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34 Satz 44 (Differentiation unter dem Integral). Sei f : Rn ×] a, b [→ R, (x, t) → f (x, t) =: ft (x). Es gelte: (i) F¨ ur alle t ∈] a, b [ ist ft ∈ L1 (φ). (ii) F¨ ur alle x ∈ Rn ist t → f (x, t) differenzierbar. , t)dφ. ∂t ft dφ = Beweis. Wir setzen F (t) := ft dφ. Sei (ti ) eine gegen t konvergente Folge in ] a, b[ \ {t}. Dann ist fti − ft dφ ti − t 1 (F (ti ) − F (t)) = ti − t (16) 1 ∈L (φ) Nun ist f¨ ur fast alle x f (x, ti ) − f (x, t) fti − ft (x) = ti − t ti − t = MW S ∂f (x, t∗i ) ≤ g(x).

Ist umgekehrt χN φ-integrierbar mit Integral = 0, so folgt aus dem Korollar 37 zum Satz von B. Levi, dass χN =φ 0, also N eine φ-Nullmenge. Satz 70. F¨ ur i = 1, 2 seien φi ein Maß auf Rni und fi : Rni → R nicht negativ. Wir definieren φ := φ1 × φ2 und f : Rn = Rn1 +n2 → R durch f (x1 , x2 ) := f1 (x1 )f2 (x2 ). Dann gilt (i) Ist fi ∈ L1 (φi ) f¨ ur i = 1, 2, so ist f ∈ L1 (φ) und f dφ = f1 dφ1 f2 dφ2 . (ii) Sind die fi bez¨ uglich φi messbar, so ist f bez¨ uglich φ messbar. Beweis. Vgl. Hausaufgaben.

Im letzteren Fall definiert man das Maß von A als φ(A) := χAdφ. (ii) F¨ ur f : Rn ⊃ B → R bezeichnen wir mit fˆ : Rn → R die triviale Erweiterung von f durch fˆ(x) := 0 f¨ ur x ∈ / B. Ist A ⊂ B φ-messbar, so nennen wir f φ-messbar auf A bzw. φ-integrierbar u ¨ber A, wenn fˆχA φ-messbar bzw. φ-integrierbar ist. Im letzteren Fall schreibt man f ∈ L1 (A, φ) und fˆχAdφ. f dφ := A Beispiel 64. Jedes I ∈ I(Rn ) ist φ-integrierbar, und die neue Definition von φ(I) ist mit der Maß-Definition vertr¨ aglich.

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