Kac's lemma
E620664
Kac's lemma is a result in ergodic theory that relates the expected return time to a set in a measure-preserving dynamical system to the inverse of the measure of that set.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Kac's lemma canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6801182 — 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: Kac's lemma Context triple: [Poincaré recurrence theorem, relatedConcept, Kac's lemma]
-
A.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
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B.
Kolmogorov zero–one law
The Kolmogorov zero–one law is a fundamental result in probability theory stating that certain events determined by the tail behavior of independent random variables must have probability either zero or one.
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C.
Kac walk
The Kac walk is a probabilistic model introduced by mathematician Mark Kac to study the approach to equilibrium in kinetic theory via a simplified random process.
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D.
Kac
Kac is a surname most notably associated with Polish-American mathematician Mark Kac, known for his work in probability theory and mathematical physics.
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E.
Khinchin–Kolmogorov theorem
The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
- 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: Kac's lemma Target entity description: Kac's lemma is a result in ergodic theory that relates the expected return time to a set in a measure-preserving dynamical system to the inverse of the measure of that set.
-
A.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
-
B.
Kolmogorov zero–one law
The Kolmogorov zero–one law is a fundamental result in probability theory stating that certain events determined by the tail behavior of independent random variables must have probability either zero or one.
-
C.
Kac walk
The Kac walk is a probabilistic model introduced by mathematician Mark Kac to study the approach to equilibrium in kinetic theory via a simplified random process.
-
D.
Kac
Kac is a surname most notably associated with Polish-American mathematician Mark Kac, known for his work in probability theory and mathematical physics.
-
E.
Khinchin–Kolmogorov theorem
The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
result in ergodic theory
ⓘ
theorem ⓘ |
| appearsIn | Mark Kac's works on probability and statistical mechanics ⓘ |
| appliesTo |
ergodic measure-preserving transformations
ⓘ
measure-preserving dynamical systems ⓘ |
| assumes |
measurable set of positive measure
ⓘ
measure-preserving transformation ⓘ probability space ⓘ |
| category |
theorems in dynamical systems
ⓘ
theorems in ergodic theory ⓘ theorems in probability theory ⓘ |
| conclusion | E(τ_A | x in A) = 1/μ(A) where τ_A is the first return time to A ⓘ |
| consequence |
recurrence properties can be quantified via measure
ⓘ
the expected number of visits to a set is proportional to its measure ⓘ |
| field |
dynamical systems
ⓘ
ergodic theory ⓘ probability theory ⓘ |
| formalSetting | (X, Σ, μ, T) with μ a probability measure and T measure-preserving ⓘ |
| hasConcept |
ergodic transformation
ⓘ
first return time ⓘ invariant measure ⓘ recurrence time ⓘ |
| holdsIn |
discrete-time dynamical systems
ⓘ
measure-preserving transformations on probability spaces ⓘ |
| implies |
the average return time to a set is inversely proportional to its measure
ⓘ
the mean recurrence time to a set is finite if the set has positive measure ⓘ |
| influenced |
applications of ergodic theory to Markov chains
ⓘ
study of return-time statistics in dynamical systems ⓘ |
| isSpecialCaseOf | results on return times in stationary processes ⓘ |
| mathematicalDomain |
measure theory
ⓘ
probability on dynamical systems ⓘ |
| namedAfter | Mark Kac NERFINISHED ⓘ |
| relatedTo |
Birkhoff ergodic theorem
NERFINISHED
ⓘ
Markov chain stationary distributions ⓘ Poincaré recurrence theorem NERFINISHED ⓘ |
| relates |
expected return time to a set
ⓘ
inverse of the measure of that set ⓘ |
| requires | μ(A) > 0 for the set A ⓘ |
| statement | For a measure-preserving transformation on a probability space, the expected return time to a measurable set A of positive measure equals 1 divided by the measure of A. ⓘ |
| typicalFormulation | If T is a measure-preserving transformation on (X, Σ, μ) and A has μ(A) > 0, then the integral over A of the first return time to A equals 1. ⓘ |
| usedFor |
applications in Markov processes
ⓘ
applications in statistical mechanics ⓘ computing expected recurrence times ⓘ understanding recurrence in ergodic systems ⓘ |
| yearIntroducedApprox | 1940s ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Kac's lemma Description of subject: Kac's lemma is a result in ergodic theory that relates the expected return time to a set in a measure-preserving dynamical system to the inverse of the measure of that set.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.