Billingsley Dimension in Probability Spaces
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892 Helmut Cajar
Billingsley Dimension in Probability Spaces
Springer-Verlag Berlin Heidelberg New York 1981
Author
Helmut Cajar Mathematisches Institut A, Universitat Stuttgart Pfaffenwaldring 57, 7000 Stuttgart 80 Federal Republic of Germany
AMS Subject Classifications (1980): lOK50, 28099, 60F15, 60JlO
ISBN 3-540-11164-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11164-6 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1981
2141/3140-543210
D 93
TABLE OF CONTENTS Introduction Chapter § 1
l.A l.B
P-dimension
7
Preliminaries, notation, terminology General ities The sequence space Afi .
7 7 9
§ 2
The P-dimension of sets
13
§ 3
Connections between P-dimension, Billingsley dimension and Hausdorff dimension . . . . . .
30
§ 4
The quasimetric q and the metric q* The quasimetric q The metric q* .
43
The P-dimension of probability measures
47
The Billingsley dimension of saturated sets
52
Introductory arguments The saturated subsets of X (IT, d) as sequence space Markov measures . . . . . . Generalizations with respect to P Special functions . . . .
52 52
60
§ 7
The Billingsley dimension of the smallest saturated sets
62
§ 8
The Billingsley dimension of saturated sets
75
§ 9
The Billingsley dimension of special sets.
84
4.A 4.B § 5
Chapter II § 6
6.A 6.B 6.C 6.D 6.E
38 38
54
57 59
References
102
Index of Symbols
105
Index of Terms
106
Introduction A number of authors (see, e.g., Besicovitch [7], Knichal [32], Eggleston [23], [24], Volkmann [44], [45], [47], [49] and Cigler [19]) have computed the Hausdorff dimensions (h-dim) of sets of real numbers characterized by digit properties of their g-adic representations. A detailed comparison of the results of these papers shows the following phenomenon: If the Hausdorff dimens i on of some non-denumerable uni on of sets t-1 a , a E I, of the type under consideration as well as the Hausdorff dimensions of the individual sets Ma are known, possibly from different sources, then the relation h-dim( U M) = sup h-dim(M a ) aEI a aEI
(SUP)
holds, while this equality is, in general, only true for denumerable unions. In the papers cited above each real number r E [0,1] is expressed by its g-adic expansion r
00
e-
L
~ =
1
i =1 9 1
(e l , e 2 , ... ), e i E {O,I, ... , g-I}
for some fixed integer g ~ 2. The study of dimensions is not affected here by the fact that this expansion is ambiguous for denumerably many r. Then the relative frequency hn(r, j) by which the digit j occurs in the finite sequence (e I , e 2 , ... , en) is introduced. The sets under i
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