Standard Borel spaces are particular measurable spaces, well suited for a number of constructions in traditional probability theory, such as for forming conditional probability.
The real line is in particular a standard Borel space.
From the point of view of categorical probability they form a particularly well-behaved Markov category, BorelStoch.
A topological space is called completely separably metrizable if the topology can be metrized by a complete and separable metric. These spaces are also called Polish spaces, see there for more information.
A measurable space is called a standard Borel space if it can be written as a Polish space with its Borel sigma-algebra.
By analogy, a measure space (for example a probability space) is called a standard Borel measure space if and only if its underlying measurable space is standard Borel as above.
The real line is a standard Borel space.
Every finite or countable set, equipped with the discrete sigma-algebra, is a standard Borel spaces.
Up to isomorphism of measurable spaces, these are the only examples. Note in particular that all finite powers of the real line and all intervals, as measurable spaces, are isomorphic to the real line.
The category BorelMeas (or Pol) has as objects standard Borel measurable spaces, and as morphisms measurable functions. (Sometimes, depending on the author, Pol denotes the category of completely metrizable topological spaces and continuous maps.)
The category BorelStoch has as objects standard Borel spaces, and as morphisms Markov kernels between them.
The category of couplings has as objects standard Borel measure (not just measurable) spaces, and either equivalence classes of Markov kernels between them (under almost sure equality), or couplings between them.
Finite and countable products (in the category Meas, i.e. with the tensor product sigma-algebra) of standard Borel spaces are again standard Borel.
Measurable subsets of standard Borel spaces, with the induced sigma-algebra, are again standard Borel.
Measurable retracts of a standard Borel space are again standard Borel.
The sigma-algebra of a standard Borel space is countably generated and separating.
This has a number of consequences, for example:
Every Markov kernel between standard Borel probability spaces admits a Bayesian inverse.
Disintegration theorem: For every sub-sigma-algebra of a standard Borel space, conditional expectation gives rise to a regular conditional distribution.
Standard Borel spaces admit countably infinite tensor products.
The Giry monad preserves standard Borel spaces: if $X$ is a standard Borel space, its functorial image $P X$ is standard Borel too. Therefore the Giry monad restricts to a monad on Pol, usually known under the same name.
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Last revised on March 22, 2024 at 17:40:36. See the history of this page for a list of all contributions to it.