In mathematics, a Maharam algebra is a complete Boolean algebra with a continuous submeasure (defined below). 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.
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Further reading