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 x_n is a decreasing sequence with intersection 0, then the sequence m(x_n) 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, Second Series, 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