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

Tiexin Guo, Xiaohuan Mu, Qiang Tu

Abstract

First, we prove that a random metric space can be isometrically embedded into a complete random normed module, as an application of which, it is easy to see that the notion of $d$-$σ$-stability introduced for a nonempty subset of a random metric space can be regarded as a special case of the notion of $σ$-stability introduced for a nonempty subset of 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 $L^{\infty}$-module is an $L^{p}$-normed $L^{\infty}$-module iff it is generated by a complete random normed module, from which it is easily seen that the gluing property of an $L^{p}$-normed $L^{\infty}$-module can be derived from the $σ$-stability of the generating random normed module, as applications the known and new basic facts of module duals for $L^{p}$-normed $L^{\infty}$-modules can be obtained, in a simple and direct way, from the theory of random conjugate spaces of random normed modules. Third, we prove that a random normed space is order complete iff it is complete with respect to the $(\varepsilon,λ)$-topology, 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 $X\times B$ of a nonempty set $X$ and a complete Boolean algebra $B$ is regular iff it can be induced by a $B$-valued Boolean metric $d$ 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.

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 of which, it is easy to see that the notion of --stability introduced for a nonempty subset of a random metric space can be regarded as a special case of the notion of -stability introduced for a nonempty subset of a random normed module, as another application we give the final version of the characterization for a --stable random metric space to be stably compact. Second, we prove that an -module is an -normed -module iff it is generated by a complete random normed module, from which it is easily seen that the gluing property of an -normed -module can be derived from the -stability of the generating random normed module, as applications the known and new basic facts of module duals for -normed -modules can be obtained, in a simple and direct way, from the theory of random conjugate spaces of random normed modules. Third, we prove that a random normed space is order complete iff it is complete with respect to the -topology, as an application it is proved that the -decomposability of an order complete random normed space is exactly its --stability. Finally, we prove that an equivalence relation on the product space of a nonempty set and a complete Boolean algebra is regular iff it can be induced by a -valued Boolean metric on , as an application it is proved that a nonempty subset of a Boolean set is universally complete iff it is a -stable set defined by a regular equivalence relation.
Paper Structure (8 sections, 22 theorems, 4 equations)

This paper contains 8 sections, 22 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. On $\sigma$-stability and $d$-$\sigma$-stability in random functional analysis
  3. On the gluing property and $\sigma$-stability
  4. On $d$-decomposability and $d$-$\sigma$-stability in order complete random normed spaces
  5. On universal completeness and $B$-stability
  6. On $\sigma$-stability and the gluing property
  7. On $d$-decomposability and $d$-$\sigma$-stability in order complete $RN$ spaces
  8. On universal completeness and $B$-stability

Key Result

Proposition 1.1

$\bar{L}^0(\mathcal{F})$ is a complete lattice under the partial order $\leq$: $\xi \leq \eta$ iff $\xi^0(\omega) \leq \eta^0(\omega)$ for almost all $\omega$ in $\Omega$ (briefly, a.e.), where $\xi^0$ and $\eta^0$ are arbitrarily chosen representatives, respectively. We always denote by $\bigvee A$

Theorems & Definitions (69)

  • Proposition 1.1
  • Remark 1.2
  • Theorem 1.3
  • proof
  • Remark 1.4
  • Remark 1.5
  • Proposition 1.6
  • Definition 1.7
  • Example 1.8
  • Example 1.9
  • ...and 59 more