A structure theorem for sum form functional equations (Q675215)

The author considers the functional equation \[ \sum^k_{i=1} \sum^l_{j=1} \Biggl[ f_{ij}(x_iy_i)- \sum^N_{t=1} g_{it}(x_i) h_{jt}(y_j) \Biggr]=0 \qquad (X\in\Gamma^0_k,\;Y\in\Gamma^0_l), \] where \(k,l\geq 2\), \(N\geq 0\) are fixed integers, \(f_{ij},g_{it},h_{jt}: ]0,1[\to\mathbb{C}\) \((i=1,\dots, k\); \(j=1,\dots,l\); \(t=1,\dots, N)\) are (given or unknown) functions and \[ \Gamma^0_n= \Biggl\{ X=(x_1,\dots, x_n)\mid x_i>0\;(i=1,\dots,n);\;\sum^n_{i=1} x_i=1\Biggr\} \] is a set of all complete \(n\)-ary probability distributions with positive probabilities and proves that if \(f_{ij}\) are measurable, then each \(f_{ij}\) is in a linear space spanned by the functions \(x^{\lambda_j} (\log x)^k\) (for \(x\in ]0,1[\), \(j=1,2,\dots, n\), \(k=0,\dots, m_j-1)\) where \(\lambda_j\) are distinct complex numbers and \(m_j\) are natural numbers.