The relations among the notions of various kinds of stability and their applications

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.