Sigma-algebra

A Sigma-algebra of sets, or $\sigma$-algebra is very similar to a $\sigma$-ring of sets.

Like how ring of sets and algebra of sets differ, the same applies to $\sigma$-ring compared to $\sigma$-algebra

Definition

A non empty class of sets $S$ is a $\sigma$-algebra if

• if $$A\in S$$ then $$A^c\in S$$
• if $$\{A_n\}_{n=1}^\infty\subset S$$ then $$\cup^\infty_{n=1}A_n\in S$$

That is it is closed under complement and countable union

First theorems

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The intersection of $\sigma$-algebras is a $\sigma$-algebra

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TODO: Proof - see PTACC page 5, also in Halmos AND in that other book

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See also

• Types of set algebras