 With Eq. (B9), this property gives A(Sm)<A(T)< A(Sm+1) and with Eq. (B5): mA(S) )<nA(T)<(m+1)A(S) (B10) E being made arbitrarily small. Combining Eqs. (B8) and (B10), we get (B11) This result shows that, as n becomes large (E 0), the distance between the function m/n and the two quantities log T/ log S and A(T)/A(S) vanishes. The only possibility of verifying this property is that we have: (B12) A(T)= K log T, (B13) where K is an arbitrary constant, which must be positive to satisfy condition (2). We can also derive this result more formally using the previously established relation m = E(u) = E(n log T/logS), which gives: (B14) Shannon's entropy There exist two possible cases: u is an integer: then E(u) = U = n log T / log S, and the first inequality in Eq. (B12) intrinsically holds regardless of the size of N. The second inequality converts to (B15) which, in the limit of large n, leads to the conclusion expressed in Eqs. (B12) and (B13); is not integer: then E(u) = u+x = n log T/ log S + X, where 0 x < 1. In this case, the first inequality intrinsically holds regardless of the size of n, and the second inequality converts to (B16) which leads to the same conclusion.2 Step 3 Assume that the source now contains N equiprobable possibilities. Its information mea- sure is, therefore, H(1/N, 1/N,..., , 1/N) = A(N) = K log N. We can arbitrarily group these possibilities into m subgroups having Ni elements each, and whose associated prob- abilities are pi = ni/N. According to condition (3), if we define a first partition of two subgroups of length n and N2 = N - N1, with probabilities p1 = n1/N and p2 = n2/N, respectively, we have A(N)=H(p1,p2)+p1A(n1)+p2A(n2). (B17) In the right-hand development of Eq. (B17), the first term corresponds to the information provided by this partition and the second two terms corresponds to the information contained in the two subgroups (weighted by their respective occurrence probabilities p1, p2). Note that if any subgroup has only one element (e.g., N1 = 1), the corresponding informa