Bulletin of the American Mathematical Society

The AMS does not provide abstracts of book reviews. You may download the entire review from the links below.

Book Information

Author(s): G. Edgar
Title: Integral, probability, and fractal measures
Additional book information: Springer-Verlag, New York, 1998, x + 286, $39.95, ISBN 0-387-98205-1

1.: M. Arbeiter & N. Patzschke, Random self-similar multifractals, Math. Nachr. 181 (1996), 5-42. MR 97j:28016
2.: M. Barnsley, Fractals Everywhere, Academic Press, Boston, MA, 1988. MR 90e:58080
3.: T. Bedford, Crinkly Curves, Markov Partitions and Box Dimensions in Self-Similar Sets, Ph.D. dissertation, University of Warwick, 1984.
4.: A. Besicovitch, On the fundamental properties of linearly measurable plane sets of points, Math. Ann. 98 (1928), 422-464.
5.: A. Besicovitch, On the fundamental properties of linearly measurable plane sets of points II, Math. Ann. 115 (1938), 296-329.
6.: A. Besicovitch, On the fundamental properties of linearly measurable plane sets of points III, Math. Ann. 116 (1939), 349-357.
7.: T. Bedford & A. Fisher, Analogues of the Lebesgue density theorem for fractal sets of reals and integers., Proc. Lond. Math. Soc. 64 (1992), 95-124. MR 92j:58058
8.: A. S. Besicovitch & P. A. P. Moran, The measure of product and cylinder sets, Jour. Lond. Math. Soc. 20 (1945), 110-120. MR 8:18f
9.: G. Brown, G. Michon & J. Peyriére, On the multifractal analysis of measures, J. Statist. Phys. 66 (1992), 775-790. MR 93c:58120
10.: C. Carathéodory, Über das lineare Maß von Punktmengen - eine Verallgemeinerung des Längenbegriffs, Nachrichten der K. Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse (1914), 404-426.
11.: R. Cawley & R. D. Mauldin, Multifractal decomposition of Moran fractals, Advances in Mathematics 92 (1992), 196-236. MR 93b:58085
12.: C. D. Cutler, The Hausdorff distribution of finite measures in Euclidean space, Can. J. Math. 38 (1986), 1459-1484. MR 88b:28013
13.: C. D. Cutler, Measure disintegrations with respect to $\sigma$ -stable monotone indices and pointwise representation of packing dimension, Proceedings of the 1990 Measure Theory Conference at Oberwolfach. Supplemento Ai Rendiconti del Circolo Mathematico di Palermo, Ser. II, No. 28 (1992), 319-340. MR 93h:28005
14.: C. D. Cutler, Connecting ergodicity and dimension in dynamical systems, Ergod. Theory and Dynam. Syst. 10 (1990), 451-462. MR 91h:58063
15.: G. Edgar, Measure, Topology, and Fractal Geometry, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1990. MR 92a:54001
16.: G. Edgar, Classics on Fractals, Addison-Wesley, Menlo Park, CA, 1992.
17.: G. A. Edgar & R. D. Mauldin, Multifractal Decompositions of Digraph Recursive Fractals, Proc. London Math. Soc. 65 (1992), 604-628. MR 93h:28010
18.: K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985. MR 88d:28001
19.: K. J. Falconer, Random fractals, Math. Proc. Camb. Phil. Soc. 100 (1986), 559-582. MR 88e:28005
20.: K. J. Falconer, Fractal Geometry-Mathematical Foundations and Applications, John Wiley & Sons, 1990. MR 92j:28008
21.: K. J. Falconer, Techniques in Fractal Geometry, John Wiley & Sons, 1997. MR 99f:28013
22.: K. J. Falconer & J. D. Howroyd, Projection theorems for box and packing dimensions, Math. Proc. Cambridge Philos. Soc. 119 (1996), 287-295. MR 96i:28006
23.: K. J. Falconer & J. D. Howroyd, Packing dimensions of projections and dimension profiles, Math. Proc. Cambridge Philos. Soc. 121 (1997), 269-286. MR 98j:28004
24.: K. J. Kenneth & P. Mattila, The packing dimension of projections and sections of measures, Math. Proc. Cambridge Philos. Soc. 119 (1996), 695-713. MR 96m:28003
25.: K. J. Falconer & M. Järvenpää, Packing dimensions of sections of sets, Math. Proc. Cambridge Philos. Soc. 125 (1999), 89-104. MR 99g:28012
26.: H. Federer, Geometric Measure Theory, Die Grundlehren der Mathematischen Wissenschaften, Band 153, Springer-Verlag, New York, 1969. MR 41:1976
27.: J. Geronimo & D. Hardin, An exact formula for the measure dimensions associated with a class of piecewise linear maps, Constr. Approx. 5 (1989), 89-98. MR 90d:58076
28.: S. Graf, Statistically self-similar fractals, Probab. Th. Rel. Fields 74 (1987), 357-394. MR 88c:60038
29.: H. Haase, On the dimension of product measures, Mathematika 37 (1990), 316-323. MR 92b:28007
30.: F. Hausdorff, Dimension und äußeres Maß, Mathematische Annalen 79 (1918), 157-179.
31.: T. Hawkins, Lebesgue's Theory of Integration. Its Origins and Development, Chelsea, 1975. MR 54:2904
32.: T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia & B. J. Shraiman, Fractal measures and their singularities: The characterization of strange sets, Phys. Rev. A 33 (1986), 1141-1151. MR 87h:58125a
33.: X. Hu & S. J. Taylor, Fractal properties of products and projections of measures in $\mathbb{R} ^{d}$ , Math. Proc. Camb. Phil. Soc. 115 (1994), 527-544. MR 95f:28013
34.: J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. MR 82h:49026
35.: K.-S. Lau & S.-M. Ngai, Multifractal measures and a weak separation condition, Advances in Mathematics 141 (1999), 45-96. MR 2000d:28010
36.: K.-S. Lau & S.-M. Ngai, $L^{q}$ -spectrum of Bernoulli convolutions associated with the golden ratio, Studia Math. 131 (1998), 225-251. MR 99f:28008
37.: B. Mandelbrot, Les Objects fractales: forme, hasard et Dimension, Flammarion, 1975.
38.: B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, 1982. MR 84h:00021
39.: P. Massopust, Fractal Functions, Fractal Surfaces, and Wavelets, Academic Press, San Diego, CA, 1994. MR 96b:28007
40.: J. Marstrand, Some fundamental geometric properties of plane sets of fractional dimension, Proc. Lond. Math. Soc. 4 (1954), 257-302. MR 16:121g
41.: J. M. Marstrand, The dimension of Cartesian product sets, Proc. Lond. Math. Soc. 5 (1954), 198-206. MR 15:691g
42.: J. Marstrand, Hausdorff two-dimensional measure in -space, Proc. Lond. Math. Soc. 11 (1961), 91-108. MR 23:A994
43.: J. Marstrand, The $(\phi ,s)$ regular subsets of -space, Trans. Amer. Math. Soc. 113 (1964), 369-392. MR 29:3613
44.: P. Mattila, Hausdorff regular and rectifiable sets in -space, Trans. Amer. Math. Soc. 205 (1975), 263-274. MR 50:10209
45.: P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics, Vol. 44, Cambridge University Press, 1995. MR 96h:28006
46.: C. McMullen, The Hausdorff dimension of general Sierpinski carpets, Nagoya Math. J. 96 (1984), 1-9. MR 86h:11061
47.: P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proceedings of the Cambridge Philosophical Society 42 (1946), 15-23. MR 7:278f
48.: P. Mörters & D. Preiss, Tangent measure distributions of fractals measures, Math. Ann. 312 (1998), 53-93. MR 99i:28013
49.: R.D. Mauldin & M. Urbanski, Dimension and measures in infinite iterated function systems, Proc. Lond. Math. Soc. 73 (1996), 105-154. MR 97c:28020
50.: R. D. Mauldin & S. C. Williams, Random recursive constructions: Asymptotic geometric and topological properties, Trans. Am. Math. Soc. 295 (1986), 325-346. MR 87j:60027
51.: R. D. Mauldin & S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Am. Math. Soc. 309 (1988), 811-829. MR 89i:28003
52.: L. Olsen, A multifractal formalism, Advances in Mathematics 116 (1995), 82-196. MR 97a:28006
53.: L. Olsen, Random Geometrically Graph Directed Self-Similar Multifractals, Pitman Research Notes in Mathematics Series, Vol. 307, Longman Scientific & Technical, 1994. MR 95j:28006
54.: Y. Peres, The Packing Measure of Self-Affine Carpets, Math. Proc. Cambridge Phil. Soc. 115 (1994), 437-450. MR 95e:28001
55.: Y. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. MR 99b:58003
56.: D. Preiss, Geometry of measures in $\mathbb{R} ^{n}$ : Distribution, rectifiability, and densities, Ann. of Math. 125 (1987), 537-643. MR 88d:28008
57.: Y. Peres & W. Schlag, Smoothness of projections, Bernoulli convolutions and the dimension of exceptions, preprint (1998).
58.: Y. Peres & B. Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof, Math. Res. Lett. 3 (1996), 231-239. MR 97f:28006
59.: X. S. Raymond & C. Tricot, Packing regularity of sets in -space, Math. Proc. Camb. Phil. Soc. 103 (1988), 133-145. MR 88m:28002
60.: C. A. Rogers, Hausdorff Measures, Second Edition, Cambridge University Press, London-New York, 1998. MR 2000b:28009
61.: B. Solomyak, On the random series $\sum -\lambda \sp{n}$ (an Erdös problem), Ann. of Math. 142 (1995), 611-625. MR 97d:11125
62.: K. Simon & B. Solomyak, Correlation dimension for self-similar Cantor sets with overlaps, Fund. Math. 155 (1998), 293-300. MR 99g:28036
63.: S. J. Taylor & C. Tricot, Packing measure, and its evaluation for a Brownian path, Trans. Amer. Math. Soc. 288 (1985), 679-699. MR 87a:28002
64.: C. Tricot, Two definitions of fractional dimension, Math. Proc. Camb. Phil. Soc. 91 (1982), 57-74. MR 84d:28013
65.: C. Tricot, Curves and Fractal Dimension, Springer-Verlag, New York, 1995. MR 95i:28005
66.: L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergod. Th. & Dynam. Sys. 2 (1982), 109-124. MR 84h:58087

Reviewer(s):
Lars Olsen
Affiliation: University of St. Andrews
Email: lo@st-and.ac.uk

MSC (2000): Primary 28A80
PII: S 0273-0979(00)00878-8
Posted: June 27, 2000
Copyright of article: Copyright 2000, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-9485(e) ISSN 0273-0979(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues: 1891 - Present Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS