Sigma Algebra

Last updated Nov 1, 2022

Let $X$ be a Nonempty Set. Then $\mathcal{M} \subset \mathcal{P}(X)$ is a Sigma Algebra on $X$ if

1. $X \in \mathcal{M}$
2. If $A \in \mathcal{M}$ then $A^{C} \in \mathcal{M}$
3. Suppose $(A_n){n \in \mathbb{N}} \subset \mathcal{M}$. Then $\bigcup\limits{n \in \mathbb{N}} A_{n} \in \mathcal{M}$

1. $\mathcal{M}$ is closed under Countable Set Intersection

   Proof: This follows from De Morgan’s Law. Suppose $(A_n){n \in \mathbb{N}} \subset \mathcal{M}$. Then $$\bigcap\limits{n \in \mathbb{N}} A_{n} = \Big( \bigcup\limits_{n \in \mathbb{N}} A_{n}^{C} \Big)^{C} \in \mathcal{M}$$