The envelope problem, and a possible solution

[AaronBrown]

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I'm still a little bit confused about your stance here. Let me clarify some things from my side first. When I say "probability theory", I am talking about the branch of mainstream measure theory that studies sets, sigma-algebras on those sets, and countably-additive, non-negative measures on those sigma-algebras whose total mass is 1. This approach to probability theory was introduced by Kolmogorov. Perhaps you could elaborate on the four theories you mentioned, with links if possible, and explain their relationship to what I am calling "probability theory".

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That's easy enough. You are giving Kolmogorov credit for what I called Von Neumann's formulation. Many people contributed to all four systems, I didn't mean to exclude anyone, I just used familiar tag-lines.

Jimmy Savage's The Foundation of Statistics not only gives his formulation, it has clear explanations and contrasts with the other three, plus an extensive annotated bibliography.

Everyone uses sets, sigma-algebras and measures. But there is controversy over concepts like shrinkage and resampling, which do not adapt well to those tools.