Maharam algebra

In mathematics, a Maharam algebra is a complete Boolean algebra with a continuous submeasure (defined below). They were introduced by Dorothy Maharam (1947).

Definitions

A continuous submeasure or Maharam submeasure on a Boolean algebra is a real-valued function m such that

• ${\displaystyle m(0)=0,m(1)=1,}$ and ${\displaystyle m(x)>0}$ if ${\displaystyle x\neq 0}$.
• If ${\displaystyle x\leq y}$, then ${\displaystyle m(x)\leq m(y)}$.
• ${\displaystyle m(x\vee y)\leq m(x)+m(y)-m(x\wedge y)}$.
• If ${\displaystyle x_{n}}$ is a decreasing sequence with greatest lower bound 0, then the sequence ${\displaystyle m(x_{n})}$ 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.

Michel Talagrand (2008) 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

• 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
• Talagrand, Michel (2008), "Maharam's Problem", Annals of Mathematics, Second Series, 168 (3): 981–1009, doi:10.4007/annals.2008.168.981, JSTOR 40345433, MR 2456888, Zbl 1185.28002
• Velickovic, Boban (2005), "ccc forcing and splitting reals", Israel Journal of Mathematics, 147: 209–220, doi:10.1007/BF02785365, MR 2166361, Zbl 1118.03046