What comes to my mind are Bochner integrals and random elements. I'm not sure how much integrability theory one can develop outside of a Banach space, although you can get interesting fractal type integrals when dealing with Hausdorff measure. Integrability theory is really just an extension of measure theory, which was pinned down in painstaking detail by Lebesgue, Caratheodory, Perron, Henstock, and Kurzweil (no relation to the singularity Kurzweil). The Henstock-Kurzweil (HK) integral is the most generalized integral over the reals and complexs that preserves certain nice properties, like the fundamental theorem of calculus. The name of the game in integration theory was never an attempt to find the most abstract workable definitions of integration, but rather to see under what general assumptions you could get physically meaningful results, like mean value theorem or fundamental theorem of calculus, to hold. Complex integration theory, especially in higher dimensions shattered a lot of the preconceived notions of how functions should behave.
In looking up surreal numbers, it appears that Conway and Knuth invented them. I was surprised to learn that the hyperreal numbers (developed by Abraham Robinson) are contained in the surreals. To my knowledge, which is a bit limited because I focus more on applied math and so I am probably not as familiar with the literature on something like surreal numbers as other LWers may be, there hasn't been much work, if any, on defining an integral over the surreals. My guess, though, is that such an integral would wind up being an unsatisfyingly trivial extension of integration over the regular reals, as is the case for hyperreals.
I'll definitely take a look at Kruskal's papers and see what he's come up with.
I was surprised to learn that the hyperreal numbers (developed by Abraham Robinson) are contained in the surreals.
Every ordered field is contained within the surreals, which is why I find them promising for utility theory. The surreals themselves are not a field but a Field, since they form a proper class.
Expected utility can be expressed as the sum ΣP(Xn)U(Xn). Suppose P(Xn) = 2-n, and U(Xn) = (-2)n/n. Then expected utility = Σ2-n(-2)n/n = Σ(-1)n/n = -1+1/2-1/3+1/4-... = -ln(2). Except there's no obvious order to add it. You could just as well say it's -1+1/2+1/4+1/6+1/8-1/3+1/10+1/12+1/14+1/16-1/5+... = 0. The sum depends on the order you add it. This is known as conditional convergence.
This is clearly something we want to avoid. Suppose my priors have an unconditionally convergent expected utility. This would mean that ΣP(Xn)|U(Xn)| converges. Now suppose I observe evidence Y. ΣP(Xn|Y)|U(Xn)| = Σ|U(Xn)|P(Xn∩Y)/P(Y) ≤ Σ|U(Xn)|P(Xn)/P(Y) = 1/P(Y)·ΣP(Xn)|U(Xn)|. As long as P(Y) is nonzero, this must also converge.
If my prior expected utility is unconditionally convergent, then given any finite amount of evidence, so is my posterior.
This means I only have to come up with a nice prior, and I'll never have to worry about evidence braking expected utility.
I suspect that this can be made even more powerful, and given any amount of evidence, finite or otherwise, I will almost surely have an unconditionally convergent posterior. Anyone want to prove it?
Now let's look at Pascal's Mugging. The problem here seems to be that someone could very easily give you an arbitrarily powerful threat. However, in order for expected utility to converge unconditionally, either carrying out the threat must get unlikely faster than the disutility increases, or the probability of the threat itself must get unlikely that fast. In other words, either someone threatening 3^^^3 people is so unlikely to carry it out to make it non-threatening, or the threat itself must be so difficult to make that you don't have to worry about it.