Aufgabe 3.5

I-Math UZH
Ana-3 Hs 2012
Prof. T. Kappeler
Analysis III – Serie 3 – Musterlösung
Abgabe: Freitag, 12.10.2012, 10 Uhr, in der Vorlesung
* Aufgabe 5
Sei X = [a, b] ein endliches, abgeschlossenes Intervall in R, und bezeichne A die
σ -Algebra {A ∈ B(R) : A ⊆ [a, b]} sowie λ : A → [0, ∞) die Einschränkung des Borel Masses auf
[a, b].
Zeige, dass für jede stetige, nicht negative Funktion f : [a, b] → R gilt
Zb
Z
f dλ = R –
f (x) dx,
a
wobei R –
Rb
a
f (x) dx das Riemann-Integral von f über [a, b] bezeichnet.
///// Suppose f : [a, b] → R is bounded and let J = (In )1ànàm be a finite decomposition of [a, b] into
up to boundary points pairwise disjoint subintervals. Then the lower sum of f with respect to J is
defined by
X
Σ∗ (f , J) Í
f∗ (I)|I|,
f∗ (I) Í inf f (x).
x∈I
I∈I
Since refinements of J only increase this sum, the supremum of all lower sums
I∗ (f ) Í sup Σ∗ (f , J) : J decomposition of [a, b]
exists and is finite, as Σ∗ (f , J) à kf k∞ (b − a) for each decomposition. We call I∗ (f ) the lower
Riemann-sum of f . Similarly, with
X
Σ ∗ (f , J) Í
f ∗ (I)|I|,
f ∗ (I) Í sup f (x),
I∈I
x∈I
and
I ∗ (f ) Í inf Σ ∗ (f , J) : J decomposition of [a, b]
the upper Riemann-sum is defined. Note that for arbitrary decompositions J and J 0 of [a, b], we
have
˜ à Σ ∗ (f , J)
˜ à Σ ∗ (f , J 0 ),
Σ∗ (f , J) à Σ∗ (f , J)
where J˜ denotes the refinement subordinated to J and J 0 . Thus we have
I∗ (f ) à I ∗ (f ).
A function where both sums coincide is said to be Riemann-integrable and
Zb
a
f (x) dx Í I∗ (f ) = I ∗ (f ).
Now assume f is also continuous, then f is uniformly continuous on [a, b]. For any ε > 0 we thus
find a δ > 0, such that f varies at most ε on each interval Iδ ⊆ [a, b] of length δ, that is
f ∗ (Iδ ) à f∗ (Iδ ) + ε.
17. Oktober 2012 11:11
Serie 3
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I-Math UZH
Ana-3 Hs 2012
Prof. T. Kappeler
Consequently, as we can decompose [a, b] into intervals of length less or equal to δ, we have
I∗ (f ) à I ∗ (f ) à I∗ (f ) + ε.
Since ε was arbitrary, we conclude I∗ (f ) = I ∗ (f ) so each continuos function is integrable.
Now with f given as above, the Riemann-integral exists, i.e. there exists a sequence of decompositions Jn , such that
Zb
a
f (x) dx = lim J∗ (f , Jn ).
n→∞
For any such decomposition Jn of [a, b] we define
ϕJn Í
X
f∗ (I)1I ,
I∈Jn
then ϕJn is simple, ϕJn à ϕJn+1 à f and ϕJn → f . Moreover,
Z
X
ϕJn dλ =
f∗ (I)λ(I) = Σ∗ (f , Jn ).
I∈Jn
Passing to the limit and using the monotone convergence theorem then gives the claim.
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