Additive measures [sect:1-1]
Let S be a set and B⊂P(S) a nonempty
system of subsets
1 of S .
Then
B is called a ring over S , if
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==> E1 ∖E2 ∈B and E1∪E2 ∈B |
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(then also E1∩E2 = E1∖(E1∖E2) ∈B ).
One also calles B a
ring of subsets of S ,
or a
BOOLEan ring
.
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Supplement:
In the case, that the entire set S belongs to B ,
that is
one calls B an
algebra of subsets of S ,
or a
BOOLEan algebra
.
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A map
where for out purpose Z = R or Z = R∪{+∞} ,
we call a set function.
A set function μ on a ring B is called
additive measure,
if the following holds:
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E1,E2∈B and E1∩E2=∅ ==> μ(E1∪E2) = μ(E1) + μ(E2) .
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(If Z = R∪{+∞} , this equation has to be understood
in the sense, that +∞= a + ∞ for each a∈Z ).
Moreover, we call μ
monotone, if
Let us collect some simple consequences of this definition:
Properties of additive measures.
Let B be a ring of subsets of S and
μ an additive measure on B . Then the following holds:
- [sect:1-1-(ii)]
E1,..., Em ∈B ==> ∩ j=1m Ej , ∪ j=1m Ej ∈B
- [sect:1-1-(iii)]
μ(∅) = 0
- [sect:1-1-(iv)]
For in pairs disjoint subsets E1,..., Em of S
- [sect:1-1-(v)]
For all subsets E1,..., Em of S
where
BI := {x∈S ; x∈Ej , falls j∈I
und x∉Ej , falls j∉I} .
- [sect:1-1-(vi)]
μ≧0 ==> μ monoton .
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