Misplaced Pages

Maharam algebra

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
This article includes inline citations, but they are not properly formatted. Please improve this article by correcting them. (May 2024) (Learn how and when to remove this message)

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


Stub icon

This algebra-related article is a stub. You can help Misplaced Pages by expanding it.

Categories: