By N. Bourbaki
Intégration 5
Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements.
Ce cinquième chaptire du Livre d’Intégration, sixième Livre des éléments de mathématique, traite notamment d’une generalisation du théorème des Lebesgue-Fubini et du théorème de Lebesque-Nikodym.
Il contient également des notes historiques.
Ce quantity est une réimpression de l’édition de 1967.
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1 , f o r every X > 0 , F(A x+£) i s d e f i n e d f o r m x L e b e s g u e - a . e . F). '3) V U= ^ where { T p . ^i MVUII - We write VU^T p=l ' *> c and its p;U C and S |« P=l | < + co. 1) as an absolutely convergent infinite series. series together. ,m}. ,cu } € [k;m], we shall always write 26 If GENERALIZED DYSON SERIES AND FEYNMANlS OPERATIONAL CALCULUS {a 27 k + l ' * • ' 'am} " U , . . ,m}\{alf . . ,ak>. Now, i n view of ( 2 . 2 ) and ( 2 . 4) F(x) = - n [ / 6 ( s , x ( s ) ) d u u ( s) + I a)p u 8u ( T p u u-1 (0,t) u p-1 ' ' - m z Z k=0 { a 1 , .
W. dsn-e ° I5lfl_e ° *| n (f) 1 1 / m / T-I m . -T(H n /X) (e n T m -i T , . n -T(H0/X) ° T+i _ n m < (l ? ) n / ~ 1 m [ 9 . d8n m -(t-x)(Hn/X) ^ P - « ° _, _ m n n (j) f ... d Sn | Z f &l> d s ^ . d s ^ 1 n . dsn i n _ _ m n 2 < (5)n / / ... 41) = (j) J J (=) - J J . Hence ( 4 . 3 8 ) i s e s t a b l i s h e d and ( 4 . 3 7 ) h o l d s f o r a l l A c C + . REMARK 4 . 6 . The p r o o f above r e v e a l s r a t h e r n i c e l y some of t h e i n t u i t i v e ideas underlying the present theory.
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