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

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 - σ -stability in a random metric space can be regarded as a special case of the notion of σ -stability in a random normed module; as another application we give the final version of the characterization for a d - σ -stable random metric space to be stably compact. Second, we prove that an L p -normed L ∞ -module is exactly generated by a complete random normed module so that the gluing property of an L p -normed L ∞ -module can be derived from the σ -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 ( ε , λ ) -complete, as an application it is proved that the d -decomposability of an order complete random normed space is exactly its d - σ -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.