A representation theorem for entropies with the branching property

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 {s_i}, with s_i ↑ 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.