Gordon Watson wrote:In the meantime, you might like to consider the nature of my functions AND comment thereon: for they are required to covert Bell's continuous variable λ into discrete outcomes.
Bell wrote “integral rho(lambda) d lambda ... “ [formula for expectation value, in absolutely continuous case]
He could just as well have written “sum p(lambda) ...” [formula for expectation value, in discrete case]
He could have written “integral P(d lambda) ...” [formula for expectation value, *all cases*, using notation from measure theory, following Borel, Kolmogorov, ...].
He did it his way because he was talking to physicists in 1964. He mentioned that of course his proof did not need the assumption of a probability density and lambda in R^something. He says he writes it that way just for convenience, and says of course it goes through in the discrete case.
In CHSH we have four binary hidden variables - the counterfactual outcomes of Alice and Bob to each of two settings. So we need a probability space with 16 outcomes only. This has been known for half a century...