Maharam algebra
In mathematics, a Maharam algebra is a complete Boolean algebra with a continuous submeasure. They were introduced by Maharam (1947).
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, m(x) > 0 if x ≠ 0.
- If x < y then m(x) < m(y)
- m(x ∨ y) ≤ m(x) + m(y)
- If xn is a decreasing sequence with intersection 0, then the sequence m(xn) has 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 measure zero sets is complete it is a Maharam algebra.
References
- Balcar, Bohuslav; Jech, Thomas (2006), "Weak distributivity, a problem of von Neumann and the mystery of measurability", Bulletin of Symbolic Logic, 12 (2): 241–266, doi:10.2178/bsl/1146620061, MR 2223923, Zbl 1120.03028
- Maharam, Dorothy (1947), "An algebraic characterization of measure algebras", Annals of Mathematics (2), 48: 154–167, doi:10.2307/1969222, JSTOR 1969222, MR 0018718, Zbl 0029.20401
- Velickovic, Boban (2005), "ccc forcing and splitting reals", Israel J. Math., 147: 209–220, doi:10.1007/BF02785365, MR 2166361, Zbl 1118.03046
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