Dynkin's π-λ theorem

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 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 $\mathcal{D}$.

Going deeper

For a student

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.