20.1.2 The Lebesgue Integral For Nonnegative Functions
Here is the definition of the Lebesgue integral of a function which is measurable and has values in
.
Definition 20.1.3 Let
be a measure space and suppose f : Ω
→ is measurable. Then
define
∫ ∫ ∞
fdμ ≡ μ ([f > λ])dλ
0
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which makes sense because λ → μ
is nonnegative and decreasing.
Note that if f ≤ g, then ∫
fdμ ≤∫
gdμ because μ
≤ μ.
For convenience ∑
i=10ai ≡ 0.
Lemma 20.1.4 In the situation of the above definition,
∫
∞∑
fdμ = suhp>0 μ([f > hi])h
i=1
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Proof: Let m
∈ ℕ satisfy
R − h < hm ≤ R. Then lim
R→∞m =
∞ and so
∫ ∫ ∫
∞ hm (h,R)
fdμ ≡ 0 μ ([f > λ])dλ = suMpsuRp 0 μ([f > λ])∧ M dλ
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m(∑h,R)
= supsup sup (μ([f > kh ])∧ M )h
M R>0 h>0 k=1
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The sum is just a lower sum for the integral ∫
0hmμ
∧Mdλ. Hence, switching the order of the
sups, this equals
m(h,R)
supsupsup ∑ (μ([f > kh])∧ M )h
R>0h>0 M k=1
m (h,R)
= supsup lim ∑ (μ ([f > kh])∧M ) h
R>0h>0M → ∞ k=1
m(∑R,h) ∑∞
= supsup (μ ([f > kh]))h = sup (μ([f > kh]))h.■
h>0 R k=1 h>0k=1
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