Separable sigma algebra

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In mathematics, σ-algebras are usually studied in the context of measure theory. A separable σ-algebra (or separable σ-field) is a σ-algebra ${\displaystyle {\mathcal {F}}}$ which is a separable space when considered as a metric space with metric ${\displaystyle \rho (A,B)=\mu (A\triangle B)}$ for ${\displaystyle A,B\in {\mathcal {F}}}$ and a given measure ${\displaystyle \mu }$ (and with ${\displaystyle \triangle }$ being the symmetric difference operator).{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} Note that any σ-algebra generated by a countable collection of sets is separable, but the converse need not hold (for example, the Lebesgue σ-algebra is separable but not countably generated). {{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}