Definition
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,
In terms of the push forward, this states that f∗(μ) = μ.
The collection of measures (usually probability measures) on X that are invariant under f is sometimes denoted Mf(X). The collection of ergodic measures, Ef(X), is a subset of Mf(X). Moreover, any convex combination of two invariant measures is also invariant, so Mf(X) is a convex set; Ef(X) consists precisely of the extreme points of Mf(X).
In the case of a 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 φt : X → X. Explicitly, μ is invariant if and only if
Put another way, μ is an invariant measure for a sequence of random variables (Zt)t≥0 (perhaps a Markov chain or the solution to a stochastic differential equation) if, whenever the initial condition Z0 is distributed according to μ, so is Zt for any later time t.
Read more about this topic: Invariant Measure
Famous quotes containing the word definition:
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)