On a functional equation arising in information theory
The author obtains the measurable solutions of \[ (1)\quad \sum^{n}_{i=1}\sum^{m}_{j=1}I(p_ iq_ j,v_ iw_ j)=\sum^{n}_{i=1}G(p_ i)\sum_{j}I(v_ j,w_ j)+\sum_{j}L(q_ j)\sum_{i}I(p_ i,v_ i) \] holding for some fixed pair (m,n), m,n\(\geq 3\) with \(I(0,0)=0=G(0)=L(0)\) where G,L: [0,1]\(\to {\mathbb{R}}\) are measurable functions and \(I: J\to {\mathbb{R}}\) is measurable in each variable. The equation (1) is a generalization of the well known sum form \[ (2)\sum^{n}_{i=1}\sum^{m}_{j=1}I(p_ iq_ j,v_ iw_ j)=\sum_{i}I(p_ i,v_ i)+\sum_{j}I(q_ j,w_ j) \] investigated by several authors including the reviewer. The measurable solutions are too complicated to be reproduced and they are nine in number.
representation of generalized additive measures of information
scientific article; zbMATH DE number 3944445