Measure-preserving transformations and systems

Depends on

• Measure spaces
• A measurable function between measurable spaces

Used by

• An orientation-reversing homeomorphism need not be quasiconformal Counterexample
• The unitary dual need not be Hausdorff Counterexample
• Asymptotically mean stationary sources Definition
• Ergodic partial sums, time averages, and the invariant L2 subspace Definition
• Finite-alphabet stationary source Definition
• Invertible measure-preserving systems Definition
• Reiter's condition (P1) Definition
• Stationary process and canonical path shift Definition
• Strict and mod-null invariant sigma-algebras Definition
• Strong and weak mixing on a probability space Definition
• The Koopman operator Definition
• The one-dimensional torus and its normalized Haar integral Definition
• Unique ergodicity Definition
• A circle representation with an averaged orthogonal weight form Example
• A preserving permutation on two equal atoms Example
• A symmetric random walk on the circle Example
• Affine flux reduces the entropy semigroup to translation Example
• Compact groups have a constant Reiter net Example
• Matching C¹ pieces across a hyperplane have no jump derivative Example
• Newton shell theorem from harmonic mean values Example
• Opposed most powerful tests for two normal alternatives Example
• The positive-type Gaussian on the real line and its cyclic model Example
• False: Poincare recurrence needs no finite total measure False statement
• A Reiter net has an invariant-mean cluster point Lemma
• Averaging a Hermitian form unitarizes a finite-dimensional compact-group representation Lemma
• Haar averaging projects contractively onto the bounded intertwiners Lemma
• Monomial integrals on the sphere and orthonormality on the distinguished torus Lemma
• Monomials form complete orthogonal systems of the Bergman spaces of the disc, the ball and the polydisc Lemma
• No-return sets have disjoint null preimage towers Lemma
• Operators commuting with a generating family of multiplications Lemma
• The maximal ergodic inequality on a probability space Lemma
• Triangular averages of finite-past information converge to the infinite-past average Lemma
• Compact and locally compact abelian groups are amenable Proposition
• Compositions, iterates and completions preserve invariance Proposition
• A two-sided normal alternative need not have a UMP test Theorem
• Integral invariance under measure-preserving maps Theorem
• Isotypic projections are mutually orthogonal equivariant projections Theorem
• Iwasawa decomposition and Haar integration formula for SL2(R) Theorem
• Krylov–Bogolyubov existence of an invariant probability Theorem
• Measure preservation can be checked on a generating pi-system Theorem

…and 5 more results.

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• E–W Definition 2.1, pp.13–14 (standard reference, not scraped)