Standard Borel space information

a standard Borel space is the Borel space associated with a Polish space. Except in the case of discrete Polish spaces, the standard Borel space is unique...

complement. Borel sets are named after Émile Borel. For a topological space X, the collection of all Borel sets on X forms a σ-algebra, known as the Borel algebra...

In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets...

Borel space may refer to: any measurable space a measurable space that is Borel isomorphic to a measurable subset of the real numbers Standard Borel space...

Polish spaces, there is a Borel isomorphism; that is, a bijection that preserves the Borel structure. In particular, every uncountable Polish space has the...

mathematics, a Borel isomorphism is a measurable bijective function between two standard Borel spaces. By Souslin's theorem in standard Borel spaces (which says...

Nowadays standard probability spaces may be (and often are) treated in the framework of descriptive set theory, via standard Borel spaces, see for example...

relation on a standard Borel space X is a Borel equivalence relation E with countable classes, that can, in a certain sense, be approximated by Borel equivalence...

separable Hilbert spaces. Note that if the measure spaces (X, μ) is a standard measure space (that is X − N is a standard Borel space for some null set...

topological space (in most examples it does). A Borel space is standard if and only if it is isomorphic to the underlying Borel space of a Polish space; all...

standard Borel space if it is Borel-isomorphic to a Borel subset of a Polish space. Kuratowski's theorem then states that two standard Borel spaces X and Y are...

invariant descriptive set theory, countable Borel relations are a class of relations between standard Borel space which are particularly well behaved. This...

on the σ-algebra of Borel sets of a Hausdorff topological space X that is finite on all compact sets, outer regular on all Borel sets, and inner regular...

representations are of type I) if and only if the Borel structure of its dual is a standard Borel space. He has written numerous survey articles connecting...

if it is closed and bounded; this is the Heine–Borel theorem. As a Euclidean space is a metric space, the conditions in the next subsection also apply...

terms of a locally compact second countable (lcsc) group G, a standard Borel space X and a Borel group action G × X → X , ( g , x ) ↦ g ⋅ x . {\displaystyle...

completion of a non-complete Borel measure. The Borel measure is not complete. One simple construction is to start with the standard Cantor set K , {\displaystyle...

Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets. There are several inequivalent definitions...

Lie algebra. These are, respectively, the standard Borel subgroup B of the Lie group GLn and the standard Borel subalgebra b {\displaystyle {\mathfrak {b}}}...

example the Borel algebra of Ω, which is the smallest σ-algebra that makes all open sets measurable. Kolmogorov's definition of probability spaces gives rise...

In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R n {\displaystyle R^{n}} , closely related to the normal distribution...

self-adjoint operator T on a Hilbert space H, there corresponds a unique resolution of the identity E on the Borel sets of R, such that ⟨ T x , y ⟩ = ∫...