Separable sigma algebra

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 $\mathcal{F}$ which is a separable space when considered as a metric space with metric $\rho(A,B) = \mu(A \triangle B)$ for $A,B \in \mathcal{F}$ and a given measure $\mu$ (and with $\triangle$ being the symmetric difference operator). 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 (since every Lebesgue measurable set is equivalent to some Borel set) but not countably generated (since its cardinality is higher than continuum).

单词	Separable measure space
释义	Separable measure space 英语百科 Separable sigma algebra 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 $\mathcal{F}$ which is a separable space when considered as a metric space with metric $\rho(A,B) = \mu(A \triangle B)$ for $A,B \in \mathcal{F}$ and a given measure $\mu$ (and with $\triangle$ being the symmetric difference operator). 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 (since every Lebesgue measurable set is equivalent to some Borel set) but not countably generated (since its cardinality is higher than continuum).
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Separable measure space

Separable sigma algebra