Suppose there is a probability distribution $\pi(x,y) A\times A \to \Delta (B)$. where $\Delta(B)$ is the set of probability distributions over $B$ and $\pi_b(x,y)$ gives the probability of $b\in B$ occurring from $\pi(x,y)$.
Suppose there are two more probability distributions, $\nu: A\to \mathbb{R}$ and and $\kappa: B\to\mathbb{R}$.
Assume the following property is true: $$ \kappa (b) = \sum_{x\in A} \nu(x)\pi_b(x,y)\quad \text{for every } y $$
Is $\kappa$ unique? It clear is when $\nu$ is held fixed, but i'm unsure if there can be another $\kappa,\nu$ pair that makes the $=$ true.
If it's not true (i.e. there is another pair that makes it true) I'm sure someone can give an easy counterexample.