Step functions [sect:1-3]
Let S be a set, B⊂P(S) a ring,
and μ : B → [0, ∞[ an additive measure.
Consider mappings f : S → Y , where Y = Rk
is an EUCLIDean space
with norm
y |→ |y| := sqrt(∑ i=1k |yi|2)
for y=(yi)i=1,...,k∈Rk .
(in general Y might be any normed space
with norm y |→ |y| ).
The space of step functions with values
in Y is defined as
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T(μ; Y) := { f : S → Y ;
f(S) is a finite set , f-1({y })∈B for y≠0 } .
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Each f ∈T(μ; Y) has (up to a rearrangement of numeration) a unique representation
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f(x) =
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ΧEj (x) yj ,
m∈N∪{0},
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with m different values
yj ≠0 and Ej := f-1(yj) ∈B .
Here f=0 if m=0 .
| | Treppenfunktion [fig:treppenfunktion] |
The elementary integral for f ∈T(μ;Y) is defined by
(using this unique representation)
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f dμ:=
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μ(Ej) yj
=
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μ(f-1({y})) y .
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Moreover, we define
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