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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. 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. 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.

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Pushforward measure

In measure theory, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a measure from one measurable space to another using a measurable function. Given measurable spaces and , a measurable mapping and a measure , the pushforward of is defined to be the measure given by for This definition applies mutatis mutandis for a signed or complex measure. The pushforward measure is also denoted as , , , or .

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In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a rotation or shear mapping. For a fixed positive real number a, the mapping is the squeeze mapping with parameter a. Since is a hyperbola, if u = ax and v = y/a, then uv = xy and the points of the image of the squeeze mapping are on the same hyperbola as (x,y) is.

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