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content/Definitions/Measure Theory/Sigma-Algebra/Borel Sigma-Algebra.md

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+**Borel $\sigma$-algebra** on a [[Topological Space|topological space]] $X$ is the one generated by $\tau$.

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

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 # 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. $A \in M \implies A^{\complement} \in M$ ($X^{\complement} = \emptyset \in M$)
-3. $A_{n} \in M \implies \cup^{\infty}_{n=1} A_{n} \in M$ ($\implies \cap^{\infty}_{n=1} A_{n} = (\cup^{\infty}_{n=1}A^{\complement})^{\complement} \in M)$)
+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]]