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Standard Borel space
In mathematics, a standard Borel space is the Borel space associated to a Polish space. Discounting Borel spaces of discrete Polish spaces, there is, up to isomorphism, only one standard Borel space.
Contents
1 Formal definition
2 Properties
3 Kuratowski's theorem
4 References
Formal definition
A measurable space (X, Σ) is said to be "standard Borel" if there exists a metric on X which makes it a complete separable metric space in such a way that Σ is then the Borel σ-algebra.[1]
Standard Borel spaces have several useful properties that do not hold for general measurable spaces.
Properties
- If (X, Σ) and (Y, Τ) are standard Borel then any bijective measurable mapping f:(X,Σ)→(Y,T){displaystyle f:(X,Sigma )rightarrow (Y,mathrm {T} )}
is an isomorphism (i.e., the inverse mapping is also measurable). This follows from Souslin's theorem, as a set that is both analytic and coanalytic is necessarily Borel.
- If (X, Σ) and (Y, Τ) are standard Borel spaces and f:X→Y{displaystyle f:Xrightarrow Y}
then f is measurable if and only if the graph of f is Borel.
- The product and direct union of a countable family of standard Borel spaces are standard.
- Every complete probability measure on a standard Borel space turns it into a standard probability space.
Kuratowski's theorem
Theorem. Let X be a Polish space, that is, a topological space such that there is a metric d on X which defines the topology of X and which makes X a complete separable metric space. Then X as a Borel space is Borel isomorphic to one of
(1) R, (2) Z or (3) a finite space. (This result is reminiscent of Maharam's theorem.)
It follows that a standard Borel space is characterized up to isomorphism by its cardinality,[2] and that any uncountable standard Borel space has the cardinality of the continuum.
Borel isomorphisms on standard Borel spaces are analogous to homeomorphisms on topological spaces: both are bijective and closed under composition, and a homeomorphism and its inverse are both continuous, instead of both being only Borel measurable.
References
^ Mackey, G.W. (1957): Borel structure in groups and their duals. Trans. Am. Math. Soc., 85, 134-165.
^ Srivastava, S.M. (1991), A Course on Borel Sets, Springer Verlag, ISBN 0-387-98412-7.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
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