In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular...

In mathematics, a quasi-invariant measure μ with respect to a transformation T, from a measure space X to itself, is a measure which, roughly speaking...

In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral...

theory, a Sinai–Ruelle–Bowen (SRB) measure is an invariant measure that behaves similarly to, but is not an ergodic measure. In order to be ergodic, the time...

necessary for the invariant measure to be ergodic; hence the notions of "ergodicity" for a Markov chain and the associated shift-invariant measure are different...

system (see Invariants of tensors). The singular values of a matrix are invariant under orthogonal transformations. Lebesgue measure is invariant under translations...

Lebesgue measure cannot be straightforwardly extended to all infinite-dimensional spaces due to a key limitation: any translation-invariant Borel measure on...

mathematics, the Krylov–Bogolyubov theorem (also known as the existence of invariant measures theorem) may refer to either of the two related fundamental theorems...

strictly invariant sets, and by I ~ {\displaystyle {\tilde {\mathcal {I}}}} the sigma-algebra of almost surely invariant sets. Given a measure-preserving...

the trivial measure on some measurable space (X, Σ). A measure ν is the trivial measure μ if and only if ν(X) = 0. μ is an invariant measure (and hence...

harmonic measure of B on ∂D is invariant under all rotations of D about x and coincides with the normalized surface measure on ∂D. The harmonic measure satisfies...

have a natural measure, such as the Liouville measure in Hamiltonian systems, chosen over other invariant measures, such as the measures supported on periodic...

transformation map T {\displaystyle T} ; the measure μ {\displaystyle \mu } can now be understood as an invariant measure; it is just the Frobenius–Perron eigenvector...

functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive measure (or mean) on subsets...

translation-invariant operator. The mapping from a polynomial function to the polynomial degree is a translation-invariant functional. The Lebesgue measure is...

modern form, in his geometric invariant theory. In large measure due to the influence of Mumford, the subject of invariant theory is seen to encompass the...

a measure μ on X that the map f leaves unchanged, a so-called invariant measure, i.e one for which f∗(μ) = μ. One can also consider quasi-invariant measures...

^{n}} is necessarily a Lebesgue measure. The Borel measure is translation-invariant, but not complete. The Haar measure can be defined on any locally compact...

by the invariant measure. It can be visualized as the behavior of a point-cloud or dust-cloud under repeated iteration. The invariant measure is an eigenstate...

is characterized by a quantity called an invariant measure or distribution function, and the invariant measure of the attractor is reproducible regardless...

descriptions as a fallback Haar measure – Left-invariant (or right-invariant) measure on locally compact topological group Lebesgue measure – Concept of area in...

other measure with these properties extends the Lebesgue measure. Circular angle measure is invariant under rotation, and hyperbolic angle measure is invariant...

thus the Bernoulli measure is a Haar measure; it is an invariant measure on the product space. Instead of the probability measure P : B → R {\displaystyle...

important convergence results. In short, we need the existence of invariant measure and Harris recurrent to establish the Law of Large Numbers of MCMC...

p(x)\log {\frac {p(x)}{q(x)}}\,dx} where q(x), which Jaynes called the "invariant measure", is proportional to the limiting density of discrete points. For...

{d}{dt}}J_{z}=0\,,} in other words angular momentum is conserved. Axial symmetry Invariant measure Isotropy Maxwell's theorem Rotational symmetry Stenger, Victor J....

by many authors. It was introduced by Alfréd Rényi in 1957, and an invariant measure for it was given by Alexander Gelfond in 1959 and again independently...

dependent on the observer. A different term, proper distance, provides an invariant measure whose value is the same for all observers. Proper distance is analogous...

properties that make the discrete entropy a useful measure of uncertainty. In particular, it is not invariant under a change of variables and can become negative...

unique invariant probability measure μ f {\displaystyle \mu _{f}} of maximal entropy for f, called the equilibrium measure (or Green measure, or measure of...