π-Systems and Uniqueness of Measures
Two models give the same probabilities to every interval. Must they be the same model? The answer is yes — and the reason is the single most reused proof pattern in probability theory.
In this subchapter
1.4.1The question: how much agreement is total agreement?
Two quants model the same asset. Their probability measures P₁ and P₂ on (ℝ, ℬ(ℝ)) agree on every interval: for all a < b, P₁((a,b)) = P₂((a,b)). Have they built the same model — the same number for every Borel event — or could they still disagree on some complicated Borel set nobody happened to check?
Intervals are not a σ-algebra; agreement on them is not closure-flavored agreement. Yet the answer is: agreement on intervals is total agreement. The proof needs two structural definitions — families that close under finite overlap, and families that close under the set-bookkeeping half of the σ-algebra rules — and one theorem binding them. The pattern is called the good sets principle, and once you see it you will meet it everywhere: distribution functions determine laws (4.2), product measures exist and are unique (6.1), independence is checkable on generators (6.1), the Kolmogorov extension theorem (8.3) — all instances of the machinery built here.
1.4.2π-systems: families closed under overlap
A family 𝒫 of subsets of Ω is a π-system if it is closed under finite intersections: whenever A, B ∈ 𝒫, also A ∩ B ∈ 𝒫. (Only the one operation is demanded — complements and unions are free to wander.)
The open intervals form a π-system: (a, b) ∩ (c, d) is (max(a,c), min(b,d)) or empty — either way an interval. The half-lines (−∞, t]: the intersection of two is the shorter half-line. The rectangles (a, b) × (c, d) in the plane: overlap of rectangles is a rectangle. All the natural generator families of probability are π-systems — this is not an accident, as Theorem 3 below shows.
On Ω = {1, 2, 3, 4} take 𝒜 = { {1,2}, {2,3} }. Each member is a fine event, but the intersection {1,2} ∩ {2,3} = {2} is not in the family. Not a π-system. Park this example: in §6 it becomes the star of the breaking example — the whole difference between "uniqueness holds" and "uniqueness fails" lives in that missing {2}.
1.4.3d-systems: Dynkin's half of a σ-algebra
A σ-algebra demands three clauses. Dynkin's observation: for uniqueness arguments you only need the bookkeeping clauses — those that mesh with how measures compute — plus one way to combine.
A family 𝒟 of subsets of Ω is a d-system if:
- Ω ∈ 𝒟;
- if A, B ∈ 𝒟 with A ⊆ B, then B ∖ A ∈ 𝒟;
- if A₁, A₂, … ∈ 𝒟 are pairwise disjoint, then ⋃n An ∈ 𝒟.
Clause 3 demands unions only when the pieces do not overlap — the case where additivity makes measures easy. That is the design: d-systems are the family shape that measures can compute with.
A family F is a σ-algebra if and only if it is both a π-system and a d-system.
(⇒) A σ-algebra is a d-system: complement clause gives B ∖ A = B ∩ Ac; the union clause covers disjoint unions; π: closed under finite intersections is Proposition 2 of 1.1.
(⇐) Let F be both. Complements: if A ∈ F, then A ⊆ Ω, both in F, so Ac = Ω ∖ A ∈ F by d-clause 2. Countable unions, overlaps and all: given A₁, A₂, … ∈ F, disjointify — B₁ = A₁, Bn = An ∖ (A₁ ∪ … ∪ An−1). The partial unions are in F by an induction: if Cn−1 = A₁ ∪ … ∪ An−1 ∈ F, then Bn = An ∖ (An ∩ Cn−1) ∈ F — using the π-clause to know An ∩ Cn−1 ∈ F, then d-clause 2. The Bn are disjoint and in F, so d-clause 3 puts ⋃n Bn = ⋃n An in F. Note where π was used: exactly to control the overlaps. A d-system alone cannot disjointify. ∎
1.4.4Dynkin's π–λ theorem
As in 1.2 for σ-algebras: d(𝒫) is the smallest d-system containing 𝒫 — the intersection of all d-systems that contain it (the intersection argument of Theorem 1 there runs verbatim: each d-clause survives intersection).
If 𝒫 is a π-system, then d(𝒫) = σ(𝒫). In words: the smallest d-system containing a π-system already contains everything the σ-rules force — it is the generated σ-algebra.
One inclusion is free: a σ-algebra is a d-system (Proposition 1), so d(𝒫) ⊆ σ(𝒫). For the hard direction we show d(𝒫) is a π-system; then, being both π and d, Proposition 1 makes it a σ-algebra containing 𝒫, and minimality of σ(𝒫) gives σ(𝒫) ⊆ d(𝒫). So everything hinges on: overlaps of d-system members stay in the d-system. The trick — remember it, it recurs in every measure-theory proof of this shape — is to fix one slot and probe with the other:
For A ∈ d(𝒫) define the good sets relative to A:
Step 1. If A ∈ 𝒫, then 𝒟A ⊇ 𝒫 — precisely because 𝒫 is a π-system: B ∈ 𝒫 means A ∩ B ∈ 𝒫 ⊆ d(𝒫). And a direct check of the three d-clauses shows 𝒟A is a d-system (e.g. if B ⊆ C in 𝒟A, then A ∩ (C ∖ B) = (A∩C) ∖ (A∩B), a difference of nested members of d(𝒫), hence in it). Being a d-system between 𝒫 and d(𝒫), minimality gives 𝒟A = d(𝒫). Conclusion so far: every B ∈ d(𝒫) overlaps every A ∈ 𝒫 inside d(𝒫).
Step 2. Now fix B ∈ d(𝒫) and probe again: by Step 1, every A ∈ 𝒫 lies in 𝒟B = {A : A ∩ B ∈ d(𝒫)}, which is again a d-system (same check, roles swapped). So 𝒟B ⊇ 𝒫, and minimality again: 𝒟B = d(𝒫). That says: for all A, B ∈ d(𝒫), A ∩ B ∈ d(𝒫). This is the π-property. Done. ∎
The pattern: to prove that a family has a property, define the sub-family of members that behave well relative to a fixed probe, show the sub-family is a d-system (or σ-algebra) containing the generator, and conclude by minimality that it is everything. The fixed-slot trick converts a two-quantifier statement ("for all pairs") into two one-quantifier statements. Every uniqueness proof below is this skeleton with a different adjective.
1.4.5Uniqueness of measures
Let 𝒫 be a π-system with σ(𝒫) = F, and let μ, ν be finite measures on (Ω, F) — finite meaning μ(Ω), ν(Ω) < ∞; probabilities qualify, with total mass 1. If μ(A) = ν(A) for every A ∈ 𝒫, then μ = ν on all of F.
Consider the agreement set:
𝒢 ⊇ 𝒫 by hypothesis. Now the three d-clauses, one by one:
- Ω ∈ 𝒢: μ(Ω) = ν(Ω) — for probabilities both are 1; for general finite measures this must be part of the hypothesis (see the warning below).
- If A ⊆ B in 𝒢: μ(B ∖ A) = μ(B) − μ(A) = ν(B) − ν(A) = ν(B ∖ A) — subtraction is legal because the measures are finite (this is where finiteness earns its keep, and where infinite measures would break the proof). So B ∖ A ∈ 𝒢.
- If A₁, A₂, … disjoint in 𝒢: additivity along the sequence, μ(⋃An) = ∑μ(An) = ∑ν(An) = ν(⋃An). So the union is in 𝒢.
So 𝒢 is a d-system containing the π-system 𝒫, whence d(𝒫) ⊆ 𝒢. By Theorem 2, σ(𝒫) = d(𝒫) ⊆ 𝒢 ⊆ F. Every set of F is in 𝒢: the measures agree everywhere. ∎
Apply Theorem 3 on ℝ with 𝒫 the intervals with rational endpoints — a π-system (Example 1) generating ℬ(ℝ) (Theorem 1 of 1.3). Two probabilities agreeing on those intervals agree on every Borel set. The two quants of §1: same model, guaranteed. Even better, agreement on half-lines (−∞, t] — a π-system generating ℬ by the same theorem — is enough: equal distribution functions imply equal laws. That statement becomes the definition-anchoring theorem of 4.2.
On [0, 1], there is at most one probability with P((a,b)) = b − a on all sub-intervals: intervals form a π-system generating ℬ([0,1]), and Theorem 3 pins the measure down uniquely. Note carefully what was and was not proved: uniqueness — at most one such measure. Existence — that such a measure exists at all on the Borel sets, with length on intervals extended to everything — is the entire content of Chapter 3. One number per interval sounds trivial; making it consistent on all Borel sets is the construction the next chapters carry.
1.4.6The condition is not negotiable: a breaking example
Theorem 3 has two hypotheses beyond the π-system: finiteness, and that the agreement family generates. Drop π — even with a generating family — and everything fails, on a four-point space.
Ω = {1, 2, 3, 4}, 𝒜 = { {1,2}, {2,3} } — the non-π-system of Example 2. It generates the full power set: the atoms {1}, {2}, {3}, {4} are carved out by intersections with complements, so σ(𝒜) = P(Ω). Now two probability measures, masses listed outcome by outcome:
Check the agreement: μ({1,2}) = 0.5 = ν({1,2}), and μ({2,3}) = 0.5 = ν({2,3}). They agree on the entire generating family — yet μ({1}) = 0.3 ≠ 0.2 = ν({1}): different measures. Theorem 3 does not apply, because 𝒜 is not a π-system — its one missing member, the overlap {2}, is exactly the channel through which the two measures smuggle their disagreement past the checks. And the repair writes itself: the smallest π-system containing 𝒜 adds {2}; agreeing on {1,2}, {2,3}, {2} pins down {1} = {1,2} ∖ {2} and {3} similarly, and then {4} by total mass — total agreement. π is the difference between a theorem and a counterexample.
Clause 2 of the good-sets proof subtracts: μ(B) − μ(A). For infinite measures, ∞ − ∞ is undefined, and the proof genuinely fails: two infinite measures really can agree on a π-system generating the σ-algebra and still differ. The standard repair — agree on the π-system and on a sequence of sets of finite measure whose union is the whole space — is exactly the σ-finiteness condition that appears in Chapter 6's product arguments. The safe statement to carry: for probabilities, finiteness is automatic and Theorem 3 applies as stated; for general measures, always ask "am I subtracting?" before reusing the pattern.
1.4.7Where this pattern will resurface
Close the chapter by naming the future instances, so you recognize the skeleton under new clothes:
- →Distribution functions determine laws (4.2): {X ≤ t} sets form a π-system; equal CDFs = agreement there = equal measures. The "a distribution is its CDF" folklore is Theorem 3 in disguise.
- →Independence is checkable on generators (6.1): two families generating independent σ-algebras are independent as soon as the generators are π-systems. Without π, false — for the same reason as Example 5.
- →Product measures are unique (6.1) and the Kolmogorov extension theorem holds: consistency on finite-dimensional cylinders (a π-system) forces a unique law on the infinite product space.
- →Martingale proofs (8.3): good-set arguments over σ(𝒢)-measurability reappear when we check conditional expectations are what they claim.
Uniqueness is never checked everywhere — it is inherited from a generator, provided the generator closes under overlap. π is the toll you pay for the shortcut.
Chapter 1 is complete: we know what an event is (1.1), how event collections are built (1.2), the σ-algebra the real line actually uses (1.3), and how much of a measure a generator determines (here). What is missing are the numbers themselves: the axioms a probability measure must satisfy, their immediate consequences, and the pathologies that appear the moment countable additivity is relaxed. That is Chapter 2, starting with the axioms.