Title:
Do Physical Probabilities Really Come from Uniform Measures?
Abstract:
Rubenstein (2026) argues that physical probabilities must be derived from (or grounded in) uniform measures over physical possibility space, which he calls "Necessary Uniformity." According to Rubenstein, whenever a uniform measure respects a probabilistic relation, the probabilities it assigns must be correct, based on two key premises: the Law of Large Numbers and a principle he calls 'Mismatch is not Typical.'. I argue that this thesis is false.
First, I show that Rubenstein's argument conflates the probability measure with probability values themselves, treating them as potentially independent when they are definitionally linked through expectation values. Second, his key premise—"Mismatch is not Typical"—unjustifiably restricts to uniform measures when judging typicality, thereby begging the question about whether uniformity is necessary.
I then present two mathematical results that independently refute Necessary Uniformity. First, for any uniform measure, infinitely many non-uniform measures (constructed via density functions) share the same typicality structure, making uniformity per se irrelevant. Second, the Disintegration Theorem in measure theory, together with results on product measure constructions, reveals that Rubenstein's appeal to larger spaces in search of appropriate uniform measures fails: there is neither uniqueness of measures on product spaces with given marginals, nor uniqueness in marginalization to subspaces.
Bohmian mechanics provides a decisive counterexample because the measure |Ψ|² is non-uniform over configuration space, yet successfully grounds all physical probabilities and empirical frequencies. Rubenstein's response, namely, that physical probabilities given by the Born Rule derive from a uniform measure on the product space of configuration space and Hilbert space, is inadequately supported.