An incompatibility result on non-Archimedean integration. (English) Zbl 1486.26063
Summary: We prove that a Riemann-like integral on non-Archimedean extensions of \(\mathbb{R}\) cannot assign an integral to every function whose standard part is measurable and simultaneously satisfy the fundamental theorem of calculus. We also discuss how existing theories of non-Archimedean integration deal with the incompatibility of these conditions.
MSC:
| 26E30 | Non-Archimedean analysis |
| 26A42 | Integrals of Riemann, Stieltjes and Lebesgue type |
| 28B99 | Set functions, measures and integrals with values in abstract spaces |
References:
| [1] | Berarducci, A.; Otero, M., An additive measure in o-minimal expansions of fields, Quart. J. Math., 55, 411-419 (2004) · Zbl 1065.03020 · doi:10.1093/qmath/hah010 |
| [2] | Bottazzi, E., Grid functions of nonstandard analysis in the theory of distributions and in partial differential equations, Adv. Math., 345, 429-482 (2019) · Zbl 1423.46056 · doi:10.1016/j.aim.2019.01.024 |
| [3] | Bottazzi, E., Spaces of measurable functions on the Levi-Civita field, Indagat. Math., 31, 4, 650-694 (2020) · Zbl 1454.26045 · doi:10.1016/j.indag.2020.06.005 |
| [4] | Bottazzi, E., A real-valued measure on non-Archimedean field extensions of \(\mathbb{R} \), p-Adic Num. Ultrametr. Anal. Appl., 14, 0 (2022) · Zbl 1494.26063 |
| [5] | Bottazzi, E.; Eskew, M., Integration with filters (2020) |
| [6] | Conway, J. H., On Numbers and Games (2001), Natick, Massachusetts: A. K. Peters, Ltd., Natick, Massachusetts · Zbl 0972.11002 |
| [7] | Costin, O.; Ehrlich, P.; Friedman, H., Integration on the surreals: a conjecture of Conway, Kruskal and Norton (2015) |
| [8] | Cutland, N. J., Nonstandard measure theory and its applications, Bull. London Math. Soc., 15, 529-589 (1983) · Zbl 0529.28009 · doi:10.1112/blms/15.6.529 |
| [9] | Flynnn, D.; Shamseddine, K., On integrable delta functions on the Levi-Civita field, p-Adic Num. Ultrametr. Anal. Appl., 10, 1, 32-56 (2018) · Zbl 1428.46048 · doi:10.1134/S207004661801003X |
| [10] | Fornasiero, A., Integration on Surreal Numbers (2004) |
| [11] | Gillman, L., An axiomatic approach to the integral, Amer. Math. Month., 100, 1, 16-25 (1993) · Zbl 0784.26009 · doi:10.1080/00029890.1993.11990359 |
| [12] | Kaiser, T., Lebesgue measure and integration theory on non-archimedean real closed fields with archimedean value group, Proc. London Math. Soc., 116, 2, 209-247 (2018) · Zbl 1436.03211 · doi:10.1112/plms.12070 |
| [13] | Klement, E. P.; Mesiar, R., Discrete integrals and axiomatically defined functionals, Axioms, 1, 9-20 (2012) · Zbl 1293.28008 · doi:10.3390/axioms1010009 |
| [14] | Shamseddine, K., New results on integration on the Levi-Civita field, Indagat. Math., 24, 1, 199-211 (2013) · Zbl 1300.26019 · doi:10.1016/j.indag.2012.08.005 |
| [15] | Shamseddine, K.; Berz, M., Measure theory and integration on the Levi-Civita field, Contemp. Math., 319, 369-388 (2003) · Zbl 1130.12301 · doi:10.1090/conm/319/05583 |
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