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By Dirk Ferus

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G \ N ist offen, also messbar, und damit ist auch G messbar. Weiter ist h(G \ N ) ⊂ h(G) ⊂ h(G \ N ) ∪ h(N ), und h(N ) ist nach Lemma 79 eine Nullmenge. Nach Teil A und B ist h(G \ N ) messbar, also ist auch h(G) messbar. Weiter ist f dµn = h(G) f dµn h(G \ N ) (f ◦ h)| det Dh | dµn = = Teil B G\N (f ◦ h)| det Dh | dµn . G Beispiel 89. F¨ ur einen Diffeomorphismus x : [a, b] → x([a, b]) ⊂ R und eine Regelfunktion f : x([a, b]) → R besagt die Substitutionsregel: x(b) b f (x(t)) x (t) dt. f (x) dx = x(a) a Vergleichen Sie das mit dem Transformationssatz.

Mit positiver Kantenl¨ ange, also µn (Wi ) > 0. 59 Also ist W(h(q),(1+ε)r) n n n h n (1− ) (2r) ≤ µn (h(W (q, r))) ≤ (1+ ) (2r) q W(q,r) und deshalb (1 − )n ≤ W(h(q),(1-ε)r) µn (h(W (p, r))) ≤ (1 + )n . µn (W (p, r)) Daraus folgt die Behauptung. Lemma 84 (Globale W¨ urfelverzerrung). Seien G ⊂ Rn offen und h : G → Rn ein 1 C -Diffeomorphismus. Sei W ein kompakter W¨ urfel in G. Dann ist h(W ) kompakt, also integrierbar, | det Dh | ∈ L1 (W, µn ) und | det Dh | dµn . µn (h(W )) = (25) W Vergleichen Sie das mit Lemma 80.

M • W¨ are µ1 (M ) > 0, so w¨ are nach der σ-Additivit¨at ˆ) = µ1 (] − 1, 2 [) ≥ µ1 (M µn (r + M ) = ∞. r Widerspruch! Also war die Annahme der Messbarkeit falsch. 52 6 Der Transformationssatz • H¨ aufig hat man Integrationsprobleme mit gewissen Symmetrieeigenschaften des Integrationsbereiches. Dann sind andere als die Euklidischen Koordinaten sehr hilfreich. Wir lernen in den folgenden Abschnitten, wie man daf¨ ur die Integrale umschreibt. Der Kernpunkt ist die Volumenverzerrung durch die Koordinatenabbildung.

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Analysis III [Lecture notes] by Dirk Ferus


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