Definition [definition:1-6]
Let B be a ring over the set S and
μ : B → Z , where Z=R or Z=R∪{+∞} .
(In the case Z=R∪{+∞} one has to pay attention zo the
usual conventions with respect to the value +∞ .)
- [definition:1-6-(i)]
Assume μ≧0 . Then μ is called
ϭ -subadditive
or sigma-subadditive
or
countable-subadditive, if for all A∈B and all
countable coverings (Ej)j∈N of A
with sets Ej∈B the following inequality holds:
Here (Ej)j∈N is a
countable covering of A , if
- [definition:1-6-(ii)]
μ is called
ϭ -additive
or sigma-additive
or
countable additive, if for all A, Ej∈B with
A=∪ j ∈N Ej and
disjoint sets Ej , j∈N , the following holds:
(The property, that the limit (in Z ) exists,
is part of this definition.)
|
