Maharam algebra
From HandWiki
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
- [math]\displaystyle{ m(0)=0, m(1)=1, }[/math] and [math]\displaystyle{ m(x)\gt 0 }[/math] if [math]\displaystyle{ x\ne 0 }[/math].
- If [math]\displaystyle{ x\le y }[/math], then [math]\displaystyle{ m(x)\le m(y) }[/math].
- [math]\displaystyle{ m(x\vee y)\le m(x)+m(y)-m(x\wedge y) }[/math].
- If [math]\displaystyle{ x_n }[/math] is a decreasing sequence with greatest lower bound 0, then the sequence [math]\displaystyle{ m(x_n) }[/math] 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
 |  |
