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50 Jahre Deutsche Statistische Gesellschaft Tagungen 1961

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Finalement on a bien trouv´e deux bor´eliens B et B avec B ⊂ A ⊂ B et λ(B\B ) = 0. 5 La mesure de Lebesgue sur Rd est invariante par translation, au sens o` u pour tout A ∈ B(Rd ) et tout x ∈ Rd , on a λ(x + A) = λ(A). Inversement, si µ est une mesure sur (Rd , B(Rd )) finie sur les parties born´ees et invariante par translation, il existe une constante c ≥ 0 telle que µ = cλ. Preuve. Notons σx la translation σx (y) = y − x pour tout y ∈ Rd . La mesure-image σx (λ) est d´efinie par ∀A ∈ B(Rd ), σx (λ)(A) = λ(σx−1 (A)) = λ(x + A).

N (An ). Exemple. Si (E, A) = (F, B) = (R, B(R)), et µ = ν = λ, on v´erifie facilement que λ ⊗ λ est la mesure de Lebesgue sur R2 (observer que la mesure de Lebesgue sur R2 est caract´eris´ee par ses valeurs sur les rectangles [a, b] × [c, d], toujours d’apr`es le lemme de classe monotone). Ceci se g´en´eralise en dimension sup´erieure et montre qu’il aurait suffi de construire la mesure de Lebesgue en dimension un. 3 Le th´ eor` eme de Fubini On commence par donner l’´enonc´e qui concerne les fonctions positives.

V¯p . Cons´ equences. Pour p ∈ [1, ∞[, on a : (i) L’espace Cc (Rd ) des fonctions continues a` support compact sur Rd est dense dans p L (Rd , B(Rd ), λ). On peut remplacer λ par n’importe quelle mesure de Radon sur (Rd , B(Rd )). (ii) L’ensemble des fonctions en escalier (`a support compact) est dense dans Lp (R, B(R), λ). En effet il sufit de v´erifier que toute fonction f ∈ Cc (R) est limite dans Lp de fonctions en escalier. Cela se voit en ´ecrivant f = lim n→∞ k f ( ) 1[ k , k+1 [ . n n n k∈Z Application.

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