Measure and integration Measure-preserving maps and further topics Concept
Measure-preserving transformation
Summary
A measure-preserving transformation is a measurable map between measure spaces under which the preimage of every measurable set has the same measure as the set. Equivalently, the pushforward of the first measure is the second one. A measure-preserving map of a space to itself is the basic object of ergodic theory. It models a system that evolves in time without creating or destroying mass, as in Hamiltonian mechanics.
Where this leads
26 pages build on this one.
- Ergodic decomposition
- Koopman operator
- Kingman's subadditive ergodic theorem
- Ornstein isomorphism theorem
- Kolmogorov automorphism
- Sinai's factor theorem
- Halmos–von Neumann theorem
- Mixing
- Furstenberg's multiple recurrence theorem
- Rokhlin lemma
- Weak mixing
- Ergodic transformation
- Shannon–McMillan–Breiman theorem
- Invariant measure
- Birkhoff's ergodic theorem
- Kac's lemma
- Oseledets theorem
- von Neumann mean ergodic theorem
- Spectral theory of dynamical systems
- Conservative system
- Poincaré recurrence theorem
- Maximal ergodic theorem
- Pesin theory
- Fundamental domain
- Kolmogorov–Sinai entropy
- Random dynamical system
Further reading
About this page
| Tag | 556WBDAHNP |
|---|---|
| URI | m2://556WBDAHNP |
| Kind | concept |
| Writing | W0 stub |
| MSC | 28D05, 37A05 |
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What links here
Instances
- Baker's map
- Interval exchange transformation
- Irrational rotation
- Bernoulli scheme
- Dyadic transformation
- Gauss map
Definitions
Used by
- Koopman operator
- Kingman's subadditive ergodic theorem
- Ornstein isomorphism theorem
- Kolmogorov automorphism
- Sinai's factor theorem
- Halmos–von Neumann theorem
- Mixing
- Furstenberg's multiple recurrence theorem
- Rokhlin lemma
- Weak mixing
- Ergodic transformation
- Shannon–McMillan–Breiman theorem
- Invariant measure
- Birkhoff's ergodic theorem
- Kac's lemma
- Oseledets theorem
- von Neumann mean ergodic theorem
- Spectral theory of dynamical systems
- Poincaré recurrence theorem
- Maximal ergodic theorem
- Fundamental domain
- Kolmogorov–Sinai entropy
- Random dynamical system
Results about this
- Kingman's subadditive ergodic theorem
- Furstenberg's multiple recurrence theorem
- Rokhlin lemma
- Kac's lemma
- Poincaré recurrence theorem
- Maximal ergodic theorem