Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Measure and area


Author: Herbert Federer
Journal: Bull. Amer. Math. Soc. 58 (1952), 306-378
DOI: https://doi.org/10.1090/S0002-9904-1952-09586-0
MathSciNet review: 0049289
Full-text PDF

References | Additional Information

References [Enhancements On Off] (What's this?)

  • A. S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets of points, Math. Ann. 98 (1928), no. 1, 422–464. MR 1512414, https://doi.org/10.1007/BF01451603
  • A. S. Besicovitch, B 2. On surfaces of minimum area, Proc. Cambridge Philos. Soc. vol. 44 (1948) p. 313. MR 28936
  • C. Carathéodory, C. Über das lineare Mass von Punktmengen--eine Verallgemeinerung des Längenbegriffs, Nachr. Ges. Wiss. Göttingen (1914) p. 404.
  • L. Cesari, CE 1. Su di un problema di analysis situs delle spazio ordinario, Rendiconti dell’Istituto Lombardo di Scienze e Lettere vol. 75 (1941) p. 267.
  • L. Cesari, CE 2. Caratterizzazione analitica delle superficie continue di area finita secondo Lebesgue, Annali delia Scuola Normale Superiore di Pisa (2) vol. 10 (1941) p. 253; vol. 11 (1942) p. 1. MR 18213
  • L. Cesari, CE 3. Sui punti di diramazione delle trasformazione continue e sull'area delle superficie in forma parametrica, Rendiconti di Math. e delle sue Applicazioni Roma (5) vol. 3 (1942) p. 37. MR 18215
  • L. Cesari, CE 4. Criteri di uguale continuita' ed applicazioni alla quadratura delle superficie, Annali della Scuola Normale Superiore di Pisa (2) vol. 12 (1943) p. 61. MR 17352
  • L. Cesari, CE 5. Una ugualianza fondamentale per l'area delle superficie, Atti della reale Academia d'Italia vol. 14 (1944) p. 891. MR 17356
  • Jesse Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931), no. 1, 263–321. MR 1501590, https://doi.org/10.1090/S0002-9947-1931-1501590-9
  • C. H. Dowker, D. Mapping theorems for noncompact spaces, Amer. J. Math. vol. 69 (1947) p. 200. MR 20771
  • J. Favard, FA. Une définition de la longueur et de l'aire, C. R. Acad. Sci. Paris vol. 194 (1932) p. 344.
  • H. Federer, F 1. Surface area. I, II, Trans. Amer. Math. Soc. vol. 55 (1944) p. 420. MR 10610
  • H. Federer, F 2. The Gauss-Green Theorem, Trans. Amer. Math. Soc. vol. 58 (1945) p. 44. MR 13786
  • H. Federer, F 3. Coincidence functions and their integrals, Trans. Amer. Math. Soc. vol. 59 (1946) p. 441. MR 15466
  • H. Federer, F 4. The $(\varphi,k)$ rectifiable subsets of n-space, Trans. Amer. Math. Soc. vol. 62 (1947) p. 114. MR 22594
  • H. Federer, F 5. Dimension and measure, Trans. Amer. Math. Soc. vol. 62 (1947) p. 536. MR 23325
  • H. Federer, F 6. Essential multiplicity and Lebesgue area, Proc. Nat. Acad. Sci. U.S.A. vol. 34 (1948) p. 411. MR 27837
  • H. Federer, F 7. Hausdorff measure and Lebesgue area, Proc. Nat. Acad. Sci. U.S.A. vol. 37 (1951) p. 90. MR 47128
  • Felix Hausdorff, Dimension und äußeres Maß, Math. Ann. 79 (1918), no. 1-2, 157–179 (German). MR 1511917, https://doi.org/10.1007/BF01457179
  • W. Hurewicz and H. Wallman, HW. Dimension theory, Princeton Mathematical Series, Princeton University Press, 1941. MR 6493
  • H. Lebesgue, L. Intégrale, longueur, aire, Annali di Matematica Pura ed Applicata vol. 7 (1902) p. 231.
  • E. J. McShane, Parametrizations of saddle surfaces, with application to the problem of plateau, Trans. Amer. Math. Soc. 35 (1933), no. 3, 716–733. MR 1501713, https://doi.org/10.1090/S0002-9947-1933-1501713-3
  • E. F. Moore, MEF 1. Convexly generated k-dimensional measures, Proceedings of the American Mathematical Society vol. 2 (1951) p. 597. MR 43175
  • E. F. Moore, MEF 2. Density ratios and $(\phi,1)$ rectifiability in n-space, Trans. Amer. Math. Soc. vol. 69 (1950) p. 324. MR 37894
  • C. B. Morrey, Jr., MCB 1. A class of representations of manifolds. I, II, Amer. J. Math. vol. 55(1933) p. 683; vol. 56 (1934) p. 275.
  • C. B. Morrey, Jr., MCB 2. An analytic characterization of surfaces of finite Lebesgue area. I, II, Amer. J. Math. vol. 57 (1935) p. 692; vol. 58 (1936) p. 313.
  • A. P. Morse, M. A theory of covering and differentiation, Trans. Amer. Math. Soc. vol. 55 (1944) p. 205. MR 9974
  • A. P. Morse and H. Federer, MF. Some properties of measurable functions, Bull. Amer. Math. Soc. vol. 49 (1943) p. 270. MR 7916
  • A. P. Morse and J. F. Randolph, MR. The $\phi$ rectifiable subsets of the plane, Trans. Amer. Math. Soc. vol. 55 (1944) p. 236. MR 9975
  • Hans Rademacher, Über partielle und totale differenzierbarkeit von Funktionen mehrerer Variabeln und über die Transformation der Doppelintegrale, Math. Ann. 79 (1919), no. 4, 340–359 (German). MR 1511935, https://doi.org/10.1007/BF01498415
  • Tibor Radó, The problem of the least area and the problem of Plateau, Math. Z. 32 (1930), no. 1, 763–796. MR 1545197, https://doi.org/10.1007/BF01194665
  • T. Rado, R 2. On the derivative of the Lebesgue area of continuous surfaces, Fund. Math. vol. 30 (1938) p. 34.
  • Tibor Radó, On absolutely continuous transformations in the plane, Duke Math. J. 4 (1938), no. 1, 189–221. MR 1546044, https://doi.org/10.1215/S0012-7094-38-00415-6
  • T. Rado, R 4. On continuous mappings of Peano spaces, Trans. Amer. Math. Soc. vol. 58 (1945) p. 420. MR 14419
  • T. Rado, R 5. On surface area, Proc, Nat. Acad. Sci. U.S.A. vol. 31 (1945) p. 102. MR 11712
  • T. Rado, R 6. Length and area, Amer. Math. Soc. Colloquium Publications, vol. 30, New York, 1948. MR 24511
  • A. Sard, S. The equivalence of n-measure and Lebesgue measure in E, Bull. Amer. Math. Soc. vol. 49 (1943) p. 758. MR 8837
  • H. Whitney, W. On totally differentiable and smooth functions, Pacific Journal of Mathematics vol. 1 (1951) p. 143. MR 43878


Additional Information

DOI: https://doi.org/10.1090/S0002-9904-1952-09586-0

American Mathematical Society