# IJPAM: Volume 13, No. 2 (2004)

**COMPLETION OF THE SHAPLEY ENTROPY**

OF FUZZY MEASURES

OF FUZZY MEASURES

Institute of Mathematics

Shaanxi Normal University

Xi'an 710062, P.R. CHINA

e-mail: gjwang@snnu.edu.cn

Shaanxi University of Technology

Hanzhong 723001, P.R. CHINA

e-mail: he_ning0916@sina.com.cn

**Abstract.**A fuzzy measure with complete certainty has the
minimal Shapley entropy and vice versa. This paper points out that
a fuzzy measure with complete uncertainty has the maximal Shapley
entropy but not vice versa. An example shows that fuzzy measures
with maximal Shapley entropy may far from complete uncertainty. A
complementary entropy of the Shapley entropy called the
partitional entropy is proposed which behaves well if used
together with the Shapley entropy. The sum of the two entropies is
called the absolute entropy. It is proved that a fuzzy measure
possesses complete certainty or complete uncertainty if and only
if its absolute entropy attains the minimal value or the maximal
value respectively. Moreover, extension of fuzzy measures is
discussed, the regular extension of fuzzy measures is introduced
which keeps certain basic properties of the original fuzzy
measure unchanged.

**Received: **March 6, 2004

**AMS Subject Classification: **28E10, 28D20

**Key Words and Phrases: **fuzzy measure, Shapley entropy, partitional entropy, regular extension

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2004

**Volume:** 13

**Issue:** 2