View source for Random algebra

{{Short description|Mathematical theory}}
In [[set theory]], the '''random algebra''' or '''random real algebra''' is the [[Boolean algebra]] of [[Borel set]]s of the unit interval modulo the [[ideal (set theory)|ideal]] of measure zero sets. It is used in '''random forcing''' to add '''random reals''' to a [[model (model theory)|model]] of set theory. The random algebra was studied by [[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]] of Borel sets modulo [[meager set]]s. Random forcing was introduced by {{harvtxt|Solovay|1970}}.

==See also==
*[[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]] |series=Second Series | issn=0003-486X | volume=92 | pages=1–56 | doi=10.2307/1970696}}
*{{Citation | last1=Neumann | first1=John von| author1-link=John von Neumann | title=Continuous geometry | origyear=1960 | url=https://books.google.com/books?id=onE5HncE-HgC | publisher=[[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}}