennicza (pl) Инвариантная мера (ru)
rdfs:comment	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. (en) 数学において不変測度（ふへんそくど、英: invariant measure）とは、ある函数によって保存される測度のことを言う。エルゴード理論は、力学系における不変測度についての研究である。クリロフ＝ボゴリューボフの定理は、函数と考えている空間に関するある条件の下での不変測度の存在を示すものである。 (ja) Инвариантная мера — в теории динамических систем мера, определённая в фазовом пространстве, связанная с динамической системой и не изменяющаяся с течением времени при эволюции состояния динамической системы в фазовом пространстве. Понятие инвариантной меры применяется при усреднении уравнений движения, в теории показателей Ляпунова, в теории метрической энтропии и вероятностных фрактальных размерностей. (ru) Miara niezmiennicza – miara zachowywana przez pewną funkcję. Są one szczególnym obszarem zainteresowań w studiach nad układami dynamicznymi. mówi o istnieniu miar niezmienniczych pod pewnymi warunkami względem danych: funkcji i przestrzeni. (pl)
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Link from a Wikipage to another Wikipage	Determinant Hyperbolic angle Convex set Mathematics Measure (mathematics) Geometric transformation Function (mathematics) Monoid Convex combination Pushforward measure Angle Slope Krylov–Bogolyubov theorem Measurable function Measurable space Transfer operator Haar measure Lebesgue measure Locally compact group American Mathematical Society Ergodic theory Euclidean space Ergodic (adjective) Random variable Isometry Measures (measure theory) Dynamical system (definition) Dynamical systems Area Dynamical systems Squeeze mapping If and only if Orthogonal matrix Real matrix Markov chain Special linear group Stochastic differential equation Shear mapping Probability measure Real line Borel sigma algebra Frobenius-Perron theorem
sameAs	Invariant measure Invariant measure Invariant measure Invariant measure Invariant measure Invariant measure Invariant measure
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has abstract	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. (en) 数学において不変測度（ふへんそくど、英: invariant measure）とは、ある函数によって保存される測度のことを言う。エルゴード理論は、力学系における不変測度についての研究である。クリロフ＝ボゴリューボフの定理は、函数と考えている空間に関するある条件の下での不変測度の存在を示すものである。 (ja) Инвариантная мера — в теории динамических систем мера, определённая в фазовом пространстве, связанная с динамической системой и не изменяющаяся с течением времени при эволюции состояния динамической системы в фазовом пространстве. Понятие инвариантной меры применяется при усреднении уравнений движения, в теории показателей Ляпунова, в теории метрической энтропии и вероятностных фрактальных размерностей. (ru) Miara niezmiennicza – miara zachowywana przez pewną funkcję. Są one szczególnym obszarem zainteresowań w studiach nad układami dynamicznymi. mówi o istnieniu miar niezmienniczych pod pewnymi warunkami względem danych: funkcji i przestrzeni. (pl)
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is Link from a Wikipage to another Wikipage of	Robert Zimmer Rufus Bowen Parry–Daniels map Hyperbolic angle Hyperbolic functions John von Neumann Vladimir Bogachev Dyadic transformation Dynamical systems theory Index of physics articles (I) Infinite-dimensional Lebesgue measure Invariant (mathematics) Positive real numbers Preconditioned Crank–Nicolson algorithm Essentially unique Quasi-invariant measure Glossary of areas of mathematics

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