Kakutani’s random ergodic theorem
E170224
Kakutani’s random ergodic theorem is a fundamental result in ergodic theory that extends classical ergodic theorems to sequences of randomly chosen measure-preserving transformations.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Kakutani’s random ergodic theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1477427 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Kakutani’s random ergodic theorem Context triple: [Shizuo Kakutani, hasTheoremNamedAfter, Kakutani’s random ergodic theorem]
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A.
Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
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B.
Khinchin–Kahane type inequalities
Khinchin–Kahane type inequalities are fundamental results in probability and functional analysis that bound moments or norms of random series (often with Rademacher or Gaussian coefficients) in terms of each other, providing powerful tools for studying the geometry of Banach spaces and random processes.
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C.
Poincaré–Birkhoff fixed-point theorem
The Poincaré–Birkhoff fixed-point theorem is a fundamental result in dynamical systems and topology that guarantees the existence of at least two fixed points for certain area-preserving twist maps of an annulus.
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D.
Carathéodory’s extension theorem
Carathéodory’s extension theorem is a fundamental result in measure theory that guarantees a unique extension of a pre-measure defined on an algebra of sets to a complete measure on the generated σ-algebra.
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E.
Poincaré recurrence theorem
The Poincaré recurrence theorem is a fundamental result in dynamical systems and ergodic theory stating that certain systems will, after a sufficiently long but finite time, return arbitrarily close to their initial state.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Kakutani’s random ergodic theorem Target entity description: Kakutani’s random ergodic theorem is a fundamental result in ergodic theory that extends classical ergodic theorems to sequences of randomly chosen measure-preserving transformations.
-
A.
Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
-
B.
Khinchin–Kahane type inequalities
Khinchin–Kahane type inequalities are fundamental results in probability and functional analysis that bound moments or norms of random series (often with Rademacher or Gaussian coefficients) in terms of each other, providing powerful tools for studying the geometry of Banach spaces and random processes.
-
C.
Poincaré–Birkhoff fixed-point theorem
The Poincaré–Birkhoff fixed-point theorem is a fundamental result in dynamical systems and topology that guarantees the existence of at least two fixed points for certain area-preserving twist maps of an annulus.
-
D.
Carathéodory’s extension theorem
Carathéodory’s extension theorem is a fundamental result in measure theory that guarantees a unique extension of a pre-measure defined on an algebra of sets to a complete measure on the generated σ-algebra.
-
E.
Poincaré recurrence theorem
The Poincaré recurrence theorem is a fundamental result in dynamical systems and ergodic theory stating that certain systems will, after a sufficiently long but finite time, return arbitrarily close to their initial state.
- F. None of above. chosen
Statements (33)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in ergodic theory ⓘ |
| appliesTo |
Markovian or i.i.d. random products of transformations
ⓘ
random dynamical systems ⓘ |
| assumes |
independent identically distributed random transformations in many formulations
ⓘ
integrable observable function ⓘ probability space with measure-preserving transformations ⓘ |
| author | Shizuo Kakutani ⓘ |
| concerns |
measure-preserving dynamical systems
ⓘ
random sequences of measure-preserving transformations ⓘ |
| extends |
classical ergodic theorems to random compositions
ⓘ
ergodic theorems to sequences of randomly chosen transformations ⓘ |
| field |
ergodic theory
ⓘ
measure theory ⓘ probability theory ⓘ |
| generalizes |
ergodic theorem
ⓘ
surface form:
Birkhoff’s pointwise ergodic theorem
ergodic theorem ⓘ
surface form:
von Neumann’s mean ergodic theorem
|
| guarantees | existence of almost sure limits of random ergodic averages under suitable conditions ⓘ |
| implies | pointwise convergence of random time averages for almost every point ⓘ |
| influenced |
development of stochastic ergodic theory
ⓘ
later work on random dynamical systems ⓘ |
| namedAfter | Shizuo Kakutani ⓘ |
| publishedIn | Annals of Mathematics ⓘ |
| relatedTo |
random walks on groups
ⓘ
stationary processes ⓘ subadditive ergodic theorems ⓘ |
| states | almost sure convergence of random ergodic averages ⓘ |
| topic |
almost sure behavior of random orbits
ⓘ
convergence of random compositions of transformations ⓘ |
| uses |
martingale convergence ideas in some proofs
ⓘ
tools from measure theory ⓘ tools from probability theory ⓘ |
| yearProved | 1948 ⓘ |
How these facts were elicited
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Input
Subject: Kakutani’s random ergodic theorem Description of subject: Kakutani’s random ergodic theorem is a fundamental result in ergodic theory that extends classical ergodic theorems to sequences of randomly chosen measure-preserving transformations.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.