Table Of Content

Mathematical Theory of Entropy GIAN-CARLO ROTA, Editor ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS Volume Section 1 LUIS A. SANTALO Probability Integral Geometry and Geometric Probability, 1976 2 GEORGE E. ANDREWS Number The Theory of Partitions, 1976 Theory 3 ROBERT J. McELIECE Probability The Theory of Information and Coding A Mathematical Framework for Communication, 1977 4 WILLARD MILLER, Jr. Special Symmetry and Separation of Variables, 1977 Functions 5 DAVID RUELLE Statistical Thermodynamic Formalism Mechanics The Mathematical Structures of Classical Equilibrium Statistical Mechanics, 1978 6 HENRYK MINC Linear Permanents, 1978 Algebra 7 FRED S. ROBERTS Mathematics Measurement Theory and the with Applications to Decisionmaking, Social Utility, and the Social Sciences, 1979 Sciences 8 L. C. BIEDENHARN and J. D. LOUCK Mathematics Angular Momentum in Quantum Physics: of Physics Theory and Application, 1981 9 L. C. BIEDENHARN and J. D. LOUCK Mathematics The Racah-Wigner Algebra in Quantum of Physics Theory, 1981 10 JOHN D. DOLLARD and CHARLES N. Analysis FRIEDMAN Product Integration with Application to Differential Equations, 1979 GIAN-CARLO ROTA, Editor ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS Volume Section 11 WILLIAM B. JONES and Analysis W. J. THRON Continued Fractions: Analytic Theory and Applications, 1980 12 NATHANIEL F. G. MARTIN and Real Variables JAMES W. ENGLAND Mathematical Theory of Entropy, 1981 13 GEORGE A. BAKER, JR. and Mathematics PETER GRAVES-MORRIS of Physics Pade Approximants, Vol. I: Basic Theory, 1981 14 GEORGE A. BAKER, JR. and Mathematics PETER GRAVES-MORRIS of Physics Pade Approximants, Vol. II: Extensions and Applications, 1981 Other volumes in preparation ENCYCLOPEDIA OF MATHEMATICS and Its Applications GIAN-CARLO ROTA, Editor Department of Mathematics Massachusetts Institute of Technology Cambridge, Massachusetts Editorial Board Janos D. Aczel, Waterloo Shizuo Kakutani, Yale Richard Askey, Madison Samuel Karlin, Stanford Michael F. Atiyah, Oxford J. F. C. Kingman, Oxford Donald Babbitt, U.C.L.A. Donald E. Knuth, Stanford Edwin F. Beckenbach, U.C.L.A. Joshua Lederberg, Rockefeller Lipman Bers, Columbia Andre Lichnerowicz, College de France Garrett Birkhoff, Harvard M. J. Lighthill, Cambridge Salomon Bochner, Rice Chia-Chiao Lin, M.I.T. Raoul Bott, Harvard Jacques-Louis Lions, Paris James K. Brooks, Gainesville G. G. Lorentz, Austin Felix E. Browder, Chicago Roger Lyndon, Ann Arbor A. P. Calderon, Buenos Aires Marvin Marcus, Santa Barbara Peter A. Camithers, Los Alamos N. Metropolis, Los Alamos S. Chandrasekhar, Chicago Jan Mycielski, Boulder S. S. Chern, Berkeley Steven A. Orszag, M.I.T. Hermann Chernoff, M.I.T. Alexander Ostrowski, Basle P. M. Cohn, Bedford College, London Roger Penrose, Oxford H. S. MacDonald Coxeter, Toronto Carlo Pucci, Florence Nelson Dunford, Sarasota, Florida C. R. Rao, Indian Statistical Institute F.J. Dyson, Inst, for Advanced Study Fred S. Roberts, Rutgers Harold M. Edwards, Courant Abdus Salam, Trieste Harvey Friedman, Ohio State M. P. Schutzenberger, Paris Giovanni Gallavotti, Rome Jacob T. Schwartz, Courant Andrew M. Gleason, Harvard Irving Segal, M.I.T. James Glimm, Rockefeller Olga Taussky, Caltech A. Gonzalez Dominguez, Buenos Aires Rene Thorn, Bures-sur-Yvette M. Gordon, Essex John Todd, Caltech Peter Henrici, ETH, Zurich John W. Tukey, Princeton Nathan Jacobson, Yale Stanislaw Ulam, Colorado Mark Kac, Rockefeller Veeravalli S. Varadarajan, U.C.L.A. Antoni Zygmund, Chicago GIAN-CARLO ROTA, Editor ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS Volume 12 Section: Real Variables James K. Brooks, Section Editor Mathematical Theory of Entropy Nathaniel F. G. Martin James W. England University of Virginia Swarthmore College Charlottesville, Virginia Swarthmore, Pennsylvania Foreword by James K. Brooks University of Florida, Gainesville A •• 1981 Addison-Wesley Publishing Company Advanced Book Program Reading, Massachusetts London -Amsterdam • Don Mills, Ontario • Sydney 'Tokyo CAMBRIDGE UNIVERSITY PRESS CCAamMBbrRiIdDgGeE, NUeNwIV YERoSrkIT, YM PeRlEbSoSu rne, Madrid, Cape Town, SCianmgabproidreg,e S, aNoe wPa Yuloor,k D, Melhelib, oTuorknyeo, , MMaedxriicdo, CCiatpye Town, Singapore, Sao Paulo, Delhi, Tokyo, Mexico City Cambridge University Press Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521302326 Information on this title: © 1981 - Addison - Wesley, Reading, MA 01867 ©© C19a8m1 b- rAiddgdei sUonn iv- eWrseitsyle yP,r eRsse a1d9i8n4g , MA 01867 © Cambridge University Press 1984 This publication is in copyright. Subject to statutory exception aTnhdi st op uthbeli cpartoivoins iiosn isn ocfo rpeylervigahntt. cSoulblejcetcitv eto l iscteantusitnogry a gerxeceempteiontns , naon dr etpor othdeu cptrioovni soiof nasn yo fp raerlte vmaanyt ctaokllee cptliavcee lwicietnhsoinugt tahger ewemritetnents , pneor mreipssrioodnu cotfi oCna mofb arindyg ep aUrtn mivaeyrs ittayk eP rpelsasc.e without the written permission of Cambridge University Press. First published 1981 by Addison Wesley First published 1981 by Addison Wesley First published by Cambridge University Press 1984 First published by Cambridge University Press 1984 A catalogue record for this publication is available from the British Library A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Library of Congress Cataloguing in Publication data Martin, Nathaniel F. G. MMartaitnh,e Nmaatthicaanli ethl eFo. rGy . of entropy. Mathematical theory of entropy. (Encyclopedia of mathematics and its applications; v. 12) B(Eibnlcioygclroappehdyi:a po. f mathematics and its applications; v. 12) IBnicblluidoegsr aipnhdye:x p. . 1I.n Eclnutdreosp yin (dIenxf.o rmation theory) 2. Ergodic theory. 3. S1t. aEtinsttircoaply m (eIncfhoarnmicast.i o4n. Tthoepoorlyo)g i2ca. lE drgyondaimc icthse. ory. I3.. ESntagtliasntidc,a lJ ammeecsh aWni. cIsI.. 4T.i tTleo.p IoIlIo. gSicearile ds.y namics. QI. 3E6n0g.Mla3n1d6, Jame5s1 W9.2. II. T8i1tl-e8.3 4II I. Series. Q360.M316 519.2 81-834 ISBN 978-0-521-30232-6 Hardback ISBN 978-0-521-30232-6 Hardback Cambridge University Press has no responsibility for the persistence or aCcacmurbarciyd goef UURniLvse rfsoirty e xPtreersns ahl aos rn toh irreds-ppoanrtsyib iinlitteyr nfoert twheeb psietressi srteefnecrree do r to in tahcicsu praucbyl iocaf tUioRnL, sa nfodr deoxetse rnnoatl gour atrhainrdte-ep atrhtayt ianntyer cnoent twenebt soitne ss urecfhe rwreedb stiote sin i s, othr iws ipllu brelimcaatiino, na, cacnudra dteo eosr naoptp rgouparriaantete. eI ntfhoartm aantyio cno nretegnatr doinn gs upcrhic ewse, btsraitveesl is, toimr wetiallb lreesm, aainnd, aoctchuerra ftaec otura al pipnrfooprmriaatteio. nI ngfiovremna itnio tnh irse wgaorrdki nigs cporircreecs,t tarta vel tthime etitmabel eosf, fainrsdt optrhinerti nfagc tbuuatl Cinafmorbmriadtgioen Ugniviveenr siinty t hPirse sws odroke iss ncootr rgeucta raatn tee tthhee aticmcuer aocfy f iorfs ts upcrihn tiinnfgo rbmuatt iCoanm tbhreirdegaef teUr.n iversity Press does not guarantee the accuracy of such information thereafter. To Our Wives Jo Martin and Mary England