Dynkin system

Family over a fixed universe

Record D as a family of subsets of one explicitly identified Ω, including the empty-universe case when relevant.

1. What is Ω, and how is D ⊆ P(Ω) established? definition
2. Is membership in D given by enumeration, a predicate, or a generation rule? definition

Lambda-system convention

Use Dynkin system and lambda-system for the standard measure-theoretic notion; formulations using nested differences and increasing unions require their own equivalence justification. [Oxford definition](https://ora.ox.ac.uk/objects/uuid%3A743530c5-135f-4a16-b127-18a72939462a/files/d2r36tz46z)

1. Which reference supplies the adopted definition and terminology? provenance
2. If another axiom formulation is used, what proves its equivalence over the same Ω? definition

Universe and complement obligations

Verify Ω ∈ D and A ∈ D implies Ω \ A ∈ D; together these also give ∅ ∈ D.

1. What establishes Ω ∈ D? definition
2. What general argument proves complement closure, or which member has a complement outside D? boundary

Disjoint sequence obligation

For every sequence of pairwise-disjoint members, establish that its union belongs to D; finitely many successful examples do not establish this universal claim.

1. How are membership and pairwise disjointness established for the sequence under consideration? definition
2. What proves closure for every such sequence, or supplies a sequence whose union fails membership? boundary
3. If checking a finite presentation exhaustively, why does the check cover the countable-union obligation? action

Nested differences and monotone limits

A Dynkin system supports A \ B when B ⊆ A are members, increasing countable unions, and decreasing countable intersections. Nested differences enable the disjoint decomposition used for increasing unions.

1. For a proposed difference A \ B, are both sets members and is B ⊆ A established? action
2. For a proposed limit operation, is the sequence increasing or decreasing as required? action

Intersection witness or proof

Record either a proof of binary-intersection closure or a witness against it. For example, on Ω = {1,2,3,4}, the family consisting of ∅, Ω, and all two-element subsets is a Dynkin system but excludes {1,2} ∩ {1,3} = {1}.

1. Can binary-intersection closure be proved for this D, or can two members with an excluded intersection be exhibited? boundary
2. Does a proposed overlapping union rely on an extra property that has actually been established? action

Generated family characterisation

For C ⊆ P(Ω), λ(C) is the intersection of all Dynkin systems on Ω containing C. Identifying a candidate D with λ(C) requires both containment directions.

1. What is the initial family C, and does every member lie in P(Ω)? definition
2. What establishes D = λ(C), rather than merely C ⊆ D? boundary

Sigma-algebra extension

If P is closed under binary intersections and P ⊆ D for a Dynkin system D, then σ(P) ⊆ D; consequently λ(P) = σ(P). [LSU pi-lambda theorem](https://www.math.lsu.edu/~sengupta/7360f09/DynkinPiLambda.pdf)

1. What proves that P is a pi-system and that P ⊆ D? action
2. Which target sigma-algebra is known to equal σ(P)? boundary
3. Does the conclusion establish containment in D or equality with D, and what justifies any equality? action

Property closure obligations

Define D_Q as the sets in a specified ambient sigma-algebra satisfying Q, then prove the universe, complement, and disjoint-union obligations for Q.

1. What exact property Q defines D_Q, and on which sets is Q meaningful? definition
2. Which arguments show that Q survives complements and countable disjoint unions and holds on Ω? action

Finite measure agreement

For finite measures μ and ν on the same sigma-algebra with equal total mass, the family {A : μ(A) = ν(A)} is a Dynkin system. Agreement on a generating pi-system then extends to that sigma-algebra. Infinite-measure applications need separately verified hypotheses.

1. Are μ and ν defined on the same sigma-algebra, finite, and equal on Ω? boundary
2. On which generating pi-system has their agreement been proved? provenance
3. If total mass is infinite, what precise uniqueness theorem and localisation assumptions replace the finite-measure argument? action

Check these first

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these check these first hold for the sense of Dynkin system this model covers, and on what evidence? provenance

Real-world use

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these real-world use hold for the sense of Dynkin system this model covers, and on what evidence? provenance

Failure modes and hazards

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these failure modes and hazards hold for the sense of Dynkin system this model covers, and on what evidence? provenance

Neighbouring kinds and how to tell them apart

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these neighbouring kinds and how to tell them apart hold for the sense of Dynkin system this model covers, and on what evidence? provenance