Maharam algebra

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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

  • [math]\displaystyle{ m(0)=0, m(1)=1, }[/math] and [math]\displaystyle{ m(x)\gt 0 }[/math] if [math]\displaystyle{ x\ne 0 }[/math].
  • If [math]\displaystyle{ x\le y }[/math], then [math]\displaystyle{ m(x)\le m(y) }[/math].
  • [math]\displaystyle{ m(x\vee y)\le m(x)+m(y)-m(x\wedge y) }[/math].
  • If [math]\displaystyle{ x_n }[/math] is a decreasing sequence with greatest lower bound 0, then the sequence [math]\displaystyle{ m(x_n) }[/math] 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 (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.

References