On the question of representation of ultrafilters in a product of measurable spaces

A. G. Chentsov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 2, pp. 307-319

Résumé

We study representations of ultrafilters of widely understood measurable spaces that are realized by means of generalized Cartesian products. The structure of the arising ultrafilter space is established; in more traditional measurable spaces, this structure reduces to the realization of the Stone compact space in the form of a Tikhonov product. The methods developed can be applied to the construction of extensions of abstract attainability problems with constraints of asymptotic nature; in such extensions, ultrafilters can be used as generalized elements, which admits a conceptual analogy with the Stone–Cech compactification. The proposed implementation includes the possibility of using measurable spaces, for which the set of free ultrafilters can be described completely. This results in an exhaustive representation of the corresponding Stone compact space for measurable spaces with algebras of sets. The present issue is devoted to I. I. Eremin's jubilee; the author had many discussion with him on very different topics related to mathematical investigations, and the discussions inevitably led to a deeper understanding of their essence. The author appreciates the possibility of such communication and is grateful to Eremin, who contributed significantly to the development of the mathematical science and education in the Urals.

Détail

Keywords: measurable space, Tychonoff product, ultrafilter.

Bibliographie

[1] Varga Dzh., Optimalnoe upravlenie differentsialnymi i funktsionalnymi uravneniyami, Nauka, M., 1977, 624 pp. | MR

[2] Gamkrelidze R. V., Osnovy optimalnogo upravleniya, Izd-vo Tbilis. un-ta, Tbilisi, 1975, 230 pp. | MR

[3] Krasovskii N. N., Teoriya upravleniya dvizheniem, Nauka, M., 1968, 475 pp. | MR

[4] Krasovskii N. N., Subbotin A. I., Pozitsionnye differentsialnye igry, Nauka, M., 1974, 456 pp. | MR | Zbl

[5] Subbotin A. I., Chentsov A. G., Optimizatsiya garantii v zadachakh upravleniya, Nauka, M., 286 pp. | MR

[6] Daffin R. Dzh., “Beskonechnye programmy”, Lineinye neravenstva i smezhnye voprosy, Inostr. lit., M., 1959, 263–267

[7] Golshtein E. G., Teoriya dvoistvennosti v matematicheskom programmirovanii i ee prilozheniya, Nauka, M., 1971, 351 pp. | MR

[8] Arkhangelskii A. V., “Kompaktnost”, Itogi nauki i tekhniki. Ser. “Sovremennye problemy matematiki. Fundamentalnye napravleniya”, 50, 1989, 5–128 | MR | Zbl

[9] Kelli Dzh. L., Obschaya topologiya, Nauka, M., 1981, 431 pp. | MR

[10] Edvards R., Funktsionalnyi analiz. Teoriya i prilozheniya, Mir, M., 1969, 1071 pp.

[11] Chentsov A. G., “Nekotorye konstruktsii asimptoticheskogo analiza, svyazannye s kompaktifikatsiei Stouna–Chekha”, Sovrem. matematika i ee pril. (AN Gruzii, In-t kibernetiki), 26 (2005), 119–150 | MR

[12] Chentsov A. G., “Finitely additive measures and extensions of abstract control problems”, J. Math. Sci., 133:2 (2006), 1045–1206 | DOI | MR | Zbl

[13] Chentsov A. G., “Ultrafiltry izmerimykh prostranstv kak obobschennye resheniya v abstraktnykh zadachakh o dostizhimosti”, Tr. In-ta matematiki i mekhaniki UrO RAN, 17, no. 1, 2011, 268–293

[14] Chentsov A. G., “Ob odnom predstavlenii rezultatov deistviya priblizhennykh reshenii v zadache s ogranicheniyami asimptoticheskogo kharaktera”, Tr. In-ta matematiki i mekhaniki UrO RAN, 17, no. 2, 2011, 225–239

[15] Chentsov A. G., “Ob odnom primere predstavleniya prostranstva ultrafiltrov algebry mnozhestv”, Tr. In-ta matematiki i mekhaniki, 17, no. 4, 2011, 293–311

[16] Chentsov A. G., “Filtry i ultrafiltry v konstruktsiyakh mnozhestv prityazheniya”, Vestn. Udmurt. un-ta. Matematika. Mekhanika. Kompyuternye nauki, 2011, no. 1, 113–142

[17] Chentsov A. G., “K voprosu o strukture mnozhestv prityazheniya v topologicheskom prostranstve”, Izv. In-ta matematiki i informatiki Udmurt. un-ta, 2012, no. 1(39), 147–150 | MR

[18] Bulinskii A. V., Shiryaev A. N., Teoriya sluchainykh protsessov, Fizmatlit, M., 2005, 402 pp.

[19] Neve Zh., Matematicheskie osnovy teorii veroyatnostei, Mir, M., 1969, 309 pp. | MR | Zbl

[20] Burbaki N., Obschaya topologiya, Nauka, M., 1968, 272 pp. | MR

[21] Kuratovskii K., Mostovskii A., Teoriya mnozhestv, Mir, M., 1970, 416 pp. | MR

[22] Engelking R., Obschaya topologiya, Mir, M., 1986, 751 pp. | MR

[23] Chentsov A. G., “K voprosu o predstavlenii elementov prityazheniya v abstraktnykh zadachakh o dostizhimosti s ogranicheniyami asimptoticheskogo kharaktera”, Izv. vuzov. Matematika, 2012, no. 10, 45–59 | Zbl

[24] Chentsov A. G., Elementy konechno-additivnoi teorii mery, v. I, Izd-vo UGTU-UPI, Ekaterinburg, 2008, 388 pp.