Until recently, all my knowledge of measure theory and Lebesgue integration are from Rudin's book, which focuses solely on the Lebesgue measure, its construction and nothing else. I have just put my hands on a nice book "Measure and Integration Theory" by Heinz Bauer and I'm currently enjoying it. I have encountered the definition of a Dynkin system $\mathcal D$, which is a family of subsets of a set $\Omega$ satisfying
1.) $\Omega\in\mathcal D$.
2.) If $A\in\mathcal D$, then $A^c\in\mathcal D$.
3.) For $n\in\Bbb N$, if $A_n\in\mathcal D$ are pairwise disjoint then $\bigcup_{n=1}^{\infty}A_n\in\mathcal D$.
I have some idea about what a $\sigma$-algebra is, but not about a Dynkin system. I would really appreciate if someone could give me an intuition about Dynkin systems or what they're supposed to represent. What is the characteristics of a Dynkin system that let you recognize it once you see it?
I know the $\pi$–$\lambda$ theorem and facts like a Dynkin system $\mathcal D$ is a $\sigma$-algebra if it is closed under intersection, it would be also nice if anyone could explain to me why should we expect such a result. Thank you in advance.
Best Answer
A motivation for Dynkin systems (and specially, the $\pi$-$\lambda$ Theorem) is the following problem:
For the sake of definiteness, both $\mu$ and $\nu$ are defined on a measurable space $(\Omega,\Sigma)$. A trivial answer to the previous question is “If they coincide over $\Sigma$, they are the same”. True, and useless. So let's think about this for a moment. If I have checked that $\mu(A) =\nu(A)$ for some $A\in\Sigma$, it is not necessary to check for the complement, since $$ \mu(A^c) =1-\mu(A) = 1-\nu(A) = \nu(A^c). $$ It is also immediate by $\sigma$-additivity that if they coincide on a sequence of pairwise disjoint sets $A_n$, they must coincide on their union. Hence we conclude
The above arguments can't be generalized to intersections; we can't compute $\mu(A\cap B)$ from $\mu(A)$ and $\mu(B)$. So, it would be desirable that our initial data (sets checked for coincidence) is a family of sets closed under binary intersections. Here, the $\pi$-$\lambda$ confirms this intuition: If $\Sigma$ is generated by a family $\mathcal{A}$, then it equals the smallest Dynkin system including the closure of $\mathcal{A}$ under intersections (i.e., the $\pi$-system generated by it).
Therefore, to check that $\mu$ and $\nu$ are equal, it is enough to take a family $\mathcal{A}$ such that of $\Sigma =\sigma(\mathcal{A})$ and check that the measures coincide on every finite intersection of members of $\mathcal{A}$.