Advances in Topology and Their Interdisciplinary Applications by Santanu Acharjee

Author:Santanu Acharjee
Language: eng
Format: epub
ISBN: 9789819901517
Publisher: Springer Nature Singapore

6.5 Representation Theorems with Preutility Functions

We may associate a preutility function to any nonnegative asymmetric function in the following manner. Let W be a (continuous) asymmetric function representing , then

is a (continuous) preutility function representing .

Indeed, if then , so . Conversely, if , then is impossible, otherwise by asymmetry and then . So, and finally .

For instance, let be represented by a (continuous) function . Then, for all ,

where for . So, is a (continuous) flow, and .

We may reformulate Theorem 1 with preutility functions. The next corollary is an extension of Shafer’s result11 Shafer [23] to metric spaces instead of .

Corollary 1

(Theorem 1 p. 915 in [23]) Assume (X, d) is a metric space. Let be a closed total binary relation. Then, there exists a continuous preutility function , such that for all ,

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.