Maharam algebra

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

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.

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.