Monotone Class Theorem

The monotone class theorem (Dynkin's pi-lambda theorem) is the workhorse for extending equalities of measures from a generating pi-system to the full sigma-algebra. Whenever two measures agree on a collection closed under finite intersections, they agree on everything that collection generates.

Statement

A collection $\mathcal{P}$ of subsets of $\Omega$ is a pi-system if it is closed under finite intersections: $A, B \in \mathcal{P} \Rightarrow A \cap B \in \mathcal{P}$.

A lambda-system (Dynkin system) $\mathcal{D}$ satisfies:

1. $\Omega \in \mathcal{D}$,
2. $A \in \mathcal{D} \Rightarrow A^c \in \mathcal{D}$,
3. disjoint $A_1, A_2, \ldots \in \mathcal{D} \Rightarrow \bigcup_n A_n \in \mathcal{D}$.

Dynkin's theorem: If $\mathcal{P}$ is a pi-system and $\mathcal{D}$ is a lambda-system with $\mathcal{P} \subseteq \mathcal{D}$, then $\sigma(\mathcal{P}) \subseteq \mathcal{D}$.

Equivalently, the lambda-system generated by $\mathcal{P}$ equals $\sigma(\mathcal{P})$.

Visualization

Venn structure of pi-systems vs lambda-systems:

Sets closed under:    fin. intersection?   complements?   disjoint countable union?
---------------------------------------------------------------------------
Algebra               yes                  yes            finite only
Sigma-algebra         yes                  yes            yes (all countable)
Pi-system             yes                  no             no
Lambda-system         no                   yes            yes (disjoint only)
---------------------------------------------------------------------------

Concrete example — product sigma-algebra on $\mathbb{R}^2$:

Let $\mathcal{P} = \{A \times B : A, B \in \mathcal{B}(\mathbb{R})\}$ (measurable rectangles). This is a pi-system: $(A_1 \times B_1) \cap (A_2 \times B_2) = (A_1 \cap A_2) \times (B_1 \cap B_2)$.

The theorem says $\mathcal{B}(\mathbb{R}^2) = \sigma(\mathcal{P})$. Any two measures agreeing on all rectangles must agree on all Borel sets — which is how product measures are uniquely characterised.

Uniqueness of Lebesgue measure via the theorem:

Step 1: Lebesgue measure agrees with any translation-invariant Borel
        probability measure on intervals (a,b] — a pi-system.
Step 2: Both generate the Borel sigma-algebra.
Step 3: By pi-lambda, they agree on all Borel sets.

Proof Sketch

1. Fix the pi-system $\mathcal{P}$. Let $\mathcal{D}_0 = \lambda(\mathcal{P})$ be the smallest lambda-system containing $\mathcal{P}$.

2. Show $\mathcal{D}_0$ is also a pi-system: for each fixed $B \in \mathcal{P}$, the collection $\{A : A \cap B \in \mathcal{D}_0\}$ is a lambda-system containing $\mathcal{P}$, so it contains $\mathcal{D}_0$. Repeat with $B \in \mathcal{D}_0$ to get closure under all intersections.

3. A lambda-system that is also a pi-system is a sigma-algebra (standard exercise: countable disjointification).

4. Therefore $\mathcal{D}_0$ is a sigma-algebra, and since $\mathcal{D}_0 \supseteq \mathcal{P}$ we have $\mathcal{D}_0 \supseteq \sigma(\mathcal{P})$. The reverse inclusion $\sigma(\mathcal{P}) \supseteq \lambda(\mathcal{P})$ is immediate since every sigma-algebra is a lambda-system.

Connections

The pi-lambda theorem is the key step in proving uniqueness of the product measureFundamental Theorem of Calculus$$\int_a^b f'(x)\,dx = f(b) - f(a)$$Integration and differentiation are inverse operationsRead more → and appears implicitly in the proof of Bayes' TheoremBayes' Theorem$$P(A \mid B) = \frac{P(B \mid A)\, P(A)}{P(B)}$$Reversing conditional probability to update beliefs from evidenceRead more → (where measures on a sigma-algebra generated by a pi-system of conditioning events must coincide). It also underpins Dominated Convergence TheoremDominated Convergence Theorem$$f_n \to f \text{ a.e.},\; |f_n| \leq g \in L^1 \Rightarrow \int f_n \to \int f$$Pointwise convergence plus a uniform integrable dominating bound lets you pass the limit inside the integralRead more → via the monotone class approach to extending integral identities.

import Mathlib.MeasureTheory.PiSystem

open MeasureTheory MeasurableSpace

/-- Dynkin's pi-lambda theorem: the lambda-system generated by a pi-system
    equals the sigma-algebra it generates. -/
theorem pi_lambda {α : Type*} {s : Set (Set α)} (hs : IsPiSystem s) :
    DynkinSystem.generate s = DynkinSystem.ofMeasurableSpace (generateFrom s) :=
  (DynkinSystem.generateFrom_eq hs).symm ▸ le_antisymm
    (DynkinSystem.generate_le _ (fun t ht => measurableSet_generateFrom ht))
    (DynkinSystem.le_generate _ (fun t ht => by
      rwa [DynkinSystem.generateFrom_eq hs] at ht))