Dynkin System Closed under Intersections is Sigma-Algebra

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Theorem

Let $X$ be a set, and let $\mathcal D$ be a Dynkin system on $X$.

Suppose that $\mathcal D$ satisfies the following condition:

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

That is, $\mathcal D$ is closed under intersection.


Then $\mathcal D$ 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 $\mathcal D$ is closed under arbitrary countable unions.


So let $\left({D_n}\right)_{n \in \N}$ be a sequence in $\mathcal D$.

Now define the sequence $\left({E_n}\right)_{n \in \N}$ by:

$\displaystyle E_n := D_n \cap \left({X \setminus \bigcup_{m \mathop < n} D_m}\right)$

By Dynkin System Closed under Union and $(1)$, it follows that $E_n \in \mathcal D$ 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:

$\displaystyle \bigcup_{n \mathop \in \N} E_n \in \mathcal D$

which, combined with above equality, concludes in:

$\displaystyle \bigcup_{n \mathop \in \N} D_n \in \mathcal D$

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

$\blacksquare$


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