Characterizations of Information Measures
World Scientific, 1998 - Functional equations - 281 pages
How should information be measured? That is the motivating question for this book. The concept of information has become so pervasive that people regularly refer to the present era as the Information Age. Information takes many forms: oral, written, visual, electronic, mechanical, electromagnetic, etc. Many recent inventions deal with the storage, transmission, and retrieval of information. From a mathematical point of view, the most basic problem for the field of information theory is how to measure information. In this book we consider the question: What are the most desirable properties for a measure of information to possess? These properties are then used to determine explicitly the most “natural” (i.e. the most useful and appropriate) forms for measures of information.This important and timely book presents a theory which is now essentially complete. The first book of its kind since 1975, it will bring the reader up to the current state of knowledge in this field.
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Introduction Preliminaries and Notation
The Branching Property
The Fundamental Equation of Information
Sum Form Information Measures
Basic Sum Form Functional Equations
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1-dimensional abelian group Aczel additive function an-i apply Theorem arbitrary arrive assume branching property Cauchy functional equations Chapter characteristic functional characterization cocycle coefficients completes the proof constant contradiction Daroczy defined entropies of degree equation of information exists extension F satisfies F(pq fixed form information measures function F functional equation fundamental equation given h(pq Hence i=l j=l implies In(P induction information theory integers Jarai Lebesgue measurable Lemma Let F linear linearly independent log(l logarithmic logp M(pi m)-additive Math measurable functions measurable solutions measures of information Moreover multiplicative functions multiplicative type nonconstant nonempty notation numbers obtain open domain open set polynomial positive cone projection proof of Theorem prove recursive of type Remark representation respectively satisfy the functional semigroup Shannon entropy solution of 5.2.2 solve subring Substituting sum form information suppose symmetric theory tion variable vector Verification of Step yields zero