additive (original) (raw)
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:
- ϕ(∅)=0.
- If A⊆B, then ϕ(A)≤ϕ(B).
- If A⊆B, then ϕ(B∖A)=ϕ(B)-ϕ(A).
- 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 |