Given Borel equivalence relations E and F on Polish spaces X and Y respectively, one says that E is Borel reducible to F, in symbols E ≤B F, if and only if there is a Borel function
- Θ : X → Y
such that for all x,x' ∈ X, one has
- x E x' ⇔ Θ(x) F Θ(x').
Conceptually, if E is Borel reducible to F, then E is "not more complicated" than F, and the quotient space X/E has a lesser or equal "Borel cardinality" than Y/F, where "Borel cardinality" is like cardinality except for a definability restriction on the witnessing mapping.
A measure space X is called a 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 Borel-isomorphic iff |X| = |Y|.
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- Kechris, Alexander S. (1994). Classical Descriptive Set Theory. Springer-Verlag. ISBN 978-0-387-94374-9.
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- Kanovei, Vladimir; Borel equivalence relations. Structure and classification. University Lecture Series, 44. American Mathematical Society, Providence, RI, 2008. x+240 pp. ISBN 978-0-8218-4453-3