ACIT4330-Page/content/Definitions/Measure Theory/Sigma-Algebra/Sigma-Algebra.md

Definition

A $\sigma$-algebra in a set X is a collection of subsets, so called measurable sets, of X such that (the requirements are):

1. X \in M
2. A \in M \implies A^{\complement} \in M (A^{\complement} \equiv X \setminus A) (X^{\complement} = \emptyset \in M)
3. A_{n} \in M \implies \cup^{\infty}_{n=1} A_{n} \in M (\implies \cap^{\infty}_{n=1} A_{n} = (\cup^{\infty}_{n=1}A^{\complement})^{\complement} \in M))

Related Terminologies/Functions

• Measurable
• Measure