Fractional Dimensions and Bounded Fractional Forms by R. C. Blei

By R. C. Blei

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Jj=l J-l I I = l j a 3 a . 0 . (a) ) • • • ( ! 0r 1 N II k=0 T C { 1 , . . , J } |T|=k . ®r D i, ||J . k T=(i][, . . 8) fail for some such T, and obtain the lemma. 9) || l 2 l J a. , r j^lV'^J^ ®---®r. 9) a point (GO.. , . . ,u>_) e [ 0 , 1 ] which I N I a . *. (a) ) . . 0 (a) ) | ^ $ J || N X a . r ®---®r \\„ FRACTIONAL DIMENSIONS 43 And s o , w e w r i t e S i = {1 < j < N : $ . (UK) = 1} , i=l, . . ,J , which are the required sets in this case. ,J . (n)} _. j ne3N T h e n , for every permutation be a cpartition of a of E.

R. C. BLEI J/K-dimensional Fr^chet pseudomeasures We keep all previous notation: (S ) K , J >^ K > 0 is the collection of all K-subsets of is enumerated S 01. product space = (a, , . . ) . ) , define the IN. 19) Y = x x . , J a and the corresponding product a-algebra 0 From now on we view Y = a x A. jeS ^ J a as the measurable space (Y ,0 ) . 9), define the canonical projections P : a J x x. ,(„) , and arbitrary sets ' we consider ^ ] J x x. j =l J which we call a measurable generalized rectangle in each ) J=3, K=2).

J > x< X. 25) \if 1 (E2x. xEj) = / f1(x)y(dxxE2x. -xEj) , Xx E 2 e A 2 , . . ,Ej e Aj . } respect to the signed measure Lemma 4. 9 y ( 4 . ) j=2 3 and ( 4 . 27) f l F X J~1 ^J Proof By standard convergence theorems, it suffices to check the lemma for simple functions f = f = I a xF . i i F i O F iJ = ^ if i^ J • 50 R. C. 27) are proved by induction on J: that {E. }. _. 3 is a partition of 3 £JN X0. 1) . ®r. 29) i limll <1. i y . Then °° — . b. 11 = 0 . 4 (repeatedly) to obtain a contradiction with the supposition.

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