Geodesic
Maximal linked systems on families of measurable rectangles
Vestnik rossijskih universitetov. Matematika, Tome 26 (2021) no. 133, pp. 77-104
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Linked and maximal linked systems (MLS) on $\pi$-systems of measurable (in the wide sense) rectangles are considered ($\pi$-system is a family of sets closed with respect to finite intersections). Structures in the form of measurable rectangles are used in measure theory and probability theory and usually lead to semi-algebra of subsets of cartesian product. In the present article, sets-factors are supposed to be equipped with $\pi$-systems with “zero” and “unit”. This, in particular, can correspond to a standard measurable structure in the form of semi-algebra, algebra, or $\sigma$-algebra of sets. In the general case, the family of measurable rectangles itself forms a $\pi$-system of set-product (the measurability is identified with belonging to a $\pi$-system) which allows to consider MLS on a given $\pi$-system (of measurable rectangles). The following principal property is established: for all considered variants of $\pi$-system of measurable rectangles, MLS on a product are exhausted by products of MLS on sets-factors. In addition, in the case of infinity product, along with traditional, the “box” variant allowing a natural analogy with the base of box topology is considered. For the case of product of two widely understood measurable spaces, one homeomorphism property concerning equipments by the Stone type topologies is established.
Keywords: linked systems; measurable rectangles; $\pi$-system. 
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A. G. Chentsov. Maximal linked systems on families of measurable rectangles. Vestnik rossijskih universitetov. Matematika, Tome 26 (2021) no. 133, pp. 77-104. http://geodesic.mathdoc.fr/item/VTAMU_2021_26_133_a7/

[1] A. G. Chentsov, “Bitopologicheskie prostranstva ultrafiltrov i maksimalnykh stseplennykh sistem”, Tr. IMM UrO RAN, 24, no. 1, 2018, 257–272 ; A. G. Chentsov, “Bitopological spaces of ultrafilters and maximal linked systems”, Proc. Steklov Inst. Math. (Suppl.), 305, no. suppl. 1 (2019), S24–S39 | DOI | MR | Zbl

[2] A. G. Chentsov, “Ultrafilters and maximal linked systems”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 27:3 (2017), 365–388 (In Russian) | MR | Zbl

[3] A. G. Chentsov, “Supercompact spaces of ultrafilters and maximal linked systems”, Trudy Inst. Mat. i Mekh. UrO RAN, 25, no. 2, 2019, 240–257 (In Russian)

[4] J. de Groot, “Superextensions and supercompactness”, Extension Theory of Topological Structures and its Applications, I International Symposium “Extension Theory of Topological Structures and its Applications” (Berlin, 1969), Proceedings of the Symposium, VEB Deutscher Verlag Wis., Berlin, 1969, 89–90 | MR

[5] J. van Mill, Supercompactness and Wallman spaces, Mathematical Centre Tracts, 85, Mathematisch Centrum, Amsterdam, 1977, 238 pp. | MR | Zbl

[6] M. Strok, A. Szymanski, “Compact metric spaces have binary subbases”, Fund. Math., 89:1 (1975), 81–91 | DOI | MR | Zbl

[7] V. V. Fedorchuk, V. V. Filippov, General Topology. Basic Constructions, Fizmatlit Publ., Moscow, 2006, 336 pp. (In Russian)

[8] A. V. Arkhangel'skii, “Compactness”, General topology – 2, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 50, VINITI, Moscow, 1989, 5–128 (In Russian)

[9] A. V. Bulinskiy, A. N. Shiryaev, Theory of Stochastic Processes, Fizmatlit, M., 2005, 402 pp. (In Russian)

[10] A. G. Chentsov, “K voprosu o predstavlenii ultrafiltrov v proizvedenii izmerimykh prostranstv”, Tr. IMM UrO RAN, 19, no. 2, 2013, 307–319 ; Proc. Steklov Inst. Math. (Suppl.), 284, no. suppl. 1 (2014), 65–78 | DOI | MR

[11] A. G. Chentsov, Elements of Finitely Additive Measure Theory, v. II, Ural State Technical University - UPI, Yekaterinburg, 2010, 541 pp. (In Russian)

[12] K. Kuratovsky, A. Mostovsky, Set Theory, Mir Publ., Moscow, 1970, 416 pp. (In Russian) | MR

[13] J. Warga, Optimal Control of Differential and Functional Equations, Science, Moscow, 1977, 624 pp. (In Russian)

[14] J. Neve, Mathematical Foundations of Probability Theory, Mir Publ., Moscow, 1969, 309 pp. (In Russian)

[15] A. G. Chentsov, “Filters and linked families of sets”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 30:3 (2020), 444–467 (In Russian) | MR

[16] A. G. Chentsov, “On the supercompactness of ultrafilter space with the topology of Wallman type”, Izv. IMI UdGU, 54 (2019), 74–101 (In Russian) | Zbl

[17] V. I. Bogachev, Weak Convergence of Measures, Institute for Computer Research, Moscow–Izhevsk, 2016, 396 pp. (In Russian) | MR

[18] R. Engelking, General Topology, Mir Publ., Moscow, 1986, 751 pp. (In Russian) | MR

[19] A. G. Chentsov, S. I. Morina, Extensions and Relaxations, Kluwer Acad. Publ., Dordrecht–Boston–London, 2002, 408 pp. | MR | Zbl

[20] N. Burbaki, General Topology. Basic Structures, Nauka Publ., Moscow, 1968, 272 pp. (In Russian)

[21] R. A. Alexandryan, E. A. Mirzakhanyan, General Topology, High School Publ., Moscow, 1979, 336 pp. (In Russian)