Differential- und Integralrechnung 2 by Grauert H., Fischer W.

By Grauert H., Fischer W.

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Zu j = 1, . . , N n w¨ ahle aj ∈ Aj mit | det Φ (aj )| = minx∈Aj | det Φ (x)|. Zu festem j sei h(x) := (h1 (x), . . , hn (x))T := Φ(x) − Φ (aj )x (x ∈ Aj ). Mittelwertsatz der Differentialrechnung ⇒ zu x ∈ Aj existiert ci ∈ Aj mit |hi (x) − hi (aj )| = |hi (ci )(x − aj )| = |(Φi (ci ) − Φi (aj ))(x − aj )| ≤ Φ (ci ) − Φ (aj ) ∞ |x − aj |∞ ε δ (i = 1, . . , n). ≤ M Nimmt man das Maximum u alt man ¨ber alle i = 1, . . , n, so erh¨ |Φ(x) − Φ (aj )x − Φ(aj ) + Φ (aj )aj |∞ ≤ εδ M (x ∈ Aj ) 34 3 Weitere klassische S¨atze der Integrationstheorie εδ εδ n ⇒ Φ(Aj ) ⊂ Φ(aj ) − Φ (aj )aj + Φ (aj )(Aj ) + [− M , M] .

1 (ii)) Sei α := limn→∞ fn dμ ∈ [0, ∞]. Wegen Sei 0 < c < 1 und s Stufenfunktion mit X | fn (x) ≥ cs(x)}. Da (fn )n monoton ist, Wegen cs ≤ fn auf An gilt fn dμ ≥ lim fn dμ. n→∞ ⇒ f dμ ∈ [0, ∞] existiert. fn dμ ≤ f dμ folgt α ≤ f dμ. 0 ≤ s ≤ f . Definiere An := {x ∈ folgt A1 ⊂ A2 ⊂ . . fn dμ ≥ c sdμ. An An limn→∞ fn = f , c < 1 ⇒ n∈N An = X. 4) ⇒ lim n→∞ fn dμ ≥ c sdμ. Supremum u ¨ber alle Stufenfunktionen s mit 0 ≤ s ≤ f ⇒ lim n→∞ c < 1 beliebig ⇒ α ≥ fn dμ ≥ c f dμ. f dμ ⇒ Behauptung. 7 (Addititivit¨ at des Integrals).

17 K ⊂ Uk , K kompakt, mit sn0 |K stetig und p ε λ(Uk \ K) < . 4 sn0 ∞ Setze ϕ(x) ˜ := sn0 (x), x ∈ K, 0, x ∈ U \ Uk . 18) Fortsetzung ϕ ∈ C(U ) von ϕ˜ mit |ϕ(x)| ≤ sn0 ∞ (x ∈ U ). ϕ = 0 auf U \ Uk ⇒ supp ϕ ⊂ Uk ⊂ U ⇒ ϕ ∈ C0 (U ). Es gilt sn0 − ϕ p p |sn0 − ϕ|p dλ = = U ≤ 2p sn0 ⇒ f −ϕ p ≤ f − fk p p ∞ + fk − sn0 Uk \K λ(Uk \ K) < p + sn0 − ϕ |sn0 − ϕ|p dλ ε 2 p p . ✷ < ε. Der Beweis des folgenden Satzes verwendet den Friedrichsschen3 Gl¨ attungsoperator: Sei ˜j(x) := 1 − 1−|x| 2 e 0, Dann ist ˜j ∈ C0∞ (Rn ).

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