Sigma-algebra

up:: 023 MOC Measure Theory

Given a set $X$, a $\sigma$-algebra over $X$ $\mathcal{F}\subseteq \mathcal{P}(X)$ such that

• $X\in \mathcal{F}$
• Closed under intersection: $\forall A,B\in \mathcal{F}:A\cap B\in \mathcal{F}$
• Closed under countable unions: $\forall \{A_n{\}}_{n\in \mathbb{N}}\subseteq \mathcal{F}:\bigcup _{n\in \mathbb{N}}{A}_n\in \mathcal{F}$

Properties

• $\varnothing \in \mathcal{F}$;
• $\mathcal{F}$ is also closed under countable intersections
• Elements of $\mathcal{F}$ are called measurable sets (since they are allowed to be measured via a Measure (function))

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References

• σ-algebra - Wikipedia