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Maharam Algebra
In mathematics, a Maharam algebra is a complete Boolean algebra with a continuous submeasure (defined below). They were introduced by . Definitions A continuous submeasure or Maharam submeasure on a Boolean algebra is a real-valued function ''m'' such that * m(0)=0, m(1)=1, and m(x)>0 if x\ne 0. * If x\le y, then m(x)\le m(y). * m(x\vee y)\le m(x)+m(y)-m(x\wedge y). * If x_n is a decreasing sequence with greatest lower bound 0, then the sequence m(x_n) has Limit (mathematics), limit 0. A Maharam algebra is a complete Boolean algebra with a continuous submeasure. Examples Every probability measure is a continuous submeasure, so as the corresponding Boolean algebra of measurable sets modulo Null set, measure zero sets is complete, it is a Maharam algebra. solved a long-standing problem by constructing a Maharam algebra that is not a measure algebra, ''i.e.'', that does not admit any countably additive strictly positive finite measure. References * * * * Boolean alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Boolean Algebra
In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boolean algebra ''A'' has an essentially unique completion, which is a complete Boolean algebra containing ''A'' such that every element is the supremum of some subset of ''A''. As a partially ordered set, this completion of ''A'' is the Dedekind–MacNeille completion. More generally, if κ is a cardinal then a Boolean algebra is called κ-complete if every subset of cardinality less than κ has a supremum. Examples Complete Boolean algebras *Every finite Boolean algebra is complete. *The algebra of subsets of a given set is a complete Boolean algebra. *The regular open sets of any topological space form a complete Boolean algebra. This example is of particular importance because every forcing poset can be considered as a topological spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |