What is a Dynkin system? ($\lambda$-system)

Solution 1:

A motivation for Dynkin systems (and specially, the $\pi$-$\lambda$ Theorem) is the following problem:

How many sets should I check to be sure that two probability measures $\mu$ and $\nu$ are the same?

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

If two probability measures coincide on a family $\mathcal{A}\subseteq\Sigma$, then they coincide on the Dynkin system generated by $\mathcal{A}$ (i.e., the smallest $\lambda$-system containing it).

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 $\mathcal{A}$.