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Book Information:
Author:
G. Edgar
Title:
Integral, probability, and fractal measures
Additional book information:
Springer-Verlag,
New York,
1998,
x + 286 pp.,
ISBN 0-387-98205-1,
$39.95$
1. M. Arbeiter & N. Patzschke, Random self-similar multifractals, Math. Nachr. 181 (1996), 5-42. MR 1409071
2. M. Barnsley, Fractals Everywhere, Academic Press, Boston, MA, 1988. MR 0977274
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 1132856
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 1151978
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 1155465
12. C. D. Cutler, The Hausdorff distribution of finite measures in Euclidean space, Can. J. Math. 38 (1986), 1459-1484. MR 0873419
13. C. D. Cutler, Measure disintegrations with respect to -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 1183059
14. C. D. Cutler, Connecting ergodicity and dimension in dynamical systems, Ergod. Theory and Dynam. Syst. 10 (1990), 451-462. MR 1074313
15. G. Edgar, Measure, Topology, and Fractal Geometry, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1990. MR 1065392
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 1182103
18. K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985. MR 0867284
19. K. J. Falconer, Random fractals, Math. Proc. Camb. Phil. Soc. 100 (1986), 559-582. MR 0857731
20. K. J. Falconer, Fractal Geometry-Mathematical Foundations and Applications, John Wiley & Sons, 1990. MR 1102677
21. K. J. Falconer, Techniques in Fractal Geometry, John Wiley & Sons, 1997. MR 1449135
22. K. J. Falconer & J. D. Howroyd, Projection theorems for box and packing dimensions, Math. Proc. Cambridge Philos. Soc. 119 (1996), 287-295. MR 1357045
23. K. J. Falconer & J. D. Howroyd, Packing dimensions of projections and dimension profiles, Math. Proc. Cambridge Philos. Soc. 121 (1997), 269-286. MR 1426523
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 1362950
25. K. J. Falconer & M. Järvenpää, Packing dimensions of sections of sets, Math. Proc. Cambridge Philos. Soc. 125 (1999), 89-104. MR 1645529
26. H. Federer, Geometric Measure Theory, Die Grundlehren der Mathematischen Wissenschaften, Band 153, Springer-Verlag, New York, 1969. MR 0257325
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 0982726
28. S. Graf, Statistically self-similar fractals, Probab. Th. Rel. Fields 74 (1987), 357-394. MR 0873885
29. H. Haase, On the dimension of product measures, Mathematika 37 (1990), 316-323. MR 1099779
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 0414812
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 0823474
33. X. Hu & S. J. Taylor, Fractal properties of products and projections of measures in , Math. Proc. Camb. Phil. Soc. 115 (1994), 527-544. MR 1269937
34. J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. MR 0625600
35. K.-S. Lau & S.-M. Ngai, Multifractal measures and a weak separation condition, Advances in Mathematics 141 (1999), 45-96. MR 1667146
36. K.-S. Lau & S.-M. Ngai, -spectrum of Bernoulli convolutions associated with the golden ratio, Studia Math. 131 (1998), 225-251. MR 1644468
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 0665254
39. P. Massopust, Fractal Functions, Fractal Surfaces, and Wavelets, Academic Press, San Diego, CA, 1994. MR 1313502
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 0123670
43. J. Marstrand, The regular subsets of -space, Trans. Amer. Math. Soc. 113 (1964), 369-392. MR 0166336
44. P. Mattila, Hausdorff regular and rectifiable sets in -space, Trans. Amer. Math. Soc. 205 (1975), 263-274. MR 0357741
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 1333890
46. C. McMullen, The Hausdorff dimension of general Sierpinski carpets, Nagoya Math. J. 96 (1984), 1-9. MR 0771063
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 1645950
49. R.D. Mauldin & M. Urbanski, Dimension and measures in infinite iterated function systems, Proc. Lond. Math. Soc. 73 (1996), 105-154. MR 1387085
50. R. D. Mauldin & S. C. Williams, Random recursive constructions: Asymptotic geometric and topological properties, Trans. Am. Math. Soc. 295 (1986), 325-346. MR 0831202
51. R. D. Mauldin & S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Am. Math. Soc. 309 (1988), 811-829. MR 0961615
52. L. Olsen, A multifractal formalism, Advances in Mathematics 116 (1995), 82-196. MR 1361481
53. L. Olsen, Random Geometrically Graph Directed Self-Similar Multifractals, Pitman Research Notes in Mathematics Series, Vol. 307, Longman Scientific & Technical, 1994. MR 1297123
54. Y. Peres, The Packing Measure of Self-Affine Carpets, Math. Proc. Cambridge Phil. Soc. 115 (1994), 437-450. MR 1269931
55. Y. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. MR 1489237
56. D. Preiss, Geometry of measures in : Distribution, rectifiability, and densities, Ann. of Math. 125 (1987), 537-643. MR 0890162
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 1386842
59. X. S. Raymond & C. Tricot, Packing regularity of sets in -space, Math. Proc. Camb. Phil. Soc. 103 (1988), 133-145. MR 0913458
60. C. A. Rogers, Hausdorff Measures, Second Edition, Cambridge University Press, London-New York, 1998. MR 1692618
61. B. Solomyak, On the random series (an Erdös problem), Ann. of Math. 142 (1995), 611-625. MR 1356783
62. K. Simon & B. Solomyak, Correlation dimension for self-similar Cantor sets with overlaps, Fund. Math. 155 (1998), 293-300. MR 1607450
63. S. J. Taylor & C. Tricot, Packing measure, and its evaluation for a Brownian path, Trans. Amer. Math. Soc. 288 (1985), 679-699. MR 0776398
64. C. Tricot, Two definitions of fractional dimension, Math. Proc. Camb. Phil. Soc. 91 (1982), 57-74. MR 0633256
65. C. Tricot, Curves and Fractal Dimension, Springer-Verlag, New York, 1995. MR 1302173
66. L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergod. Th. & Dynam. Sys. 2 (1982), 109-124. MR 0684248
- 1.
- M. Arbeiter & N. Patzschke, Random self-similar multifractals, Math. Nachr. 181 (1996), 5-42. MR 1409071
- 2.
- M. Barnsley, Fractals Everywhere, Academic Press, Boston, MA, 1988. MR 0977274
- 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 1132856
- 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 1151978
- 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 1155465
- 12.
- C. D. Cutler, The Hausdorff distribution of finite measures in Euclidean space, Can. J. Math. 38 (1986), 1459-1484. MR 0873419
- 13.
- C. D. Cutler, Measure disintegrations with respect to -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 1183059
- 14.
- C. D. Cutler, Connecting ergodicity and dimension in dynamical systems, Ergod. Theory and Dynam. Syst. 10 (1990), 451-462. MR 1074313
- 15.
- G. Edgar, Measure, Topology, and Fractal Geometry, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1990. MR 1065392
- 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 1182103
- 18.
- K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985. MR 0867284
- 19.
- K. J. Falconer, Random fractals, Math. Proc. Camb. Phil. Soc. 100 (1986), 559-582. MR 0857731
- 20.
- K. J. Falconer, Fractal Geometry-Mathematical Foundations and Applications, John Wiley & Sons, 1990. MR 1102677
- 21.
- K. J. Falconer, Techniques in Fractal Geometry, John Wiley & Sons, 1997. MR 1449135
- 22.
- K. J. Falconer & J. D. Howroyd, Projection theorems for box and packing dimensions, Math. Proc. Cambridge Philos. Soc. 119 (1996), 287-295. MR 1357045
- 23.
- K. J. Falconer & J. D. Howroyd, Packing dimensions of projections and dimension profiles, Math. Proc. Cambridge Philos. Soc. 121 (1997), 269-286. MR 1426523
- 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 1362950
- 25.
- K. J. Falconer & M. Järvenpää, Packing dimensions of sections of sets, Math. Proc. Cambridge Philos. Soc. 125 (1999), 89-104. MR 1645529
- 26.
- H. Federer, Geometric Measure Theory, Die Grundlehren der Mathematischen Wissenschaften, Band 153, Springer-Verlag, New York, 1969. MR 0257325
- 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 0982726
- 28.
- S. Graf, Statistically self-similar fractals, Probab. Th. Rel. Fields 74 (1987), 357-394. MR 0873885
- 29.
- H. Haase, On the dimension of product measures, Mathematika 37 (1990), 316-323. MR 1099779
- 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 0414812
- 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 0823474
- 33.
- X. Hu & S. J. Taylor, Fractal properties of products and projections of measures in , Math. Proc. Camb. Phil. Soc. 115 (1994), 527-544. MR 1269937
- 34.
- J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. MR 0625600
- 35.
- K.-S. Lau & S.-M. Ngai, Multifractal measures and a weak separation condition, Advances in Mathematics 141 (1999), 45-96. MR 1667146
- 36.
- K.-S. Lau & S.-M. Ngai, -spectrum of Bernoulli convolutions associated with the golden ratio, Studia Math. 131 (1998), 225-251. MR 1644468
- 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 0665254
- 39.
- P. Massopust, Fractal Functions, Fractal Surfaces, and Wavelets, Academic Press, San Diego, CA, 1994. MR 1313502
- 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 0123670
- 43.
- J. Marstrand, The regular subsets of -space, Trans. Amer. Math. Soc. 113 (1964), 369-392. MR 0166336
- 44.
- P. Mattila, Hausdorff regular and rectifiable sets in -space, Trans. Amer. Math. Soc. 205 (1975), 263-274. MR 0357741
- 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 1333890
- 46.
- C. McMullen, The Hausdorff dimension of general Sierpinski carpets, Nagoya Math. J. 96 (1984), 1-9. MR 0771063
- 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 1645950
- 49.
- R.D. Mauldin & M. Urbanski, Dimension and measures in infinite iterated function systems, Proc. Lond. Math. Soc. 73 (1996), 105-154. MR 1387085
- 50.
- R. D. Mauldin & S. C. Williams, Random recursive constructions: Asymptotic geometric and topological properties, Trans. Am. Math. Soc. 295 (1986), 325-346. MR 0831202
- 51.
- R. D. Mauldin & S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Am. Math. Soc. 309 (1988), 811-829. MR 0961615
- 52.
- L. Olsen, A multifractal formalism, Advances in Mathematics 116 (1995), 82-196. MR 1361481
- 53.
- L. Olsen, Random Geometrically Graph Directed Self-Similar Multifractals, Pitman Research Notes in Mathematics Series, Vol. 307, Longman Scientific & Technical, 1994. MR 1297123
- 54.
- Y. Peres, The Packing Measure of Self-Affine Carpets, Math. Proc. Cambridge Phil. Soc. 115 (1994), 437-450. MR 1269931
- 55.
- Y. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. MR 1489237
- 56.
- D. Preiss, Geometry of measures in : Distribution, rectifiability, and densities, Ann. of Math. 125 (1987), 537-643. MR 0890162
- 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 1386842
- 59.
- X. S. Raymond & C. Tricot, Packing regularity of sets in -space, Math. Proc. Camb. Phil. Soc. 103 (1988), 133-145. MR 0913458
- 60.
- C. A. Rogers, Hausdorff Measures, Second Edition, Cambridge University Press, London-New York, 1998. MR 1692618
- 61.
- B. Solomyak, On the random series (an Erdös problem), Ann. of Math. 142 (1995), 611-625. MR 1356783
- 62.
- K. Simon & B. Solomyak, Correlation dimension for self-similar Cantor sets with overlaps, Fund. Math. 155 (1998), 293-300. MR 1607450
- 63.
- S. J. Taylor & C. Tricot, Packing measure, and its evaluation for a Brownian path, Trans. Amer. Math. Soc. 288 (1985), 679-699. MR 0776398
- 64.
- C. Tricot, Two definitions of fractional dimension, Math. Proc. Camb. Phil. Soc. 91 (1982), 57-74. MR 0633256
- 65.
- C. Tricot, Curves and Fractal Dimension, Springer-Verlag, New York, 1995. MR 1302173
- 66.
- L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergod. Th. & Dynam. Sys. 2 (1982), 109-124. MR 0684248
Review Information:
Reviewer:
Lars Olsen
Affiliation:
University of St. Andrews
Email:
lo@st-and.ac.uk
Journal:
Bull. Amer. Math. Soc.
37 (2000), 481-498
Published electronically:
June 27, 2000
Review copyright:
© Copyright 2000
American Mathematical Society