Maharam algebra

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

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.

Template:Harvs 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.

References

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