zbMATH — the first resource for mathematics

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.
30G35 Functions of hypercomplex variables and generalized variables
28A10 Real- or complex-valued set functions
28A33 Spaces of measures, convergence of measures
Full Text: DOI
[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.