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

• $m(0)=0,m(1)=1,$ 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 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

• "Weak distributivity, a problem of von Neumann and the mystery of measurability", Bulletin of Symbolic Logic 12 (2): 241–266, 2006, doi:10.2178/bsl/1146620061, https://www.math.ucla.edu/~asl/bsl/1202-toc.htm 
• Maharam, Dorothy (1947), "An algebraic characterization of measure algebras", Annals of Mathematics, Second Series 48: 154–167, doi:10.2307/1969222 
• "Maharam's Problem", Annals of Mathematics, Second Series 168 (3): 981–1009, 2008, doi:10.4007/annals.2008.168.981 
• Velickovic, Boban (2005), "CCC forcing and splitting reals", Israel Journal of Mathematics 147: 209–220, doi:10.1007/BF02785365