The $L^0$-extension of an $L^\infty$-normed module

TL;DR

This paper introduces a method to embed any $L^ Infty$-normed module into a unique complete random normed module, establishing a close relationship between their properties.

Contribution

It provides a novel embedding technique that connects $L^ Infty$-normed modules with complete random normed modules, enhancing understanding of their structure.

Findings

Established a unique embedding of $L^ Infty$-normed modules into complete random normed modules

Linked properties of $E$ with those of $E_0$

Provided a framework for analyzing $L^ Infty$-normed modules via their embeddings

Abstract

In this paper, we embed each $L^\infty$-normed module $E$ into an appropriate and unique complete random normed module $E_0$ so that the properties of $E$ are closely related to the properties of $E_0$.