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

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...

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

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...

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...

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

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...

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...

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...

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...

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...

{\displaystyle S.} The Lebesgue measure on R {\displaystyle \mathbb {R} } is a complete translation-invariant measure on a σ-algebra containing the intervals...

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

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...

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...

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

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

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...

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

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...

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

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...

system and has been used as a measure of complexity in many diverse fields. In information processing, complexity is a measure of the total number of properties...

σ-finite quasi-invariant measure μ on X which is unique up to measure equivalence (that is any two such measures have the same sets of measure zero). If Φ...

fixed points, computing the total variation, and constructing an invariant measure with maximal entropy. Kneading theory provides an effective calculus...

{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....