Abstract
First, we prove that a random metric space can be isometrically embedded into a complete random normed module, as an application it is easy to see that the notion of d-\(\sigma \)-stability in a random metric space can be regarded as a special case of the notion of \(\sigma \)-stability in a random normed module; as another application we give the final version of the characterization for a d-\(\sigma \)-stable random metric space to be stably compact. Second, we prove that an \(L^{p}\)-normed \(L^{\infty }\)-module is exactly generated by a complete random normed module so that the gluing property of an \(L^{p}\)-normed \(L^{\infty }\)-module can be derived from the \(\sigma \)-stability of the generating random normed module, as applications the direct relation between module duals and random conjugate spaces are given. Third, we prove that a random normed space is order complete iff it is \((\varepsilon ,\lambda )\)-complete, as an application it is proved that the d-decomposability of an order complete random normed space is exactly its d-\(\sigma \)-stability. Finally, we prove that an equivalence relation on the product space of a nonempty set X and a complete Boolean algebra B is regular iff it can be induced by a B-valued Boolean metric on X, as an application it is proved that a nonempty subset of a Boolean set (X, d) is universally complete iff it is a B-stable set defined by a regular equivalence relation.
Similar content being viewed by others
Data availability
Not available.
References
Aliprantis, C.D., Forder, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer, Berlin (2006)
Diestel, J., Uhl, J.J., Jr.: Vector measures. (AMS) Math. Surv. 15, 1 (1977)
Drapeau, S., Jamneshan, A., Karliczek, M., Kupper, M.: The algebra of conditional sets and the concepts of conditional topology and compactness. J. Math. Anal. Appl. 437, 561–589 (2016)
Dunford, N., Schwartz, J.T.: Linear Operators (I): General Theory. Wilely, New York (1958)
Filipović, D., Kupper, M., Vogelpoth, V.: Separation and duality in locally \(L^{0}\)-convex modules. J. Funct. Anal. 12, 3996–4029 (2009)
Gigli, N.: Lecture notes on differential calculus on RCD spaces. Publ. RIMS Kyoto Univ. 54, 855–918 (2018)
Gigli, N.: Nonsmooth differential geometry-an approach tailed for spaces with Ricci curvature bounded from below. Mem. Am. Math. Soc. 251(1196), 1–159 (2018)
Gigli, N., Lučić, D., Pasqualetto, E.: Duals and pullbacks of normed modules. Isr. J. Math. arXiv:2207.04972 (2022) (to appear)
Guo, T.X.: The theory of probabilistic metric spaces with applications to random functional analysis. Master’s thesis, Xi’an Jiaotong University, China (1989)
Guo, T.X.: Random metric theory and its applications. Ph.D thesis, Xi’an Jiaotong University, China (1992)
Guo, T.X.: A new approach to probabilistic functional analysis. In: Proceedings of the First China Postdoctoral Academic Conference. The China National Defense and Industry Press, Beijing, pp. 1150–1154 (1993) (in Chinese)
Guo, T.X.: The Radon–Nikodým property of conjugate spaces and the \(w^{*}\)-equivalence theorem for \(w^{*}\)-measurable functions. Sci. China Ser. A 39, 1034–1041 (1996)
Guo, T.X.: Some basic theories of random normed linear spaces and random inner product spaces. Acta Anal. Funct. Appl. 1, 160–184 (1999)
Guo, T.X.: Relations between some basic results derived from two kinds of topologies for a random locally convex module. J. Funct. Anal. 258(9), 3024–3047 (2010)
Guo, T.X.: The relation of Banach–Alaoglu theorem and Banach–Bourbaki–Kakutani–Šmulian theorem in complete random normed modules to stratification structure. Sci. China Math. Ser. A 51, 1651–1663 (2008)
Guo, T.X.: Representation theorems of the dual of Lebesgue–Bochner function spaces. Sci. China Ser. A 43(3), 234–243 (2000)
Guo, T.X.: On some basic theorems of continuous module homomorphisms between random normed modules. J. Funct. Spaces 2013, 989102 (2013)
Guo, T.X.: Extension theorems of continuous random linear operators on random domains. J. Math. Anal. Appl. 193, 15–27 (1995)
Guo, T.X., Li, S.B.: The James theorem in complete random normed modules. J. Math. Anal. Appl. 308, 257–265 (2005)
Guo, T.X., Shi, G.: The algebraic structure of finitely generated \(L^{0}(\cal{F},\mathbb{K} )\)-modules and the Helly theorem in random normed modules. J. Math. Anal. Appl. 381, 833–842 (2011)
Guo, T.X., Wang, Y.C., Chen, G., Xu, H.K., Yuan, G.: The noncompact Schauder fixed point theorem in random normed modules and its applications. arXiv:2014.11095 (2023)
Guo, T.X., Wang, Y.C., Tang, Y.: The Krein–Milman theorem in random locally convex modules and its applications. Sci. Sin. Math. 53(12), 1–18 (2023) (in Chinese)
Guo, T.X., Wang, Y.C., Yang, B.X., Zhang, E.X.: On \(d\)-\(\sigma \)-stability in random metric spaces and its applications. J. Nonlinear Convex Anal. 21(6), 1297–1316 (2020)
Guo, T.X., You, Z.Y.: The Riesz’s representation theorem in complete random inner product modules and its applications. Chin. Ann. Math. Ser. A 17, 361–364 (1996)
Guo, T.X., Zeng, X.L.: Random strict convexity and random uniform convexity in random normed modules. Nonlinear Anal. 73, 1239–1263 (2010)
Guo, T.X., Zhang, E.X., Wang, Y.C., Guo, Z.C.: Two fixed point theorems in complete random normed modules and their applications to backward stochastic equations. J. Math. Anal. Appl. 486(2), 123644 (2020)
Guo, T.X., Zhang, E.X., Wu, M.Z., Yang, B.X., Yuan, G., Zeng, X.L.: On random convex analysis. J. Nonlinear Convex Anal. 18, 1967–1996 (2017)
Guo, T.X., Zhao, S.E., Zeng, X.L.: Random convex analysis (I): separation and Fenchel–Moreau duality in rando locally convex modules. Sci. Sin. Math. 45, 1961–1980 (2015) (in Chinese)
Guo, T.X., Zhao, S.E., Zeng, X.L.: The relations among the three kinds of conditional risk measures. Sci. China Math. 57, 1753–1764 (2014)
Haydon, R., Levy, M., Raynaud, Y.: Randomly Normed Spaces. Hermann, Paris (1991)
Ionescu Tulcea, A., Ionescu Tulcea, C.: On the lifting property (I). J. Math. Anal. Appl. 3, 537–546 (1961)
James, R.C.: Characterizations of reflexivity. Stud. Math. 23, 205–216 (1964)
Jamneshan, A., Kupper, M., Zapata, J.M.: Parameter-dependent stochastic optimal control in finite discrete time. J. Optim. Theory Appl. 186, 644–666 (2020)
Jamneshan, A., Zapata, J.M.: On compactness in \(L^0\)–modules. arXiv:1711.09785v1 (2017)
Kabanov, Y., Stricker, C.: A teacher’s note on no-arbitrage criteria, Séminaire de Probabilités, XXXV, Lect. Notes Math. vol. 1755, pp. 149–152. Springer, Berlin (2001)
Kantorovich, L.V.: The method of successive approximation for functional equations. Acta Math. 71, 63–97 (1939)
Kantorovich, L.V., Vulikh, B.Z., Pinsker, G.A.: Functional Analysis in Semiordered Spaces. Gostekhizdat, Moscow (1950) (in Russian)
Kusraev, A.G.: On Banach–Kantorovich spaces. Sibirsk. Mat. Zh. 26(2), 119–126 (1985)
Kusraev, A.G., Kutateladze, S.S.: Boolean Valued Analysis. Mathematics and Its Applications. Springer Netherlands, Amsterdam (1999)
Lučić, D., Pasqualetto, E.: The Serre–Swan theorem for normed modules. Rend. Circ. Mat. Palermo 68(2), 385–404 (2019)
Lučić, M., Pasqualetto, E., Vojnović, I.: On the reflexivity properties of Banach bundles and Banach modules. Banach J. Math. Anal. 18, 7 (2024)
Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. Elsevier/North-Holland, New York (1983); reissued by Dover Publications, New York (2005)
Wagner, D.H.: Survey of measurable selection theorems. SIAM J. Control Optim. 15, 859–903 (1977)
Weaver, N.: Lipschitz Algebras. World Scientific Publishing Co. Inc, River Edge (1999)
Wu, M.Z., Guo, T.X.: A counterexample shows that not every locally \(L^{0}\)-convex topology is necessarily induced by a family of \(L^0\)-seminorms. arXiv:1501.04400v1 (2015)
Wu, M.Z., Guo, T.X., Long, L.: The fundamental theorem of affine geometry in regular \(L^{0}\)-modules. J. Math. Anal. Appl. 507(2), 125827 (2022)
Zapata, J.M.: On the characterization of locally \(L^0\)-convex topologies induced by a family of \(L^0\)-seminorms. J. Convex Anal. 24(2), 383–391 (2017)
Acknowledgements
This research is supported by the National Natural Science Foundation of China (Grant Nos. 12371141,11971483) and the Provincial Natural Science Foundation of Hunan (Grant No. 2023JJ30642). We would like to thank the reviewers for presenting lots of valuable suggestions which considerably improve the readability of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Antonio M. Peralta.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Guo, T., Mu, X. & Tu, Q. The relations among the notions of various kinds of stability and their applications. Banach J. Math. Anal. 18, 42 (2024). https://doi.org/10.1007/s43037-024-00354-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s43037-024-00354-w
Keywords
- \(\sigma \)-Stability
- d-\(\sigma \)-Stability
- Gluing property
- d-Decomposability
- Universal completeness
- B-stability