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borel equivalence relation wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 kuratowski s theorem 2 see also 3 references toggle the table of contents borel equivalence relation add languages add links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia in mathematics a borel equivalence relation on a polish space x is an equivalence relation on x that is a borel subset of x x in the product topology 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 kuratowski s theorem edit 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 see also edit hyperfinite equivalence relation wadge hierarchy entourage topology topological space with a notion of uniform properties pages displaying short descriptions of redirect targets references edit harrington l a a s kechris a louveau oct 1990 a glimm effros dichotomy for borel equivalence relations journal of the american mathematical society 3 2 903 928 doi 10 2307 1990906 jstor 1990906 kechris alexander s 1994 classical descriptive set theory springer verlag isbn 978 0 387 94374 9 silver jack h 1980 counting the number of equivalence classes of borel and coanalytic equivalence relations annals of mathematical logic 18 1 1 28 doi 10 1016 0003 4843 80 90002 9 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 v t e measure theory basic concepts absolute continuity of measures lebesgue integration l p spaces measure measure space probability space measurable space function sets almost everywhere atom baire set borel set equivalence relation borel space carathéodory s criterion cylindrical σ algebra cylinder set 𝜆 system essential range infimum supremum locally measurable π system σ algebra non measurable set vitali set null set support transverse measure universally measurable types of measures atomic baire banach besov borel brown complex complete content logarithmically convex decomposable discrete equivalent finite inner quasi invariant locally finite maximising metric outer outer perfect pre measure sub probability projection valued radon random regular borel regular inner regular outer regular saturated set function σ finite s finite signed singular spectral strictly positive tight vector particular measures counting dirac euler gaussian haar harmonic hausdorff intensity lebesgue infinite dimensional logarithmic product projections pushforward spherical tangent trivial young maps measurable function bochner strongly weakly convergence almost everywhere of measures in measure of random variables in distribution in probability cylinder set measure random compact set element measure process variable vector projection valued measure main results carathéodory s extension theorem convergence theorems dominated monotone vitali decomposition theorems hahn jordan maharam s egorov s fatou s lemma fubini s fubini tonelli hölder s inequality minkowski inequality radon nikodym riesz markov kakutani representation theorem other results disintegration theorem lifting theory lebesgue s density theorem lebesgue differentiation theorem sard s theorem vitali hahn saks theorem for lebesgue measure isoperimetric inequality brunn minkowski theorem milman s reverse minkowski steiner formula prékopa leindler inequality vitale s random brunn minkowski inequality applications related convex analysis descriptive set theory probability theory real analysis spectral theory retrieved from https en wikipedia org w index php title borel_equivalence_relation oldid 1187921244 categories descriptive set theory equivalence mathematics hidden category pages displaying short descriptions of redirect targets via module annotated link this page was last edited on 2 december 2023 at 07 52 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search toggle the table of contents borel equivalence relation add languages add topic