I'm reading through the communication complexity book of Kushilevitz and Nisan, and in the section about fooling sets I encountered this proposition:
Let $\mu$ be a probability distribution of $X\times Y$. If any $f$-monochromatic rectange $R$ has measure $\mu(R) \leq \delta$, then $D(f) \geq \log_2 1/\delta$. [Prop. 1.24]
I understand the idea of a fooling set (find a set which takes the same function value everywhere, but mixing and matching $x$ and $y$ coordinates will change the function value). I understand that $\mu$ maps $X\times Y$ to $\mathbb{R}$, and that $\mu(X\times Y) = 1$. But I don't understand why we should be allowed to use a probability distribution at all. There's no randomness here, and even if their likelihoods of being chosen were not uniform, each pair $(x,y)$ should in theory be possible, and we need to take this into consideration because communication complexity deals with the worst-case.
I'm obviously missing something. Any help here would be appreciated.
(Also, the use of any
in the proposition is ambiguous. I assume that it means "for all", but it could also possibly mean "there exists"...)