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On quaternionic measures. (English) Zbl 07261685
The paper proposes a generalization of the notion of complex measure in the quaternionic setting. For such a generalization, the authors prove several founding results, such as:
– the finiteness of the modulus of variation (extending a Lemma originally stated by Rudin);
– a Lebesgue type decomposition;
– a Radon-Nikodym type theorem;
– the fact that the set of quaternionic measures on a given domain is a Banch space.
These results are proven in many details and are treated very well.
MSC:
30G35 Functions of hypercomplex variables and generalized variables
28A10 Real- or complex-valued set functions
28A33 Spaces of measures, convergence of measures
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[1] Agrawal, S.; Kulkarni, SH, An analogue of the Riesz-representation theorem, Novi Sad J. Math., 30, 143-154 (2000) · Zbl 1009.46012
[2] Alpay, D.; Luna-Elizarrarás, ME; Shapiro, M., Kolmogorov’s axioms for probabilities with values in hyperbolic numbers, Adv. Appl. Cliff. Algebras, 27, 2, 913-929 (2017) · Zbl 1388.60011
[3] Artstein, Z., Set-valued measures, Trans. Am. Math. Soc., 165, 103-125 (1972) · Zbl 0237.28008
[4] Benci, V.; Horsten, L.; Wenmackers, S., Non-Archimedean probability, Milan J. Math., 81, 1, 121-151 (2013) · Zbl 1411.60007
[5] Ciurea, G.: Nonstandard measure theory. Abstr. Appl. Anal. Art. ID 851080 (2014) · Zbl 07023194
[6] Colombo, F.; Gantner, J., Quaternionic Closed Operators, Fractional Powers and Fractional Diffusion Processes. Operator Theory: Advances and Applications (2019), Basel: Birkhäuser, Basel · Zbl 07093564
[7] Cutland, NJ, Nonstandard measure theory and its applications, Bull. Lond. Math. Soc., 15, 6, 529-589 (1983) · Zbl 0529.28009
[8] Diestel, J.; Faires, B., On vector measures, Trans. Am. Math. Soc., 198, 253-271 (1974) · Zbl 0297.46034
[9] Ghosh, C.; Biswas, S.; Yasin, T., Hyperbolic valued signed measures, Int. J. Math. Trends Technol., 55, 7, 515-522 (2018)
[10] Halmos, PR, Measure Theory (1950), New York: Springer, New York
[11] Hofweber, T.; Schindler, R., Hyperreal-valued probability measures approximating a real-valued measure, Notre Dame J. Form. Log., 57, 3, 369-374 (2016) · Zbl 1385.60008
[12] Kumar, R.; Sharma, K., Hyperbolic valued measures and fundamental law of probability, Glob. J. Pure Appl. Math., 13, 10, 7163-7177 (2017)
[13] Ludkowski, S.V.: Hypercomplex generalizations of Gaussian-type Measures. arXiv:1812.06326v1 [math.PR] (2018)
[14] Maitland Wright, JD, Sone-algebra-valued measures and integrals, Proc. Lond. Math. Soc. (3), 19, 107-122 (1969) · Zbl 0186.46504
[15] Rudin, W., Real and Complex Analysis (1987), Singapore: McGraw-Hill Book Company, Singapore · Zbl 0925.00005
[16] Sun, Y., On the theory of vector valued Loeb measures and integration, J. Funct. Anal., 104, 2, 327-362 (1992) · Zbl 0780.46044
[17] Tutubalin, VN, Correlation analysis of random quaternions, Vestnik Moskow. Univ. Ser. 1. Mat. Mekh., 2, 15-23 (1982)
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