# Article

Full entry | PDF   (0.2 MB)
Keywords:
entropy-like functionals; Hartley value of a piece of information; moderate $E$-entropy
Summary:
We describe a conceptual approach which provides a unified view of various entropy-like functionals on the class of semimetric spaces, endowed with a bounded measure. The entropy $E$ considered in the author's previous articles is modified so as to assume finite values for a fairly wide class of spaces which fail to be totally bounded.
References:
[H28] Hartley S.V.L.: Transmission of information. Bell System Tech. J. 7 , no. 3 (1978), 535-563.
[K80a] Katětov M.: Extensions of the Shannon entropy to semimetrized measure spaces. Comment. Math. Univ. Carolinae 21 (1980), 171-192. MR 0566249
[K80b] Katětov M.: Correction to Extensions of the Shannon entropy to semimetrized measure spaces''. Comment. Math. Univ. Carolinae 21 (1980), 825-830. MR 0597770 | Zbl 0476.94014
[K83] Katětov M.: Extended Shannon entropies I. Czechoslovak Math. J. 33 (108) (1983), 546-601. MR 0721088
[K85] Katětov M.: Extended Shannon entropies II. Czechoslovak Math. J. 35 (110) (1985), 565-616. MR 0809043
[K86a] Katětov M.: On extended Shannon entropies and the epsilon entropy. Comment. Math. Univ. Carolinae 27 (1986), 519-534. MR 0873625
[K86b] Katětov M.: On the Rényi dimension. Comment. Math. Univ. Carolinae 27 (1986), 741-753. MR 0874669
[K87] Katětov M.: On dimensions of semimetrized measure spaces. Comment. Math. Univ. Carolinae 28 (1987), 399-411. MR 0912568
[K88] Katětov M.: On the differential and residual entropy. Comment. Math. Univ. Carolinae 29 (1988), 319-349. MR 0957402
[K90] Katětov M.: On entropy-like functionals and codes for metrized probability spaces I. Comment. Math. Univ. Carolinae 31 (1990), 49-66. MR 1056171
[K92] Katětov M.: On entropy-like functionals and codes for metrized probability spaces II. Comment. Math. Univ. Carolinae 33 (1992), 79-95. MR 1173750
[KT59] Kolgomorov A., Tihomirov V.: $\varepsilon$-entropy and $\varepsilon$-capacity of sets in function spaces (in Russian). Uspehi Mat. Nauk 14 , no. 2 (1959), 3-86. MR 0112032
[PRR67] Posner E.C., Rodemich H., Rumsey H., Jr.: Epsilon entropy of stochastic processes. Ann. Math. Statist. 38 (1967), 1000-1020. MR 0211457 | Zbl 0168.18003
[R59] Rényi A.: On the dimension and entropy of probability distributions. Acta Math. Acad. Sci. Hungar. 10 (1959), 193-215. MR 0107575
[S48] Shannon C.E.: A mathematical theory of communication. Bell System Tech. J. 27 (1948), 375-423, 623-656. MR 0026286 | Zbl 1154.94303

Partner of