Separable sigma algebra
Encyclopedia
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 pseudometric
that renders it separable as a pseudometric 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 pseudometric as defined above need not to be 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 induced metric. If the measure space is separable, it can be shown that the corresponding metric space is, too.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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 pseudometric
Pseudometric space
In mathematics, a pseudometric space is a generalized metric space in which the distance between two distinct points can be zero. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space...
that renders it separable as a pseudometric space
Pseudometric space
In mathematics, a pseudometric space is a generalized metric space in which the distance between two distinct points can be zero. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space...
. The distance between two sets is defined as the measure of the symmetric difference
Symmetric difference
In mathematics, the symmetric difference of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by A\,\Delta\,B\,orA \ominus B....
of the two sets. Note that the symmetric difference of two distinct sets can have measure zero; hence the pseudometric as defined above need not to be 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 induced metric. If the measure space is separable, it can be shown that the corresponding metric space is, too.