# Sigma-Algebra is Delta-Algebra

## Theorem

A $\sigma$-algebra is also a $\delta$-algebra.

## Proof

Let $\SS$ be a $\sigma$-algebra whose unit is $\mathbb U$.

Let $A_1, A_2, \ldots$ be a countably infinite collection of elements of $\SS$.

Then:

 $\ds \forall i: \,$ $\ds \mathbb U \setminus A_i$ $\in$ $\ds \SS$ $\SS$ is closed under relative complement with $\mathbb U$ $\ds \leadsto \ \$ $\ds \bigcup_{i \mathop = 1}^\infty \paren {\mathbb U \setminus A_i}$ $\in$ $\ds \SS$ $\SS$ is closed under countable unions $\ds \leadsto \ \$ $\ds \mathbb U \setminus \bigcap_{i \mathop = 1}^\infty A_i$ $\in$ $\ds \SS$ De Morgan's Laws $\ds \leadsto \ \$ $\ds \bigcap_{i \mathop = 1}^\infty A_i$ $\in$ $\ds \SS$ $\SS$ is closed under relative complement with $\mathbb U$

Thus $\SS$ is a $\delta$-algebra.

$\blacksquare$