Maharam algebra

In mathematics, a Maharam algebra is a complete Boolean algebra with a continuous submeasure. They were introduced by Dorothy Maharam in 1947.

Definitions

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

and if.
If, then.
.
If is a decreasing sequence with greatest lower bound 0, then the sequence 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 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.