Maharam algebra

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In mathematics, a Maharam algebra is a complete Boolean algebra with a continuous submeasure. They were introduced by Maharam (1947).

Definitions[edit]

A continuous submeasure or Maharam submeasure on a Boolean algebra is a real-valued function m such that

  • m(0) = 0, m(1) = 1, m(x) > 0 if x ≠ 0.
  • If x < y then m(x) < m(y)
  • m(x ∨ y) ≤ m(x) + m(y)
  • If xn is a decreasing sequence with intersection 0, then the sequence m(xn) has limit 0.

A Maharam algebra is a complete Boolean algebra with a continuous submeasure.

Examples[edit]

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.

References[edit]