Aufgabe 3.5
Transcription
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 Seite 1 von 2 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. 17. Oktober 2012 11:11 Serie 3 ///// Seite 2 von 2