1. Fuzzy Measures and Integrals
2. Belief and Plausibility Measure
3. Possibility and Necessity Measure
4. Sugeno Measure
5. Fuzzy Integrals

2. Fuzzy Measures Fuzzy Set versus Fuzzy Measure Fuzzy Set Fuzzy Measure
Underlying
Set
Vague boundary
Crisp boundary
Vague boundary: Probability of fuzzy set
Representation
Membership value of an element in X
Degree of evidence or
belief of an element that belongs to A in X
Example
Set of large number
A degree of defection of a tree
Degree of Evidence or
Belief of an object that is tree

3. Fuzzy Measure
Axiomatic Definition of Fuzzy Measure
Note:

4. Belief and Plausibility Measure
Belief Measure
Note:
Interpretation:
Degree of evidence or certainty factor of an element in X that belongs to the crisp set A, a particular question. Some of answers are correct, but we don’t know because of the lack of evidence.

5. Belief and Plausibility Measure
Properties of Belief Measure
Vacuous Belief: (Total Ignorance, No Evidence)

6. Belief and Plausibility Measure
Other Definition
Properties of Plausibility Measure

7. How to calculate Belief
Basic Probability Assignment (BPA)
Note

8. How to calculate Belief
Calculation of Bel and Pl
Simple Support Function is a BPA such that
Bel from such Simple Support Function

9. How to calculate Belief
Bel from total ignorance
Body of Evidence

10. How to calculate Belief
Dempster’s rule to combine two bodies of evidence
Example: Homogeneous Evidence A X

11. How to calculate Belief
Example: Heterogeneous Evidence X B A X

12. How to calculate Belief
Example: Heterogeneous Evidence

13. Probability Measure
Theorem: The followings are equivalent.

14. Joint and Marginal BoE
Marginal BPA
Example 7.2

15. Possibility and Necessity Measure
Consonant Bel and Pl Measure

16. Possibility and Necessity Measure
Necessity and Possibility Measure
Consonant Body of Evidence
Belief Measure -> Necessity Measure
Plausibility Measure -> Possibility Measure
Extreme case of fuzzy measure
Note:

17. Possibility and Necessity Measure
Possibility Distribution

18. Possibility and Necessity Measure
Basic Distribution and Possibility Distribution
Ex.

19. Fuzzy Set and Possibility
Interpretation
Degree of Compatibility of v with the concept F
Degree of Possibility when V=v of the proposition p: V is F
Possibility Measure
Example

20. Summary Fuzzy Measure Plausibility Measure Probability Measure
Possibility Measure
Belief Measure
Necessity Measure

21. Sugeno Fuzzy Measure
Sugeno’s g-lamda measure
Note:

22. Sugeno Fuzzy Measure
Fuzzy Density Function

23. Sugeno Fuzzy Measure
How to construct Sugeno measure from fuzzy density

24. Fuzzy Integral
Sugeno Integral

25. Fuzzy Integral
Algorithm of Sugeno Integral

26. Fuzzy Integral Choquet Integral
Interpretation of Fuzzy Integrals in Multi-criteria Decision Making