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 which is a separable space when considered as a metric space with metric for and a given measure (and with 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).