Maharam algebra explained

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.