additive
Let ϕ be some positive-valued set function defined on an algebra of sets
𝒜. We say that ϕ is additive if, whenever A and B are disjoint sets in 𝒜, we have
ϕ(A∪B)=ϕ(A)+ϕ(B). |
Given any sequence ⟨Ai⟩ of disjoint sets in A and whose union is also in A, if we have
ϕ(⋃Ai)=∑ϕ(Ai) |
we say that ϕ is countably additive or σ-additive.
Useful properties of an additive set function ϕ include the following:
-
1.
ϕ(∅)=0.
-
2.
If A⊆B, then ϕ(A)≤ϕ(B).
-
3.
If A⊆B, then ϕ(B∖A)=ϕ(B)-ϕ(A).
-
4.
Given A and B, ϕ(A∪B)+ϕ(A∩B)=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 |