Dynkin’s lemma

Dynkin’s lemma is a result in measure theory showing that the $\sigma$ -algebra (http://planetmath.org/SigmaAlgebra) generated by any given $\pi$ -system (http://planetmath.org/PiSystem) on a set $X$ coincides with the Dynkin system generated the $\pi$ -system. The result can be used to prove that measures are uniquely determined by their values on $\pi$ -systems generating the required $\sigma$ -algebra. For example, the Borel $\sigma$ -algebra on $\mathbb{R}$ is generated by the $\pi$ -system of open intervals $(a,b)$ for $a<b$ and consequently the Lebesgue measure $\mu$ is uniquely determined by the property that $\mu((a,b))=b-a$ .

Note that this lemma generalizes the statement that a Dynkin system which is also a $\pi$ -system is a $\sigma$ -algebra.

Lemma (Dynkin).

Let $A$ be a $\pi$ -system on a set $X$ . Then $\mathcal{D}(A)=\sigma(A)$ . That is, the smallest Dynkin system containing $A$ coincides with the $\sigma$ -algebra generated by $A$ .

Proof.

As $A$ is a $\pi$ -system, the set $\mathcal{D}_{1}\equiv\{S\subseteq X:S\cap T\in\mathcal{D}(A)\text{ for every }% T\in A\}$ contains $A$ . We show that $\mathcal{D}_{1}$ is also a Dynkin system.

First, for every $T\in A$ , $X\cap T=T\in A$ so $X$ is in $\mathcal{D}_{1}$ . Second, if $S_{1}\subseteq S_{2}$ are in $\mathcal{D}_{1}$ and $T\in A$ then $(S_{2}\setminus S_{1})\cap T=(S_{2}\cap T)\setminus(S_{1}\cap T)$ is in $\mathcal{D}(A)$ showing that $S_{2}\setminus S_{1}\in\mathcal{D}_{1}$ . Finally. if $S_{n}\in\mathcal{D}_{1}$ is a sequence increasing to $S\subseteq X$ and $T\in A$ then $S_{n}\cap T$ is a sequence in $\mathcal{D}(A)$ increasing to $S\cap T$ . As Dynkin systems are closed under limits of increasing sequences this shows that $S\cap T\in\mathcal{D}(A)$ and therefore $S\in\mathcal{D}_{1}$ . So $\mathcal{D}_{1}$ is indeed a Dynkin system. In particular, $\mathcal{D}(A)\subseteq\mathcal{D}_{1}$ and $S\cap T\in\mathcal{D}(A)$ for all $S\in\mathcal{D}(A)$ and $T\in A$ .

We now set $\mathcal{D}_{2}\equiv\{S\subseteq X:S\cap T\in\mathcal{D}(A)\text{ for every }% T\in\mathcal{D}(A)\}$ which, as shown above, contains $A$ . Also, as in the argument above for $\mathcal{D}_{1}$ , $\mathcal{D}_{2}$ is a Dynkin system. Therefore, $\mathcal{D}(A)$ is contained in $\mathcal{D}_{2}$ and it follows that $S\cap T\in\mathcal{D}(A)$ for any $S,T\in\mathcal{D}(A)$ . So $\mathcal{D}(A)$ is both a $\pi$ -system and a Dynkin system.

We can now show that $\mathcal{D}(A)$ is a $\sigma$ -algebra. As it is a Dynkin system, $S^{c}=X\setminus S\in\mathcal{D}(A)$ for every $S\in\mathcal{D}(A)$ and, as it is also a $\pi$ -system, this shows that $\mathcal{D}(A)$ is an algebra of sets on $X$ . Finally, choose any sequence $A_{n}\in\mathcal{D}(A)$ . Then, $\bigcup_{m=1}^{n}A_{m}$ is a sequence in $\mathcal{D}(A)$ increasing to $\bigcup_{n}A_{n}$ which, as $\mathcal{D}(A)$ is Dynkin system, must be in $\mathcal{D}(A)$ . So, $\mathcal{D}(A)$ is a $\sigma$ -algebra and must contain $\sigma(A)$ . Conversely, as $\sigma(A)$ is a Dynkin system (as it is a $\sigma$ -algebra) containing $A$ , it must also contain $\mathcal{D}(A)$ . ∎

References

1 David Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, 1991.

Title	Dynkin’s lemma
Canonical name	DynkinsLemma
Date of creation	2013-03-22 18:33:05
Last modified on	2013-03-22 18:33:05
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	11
Author	gel (22282)
Entry type	Theorem
Classification	msc 28A12
Synonym	pi-system d-system lemma
Related topic	PiSystem
Related topic	DynkinSystem
Related topic	UniquenessOfMeasuresExtendedFromAPiSystem