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 sigma algebra that can be generated by a countable collection of sets. To learn what is meant by the σ-algebra generated by a collection of sets, refer to the article on sigma algebras.

A separable measure space has a natural metric that renders it separable as a metric space. The distance between two sets is defined as the measure of the symmetric difference of the two sets. Note that the symmetric difference of two distinct sets can have measure zero; hence the metric as defined above is not a true metric. However, if sets whose symmetric difference has measure zero are identified into a single equivalence class, the resulting quotient set can be properly metrized by the above metric. If the measure space is separable, it can be shown that the corresponding metric space is, too.