Yes, good point. Is there any study of the most general objects to which integrability theory applies? Also, are you familiar with Martin Kruskal’s work on generalizing calculus to the surreal numbers? I am having difficulty locating any of his papers.
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.
No, all of the integrability theory (w.r.t. probability measures) extends straightforwardly to complex valued functions. See this and this.
Yes, good point. Is there any study of the most general objects to which integrability theory applies? Also, are you familiar with Martin Kruskal’s work on generalizing calculus to the surreal numbers? I am having difficulty locating any of his papers.
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.
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.