Random algebra

In set theory, the random algebra or random real algebra is the Boolean algebra of Borel sets of the unit interval modulo the ideal of measure zero sets. It is used in random forcing to add random reals to a model of set theory. The random algebra was studied by John von Neumann in 1935 (in work later published as Neumann (1998, p. 253)) who showed that it is not isomorphic to the Cantor algebra of Borel sets modulo meager sets. Random forcing was introduced by Solovay (1970).

References

• Bartoszyński, Tomek (2010), "Invariants of measure and category", Handbook of set theory, 2, Springer, pp. 491–555, MR 2768686 
• Bukowský, Lev (1977), "Random forcing", Set theory and hierarchy theory, V (Proc. Third Conf., Bierutowice, 1976), Lecture Notes in Math., 619, Berlin: Springer, pp. 101–117, MR 0485358 
• Solovay, Robert M. (1970), "A model of set-theory in which every set of reals is Lebesgue measurable", Annals of Mathematics. Second Series, 92: 1–56, ISSN 0003-486X, JSTOR 1970696, MR 0265151 
• Neumann, John von (1998) [1960], Continuous geometry, Princeton Landmarks in Mathematics, Princeton University Press, ISBN 978-0-691-05893-1, MR 0120174