I'm trying to follow this mathematical explanation of Bell's theorem. The problem I find is with the assumption of a probability density for the hidden variable. That implies - and my question is: am I wrong? why? - that you can expect the same distribution of such a variable for any repetition of the experiment (measurement). Now that's inconsistent (imho) with the nature itself of the variable, that should represent an unknown law of physics, an expression whose values are clearly function of time, at least, and space. I mean, I could just imagine that distribution as a Dirac delta of a predetermined result, but function of the angle of measurement and different for each measurement (cause each one will happen in a different space-time...). I fail to see how a (unique) unitary integral of a density for the hidden variable is realistic.