Let ϕ be some positive-valued set functionMathworldPlanetmath defined on an algebra of setsMathworldPlanetmath 𝒜. We say that ϕ is additive if, whenever A and B are disjoint sets in 𝒜, we have


Given any sequence Ai of disjoint sets in A and whose union is also in A, if we have


we say that ϕ is countably additive or σ-additive.

Useful properties of an additive set function ϕ include the following:

  1. 1.


  2. 2.

    If AB, then ϕ(A)ϕ(B).

  3. 3.

    If AB, then ϕ(BA)=ϕ(B)-ϕ(A).

  4. 4.

    Given A and B, ϕ(AB)+ϕ(AB)=ϕ(A)+ϕ(B).

Title additive
Canonical name Additive
Date of creation 2013-03-22 13:00:58
Last modified on 2013-03-22 13:00:58
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 10
Author Andrea Ambrosio (7332)
Entry type Definition
Classification msc 03E20
Synonym additivity
Defines countable additivity
Defines countably additive
Defines σ-additive
Defines sigma-additive