Measure algebra

Measure algebra

In mathematics, a measure algebra is a Boolean algebra with a countably additive positive measure. A probability measure on a measure space gives a measure algebra on the Boolean algebra of measurable sets modulo null sets.

Definition

A measure algebra is a Boolean algebra B with a measure m, which is a real-valued function on B such that:

  • m(0)=0, m(1)=1
  • m(x) >0 if x≠0
  • m is countably additive: m(Σxi) = Σm(xi) if the xi are a countable set of elements that are disjoint (xi ∧ xj=0 whenever i≠j).

References

Jech, Thomas (2003), "Saturated ideals", Set Theory, Springer Monographs in Mathematics (third millennium ed.), Berlin, New York: Springer-Verlag, p. 415, doi:10.1007/3-540-44761-X_22, ISBN 978-3-540-44085-7 


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