By Jürgen Elstrodt
Dieses Lehrbuch vermittelt dem Leser ein solides Basiswissen, wie es f?r weite Bereiche der Mathematik unerl??lich ist, insbesondere f?r die reelle research, Funktionalanalysis, Wahrscheinlichkeitstheorie und mathematische Statistik. Thematische Schwerpunkte sind Produktma?e, Fourier-Transformation, Transformationsformel, Konvergenzbegriffe, absolute Stetigkeit und Ma?e auf topologischen R?umen. H?hepunkt ist die Herleitung des Rieszschen Darstellungssatzes mit Hilfe eines Fortsetzungsresultats von Kisynski und der Beweis der Existenz und Eindeutigkeit des Haarschen Ma?es. Enthalten ist ebenfalls ein Abschnitt ?ber Konvergenz von Ma?en. Der textual content wird aufgelockert durch zahlreiche mathematikhistorische Ausfl?ge und Kurzportr?ts von Mathematikern, die zum Thema des Buches wichtige Beitr?ge geliefert haben. Eine Vielzahl von ?bungsaufgaben vertieft den Stoff.
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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.
A n I , Z f|^| n n n=0 q 0 + q i + . +qh=n ^ - " V j ^ . . dy(s_ An . , , q0,J1,. ,s ) l qQ T-, < S . 2 < T, (0,t) e < . . < S. < s. 1 j . , . Jl+J2+1 < s. integers , Jl4"*--4"^ . < . . < s^ + 1