---
tag: BBHM1A36M7
kind: statement
role: theorem
name: Dynkin's π-λ theorem
msc: [28A05, 60A10]
uses: [4W9NM803BB, V5M2CKB591, 0SV8K0TDGY, 059TJN0NZ1]
about: [4W9NM803BB, V5M2CKB591]
sources:
  - url: https://en.wikipedia.org/wiki/Dynkin_system
---

## Statement

Let $X$ be a set, let $\mathcal{P}$ be a [[π-system|4W9NM803BB]] on $X$, and let $\mathcal{D}$ be a [[Dynkin system|V5M2CKB591]] on $X$ with $\mathcal{P} \subseteq \mathcal{D}$.
Let $\sigma(\mathcal{P})$ be the [[σ-algebra|0SV8K0TDGY]] generated by $\mathcal{P}$.
Then $\sigma(\mathcal{P}) \subseteq \mathcal{D}$.

## Remarks

Let $d(\mathcal{P})$ be the smallest Dynkin system that contains $\mathcal{P}$.
For a fixed $A \in \mathcal{P}$, the sets $B$ with $A \cap B \in d(\mathcal{P})$ form a Dynkin system that contains $\mathcal{P}$, so it contains $d(\mathcal{P})$.
Repeating the argument with $A \in d(\mathcal{P})$ shows that $d(\mathcal{P})$ is a π-system.
A Dynkin system that is a π-system is a σ-algebra, so $\sigma(\mathcal{P}) = d(\mathcal{P}) \subseteq \mathcal{D}$.

The main use is uniqueness of measures.
Let $\mu$ and $\nu$ be [[probability measures|059TJN0NZ1]] on $(X, \sigma(\mathcal{P}))$ that agree on $\mathcal{P}$.
The sets on which $\mu$ and $\nu$ agree form a Dynkin system that contains $\mathcal{P}$, so $\mu = \nu$ on $\sigma(\mathcal{P})$.
For example, a probability measure on the Borel sets of $\R$ is fixed by its values on the rays $(-\infty, a]$.

The hypothesis that $\mathcal{P}$ is a π-system is needed.
On $X = \set{1, 2, 3, 4}$, let $\mathcal{P} = \set{\set{1, 2}, \set{2, 3}}$ and let $\mathcal{D}$ be the family of subsets with an even number of elements.
Then $\mathcal{D}$ is a Dynkin system that contains $\mathcal{P}$, but $\set{2} = \set{1, 2} \cap \set{2, 3}$ lies in $\sigma(\mathcal{P})$ and not in $\mathcal{D}$.

## Explanations

### L1 human

To show that a property holds for every set of a σ-algebra, it is often enough to check it on a simple generating family.
The theorem says this works when the simple family is closed under intersections and the good sets form a Dynkin system.
Dynkin systems are easy to check, since they only ask for complements and disjoint unions.

### L2 human

The π-λ theorem is the workhorse behind uniqueness statements in measure theory and probability.
It shows that a distribution function determines a probability measure on $\R$, and that a measure on a product is fixed by its values on rectangles.
It also gives the independence of σ-algebras generated by independent π-systems.
The monotone class theorem plays the same role with algebras of sets in place of π-systems.