# Random algebra

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{{Short description|Mathematical theory}}
In [set theory](/source/set_theory), the '''random algebra''' or '''random real algebra''' is the [Boolean algebra](/source/Boolean_algebra) of [Borel set](/source/Borel_set)s of the unit interval modulo the [ideal](/source/ideal_on_a_set) of measure zero sets. It is used in '''random forcing''' to add '''random reals''' to a [model](/source/model_(model_theory)) of set theory. The random algebra was studied by [John von Neumann](/source/John_von_Neumann) in 1935 (in work later published as {{harvtxt|Neumann|1998|loc=p. 253}}) who showed that it is not isomorphic to the [Cantor algebra](/source/Cantor_algebra) of Borel sets modulo [meager set](/source/meager_set)s. Random forcing was introduced by {{harvtxt|Solovay|1970}}.

==See also==
*[Random number](/source/Random_number)

==References==
{{refbegin}}
*{{citation|mr=2768686 
|last=Bartoszyński|first= Tomek|author-link= Tomek Bartoszyński
|chapter=Invariants of measure and category|title= Handbook of set theory|volume= 2|pages= 491–555|publisher= Springer|year= 2010}} 
*{{citation|mr=0485358 |last= Bukowský|first= Lev|chapter= Random forcing|title= Set theory and hierarchy theory, V (Proc. Third Conf., Bierutowice, 1976)|pages= 101–117|series= Lecture Notes in Math.|volume= 619|publisher= Springer|place= Berlin|year= 1977}}
*{{Citation | last1=Solovay | first1=Robert M. | author1-link=Robert M. Solovay | title=A model of set-theory in which every set of reals is Lebesgue measurable | jstor=1970696 | mr=0265151 | year=1970 | journal=[Annals of Mathematics](/source/Annals_of_Mathematics) |series=Second Series | issn=0003-486X | volume=92 | issue=1 | pages=1–56 | doi=10.2307/1970696}}
*{{Citation | last1=Neumann | first1=John von| author1-link=John von Neumann | title=Continuous geometry | orig-date=1960 | url=https://books.google.com/books?id=onE5HncE-HgC | publisher=[Princeton University Press](/source/Princeton_University_Press) | series=Princeton Landmarks in Mathematics | isbn=978-0-691-05893-1 | mr=0120174 | year=1998}}
{{refend}}

Category:Boolean algebra
Category:Forcing (mathematics)

{{Settheory-stub}}

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