MATHEMATICAL THEORY OF COMMUNICATION 653 



where the second equality is by definition of measure in the g space the 

 third since the/ ensemble is stationary, and the last by definition of g meas- 

 ure again. 



To prove that the ergodic property is preserved under invariant operations, 

 let Si be a subset of the g ensemble which is invariant under H^, and let ^2 

 be the set of all functions / which transform into 6*1. Then 



H^Si = F'r52 = TH^S2 = ^'i 



so that H 6*1 is included in ^i for all X. Now, since 



m[H''S2] = m[Si] 



this implies 



for all X with m[S2] 5^ 0, 1. This contradiction shows that 6*1 does not exist. 



APPENDIX 6 



The upper bound, N3 < Ni -]- Nz , is due to the fact that the maximum 

 possible entropy for a power iVi + N2 occurs when we have a white noise of 

 this power. In this case the entropy power is Ni + 7V^2. 



To obtain the lower bound, suppose we have two distributions in n dimen- 

 sions p{xi) and q(xi) with entropy powers Ni and N2. What form should 

 p and q have to minimize the entropy power Ns of their convolution r{xi) : 



r{xi) = j p(yi)q{xi - y,) dyi . 



The entropy Hz of r is given by 



Hz = — \ r{xi) log r(xi) dxi . 



We wish to minimize this subject to the constraints 



Hi = - j p{xi) log p(xi) dxi 



B2 = - j q{xi) log q{xi) dxi . 

 We consider then 

 U = -j lr{x) log r(x) -f Xp{x) log p(x) + fiq(x) log q(x)] dx 



5?7 = -| [[1 + \ogr{x)\br{x) + X[l + log p{x)]bp{x) 



+ m[1 + log q{x)bq{x)\] dx. 



