# 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 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 DynkinsLemma 2013-03-22 18:33:05 2013-03-22 18:33:05 gel (22282) gel (22282) 11 gel (22282) Theorem msc 28A12 pi-system d-system lemma PiSystem DynkinSystem UniquenessOfMeasuresExtendedFromAPiSystem