ISSN:
1420-8903
Keywords:
Primary 94A17
;
Secondary 94A15, 39B40
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary Using a recent result by B. Ebanks on the functional equation $$h(x,y) + h(x + y,z) = h(x,y + z) + h(y,z)$$ we derive a representation theorem for a large class of entropy functionals that exhibit the “branching property”. LetV(Ω, F,m) be the set of probability densities on a non-atomic measure space {Ω,F,m} and $$\bar V$$ (Ω,F,m) be the set of all simple probability densities. A functional Ф: (Ω,F,m) →R ∪ { − ∞, ∞} will be said to have thebranching property, if, given any setA ∈ F and any two functionsf, g ∈ V such that at least one of Ф(f) or Ф(g) is finite andf(ω) = g(ω) whenever ω ∈ Ω/A, then $$\Phi (f) - \Phi (g) = \Psi (f_A ,g_A ),$$ wheref A is the restriction off to the setA and Ψ:L 1(A, F,m) ×L 1(A, F,m) →R ∪ {− ∞, ∞}. Theorem 1.Given Ф: V(Ω,F,m) →R ∪ {−∞, ∞}, $$\bar V$$ (Ω,F,m) →R,If (i) Фhas the branching property (ii) Фis invariant under all metric endomorphisms (iii) (continuity) for any sequence of simple functions {si}, with si ↑ f we have (with ∥ · ∥ the L1 norm) $$\Phi \left( {\frac{{s_i }}{{\parallel s_i \parallel }}} \right) \to \Phi (f)$$ then there exists h:[0, ∞) →R continuous on (0, ∞)with h(0) = 0such that Ф(f) = ∫ Ω h(f) d m. Фis said to be “recursive” if, for any set A ∈ F and any two functions, f, g ∈ V such that f(ω) = g(ω) at each ω ∈ Ω/A and p:=∫ A f d m =∫ A g d m 〉0, $$\Phi (f) - \Phi (g) = p\left[ {\Phi \left( {\frac{{f\chi _A }}{p}} \right) - \Phi \left( {\frac{{g\chi _A }}{p}} \right)} \right],$$ where ϰ A is the characteristic function of the set A. By strengthening (i) in Theorem 1 to “Ф is recursive” we obtain a new characterization of the Boltzmann—Shannon entropy.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01836445
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