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
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.