Dynkin System Closed under Intersections is Sigma-Algebra

Theorem
Let $X$ be a set, and let $\DD$ be a Dynkin system on $X$.

Suppose that $\DD$ satisfies the following condition:


 * $(1):\quad \forall D, E \in \DD: D \cap E \in \DD$

That is, $\DD$ is closed under intersection.

Then $\DD$ is a $\sigma$-algebra.

Proof
The first two conditions for a Dynkin system are identical to those for a $\sigma$-algebra.

Hence it is only required to verify that $(1)$ implies that $\DD$ is closed under arbitrary countable unions.

So let $\sequence {D_n}_{n \mathop \in \N}$ be a sequence in $\DD$.

Now define the sequence $\sequence {E_n}_{n \mathop \in \N}$ by:


 * $\displaystyle E_n := D_n \cap \paren {X \setminus \bigcup_{m \mathop < n} D_m}$

By Dynkin System Closed under Union and $(1)$, it follows that $E_n \in \DD$ for all $n \in \N$.

Lemma
For all $n \in \N$, it holds that:


 * $\displaystyle \bigcup_{k \mathop = 0}^n E_k = \bigcup_{k \mathop = 0}^n D_k$

Proof of Lemma
The lemma, combined with the definition of the $E_n$, gives immediately that for all $n \in \N$:


 * $\displaystyle E_n \in X \setminus \bigcup_{m \mathop < n} D_m = X \setminus \bigcup_{m \mathop < n} E_m$

whence the $E_n$ are pairwise disjoint.

Another consequence is that:


 * $\displaystyle \bigcup_{n \mathop \in \N} D_n = \bigcup_{n \mathop \in \N} E_n$

Now since the $E_n$ are pairwise disjoint, it follows that:


 * $\displaystyle \bigcup_{n \mathop \in \N} E_n \in \DD$

which, combined with above equality, concludes in:


 * $\displaystyle \bigcup_{n \mathop \in \N} D_n \in \DD$

Therefore, $\DD$ is closed under countable unions, making it a $\sigma$-algebra.