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**Fuzzy Measures and Integrals**

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

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**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

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Fuzzy Measure Axiomatic Definition of Fuzzy Measure Note:

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**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.

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**Belief and Plausibility Measure**

Properties of Belief Measure Vacuous Belief: (Total Ignorance, No Evidence)

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**Belief and Plausibility Measure**

Other Definition Properties of Plausibility Measure

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**How to calculate Belief**

Basic Probability Assignment (BPA) Note

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**How to calculate Belief**

Calculation of Bel and Pl Simple Support Function is a BPA such that Bel from such Simple Support Function

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**How to calculate Belief**

Bel from total ignorance Body of Evidence

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**How to calculate Belief**

Dempster’s rule to combine two bodies of evidence Example: Homogeneous Evidence A X

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**How to calculate Belief**

Example: Heterogeneous Evidence X B A X

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**How to calculate Belief**

Example: Heterogeneous Evidence

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Probability Measure Theorem: The followings are equivalent.

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Joint and Marginal BoE Marginal BPA Example 7.2

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**Possibility and Necessity Measure**

Consonant Bel and Pl Measure

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**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:

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**Possibility and Necessity Measure**

Possibility Distribution

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**Possibility and Necessity Measure**

Basic Distribution and Possibility Distribution Ex.

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**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

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**Summary Fuzzy Measure Plausibility Measure Probability Measure**

Possibility Measure Belief Measure Necessity Measure

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

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Sugeno Fuzzy Measure Fuzzy Density Function

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Sugeno Fuzzy Measure How to construct Sugeno measure from fuzzy density

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Fuzzy Integral Sugeno Integral

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Fuzzy Integral Algorithm of Sugeno Integral

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**Fuzzy Integral Choquet Integral**

Interpretation of Fuzzy Integrals in Multi-criteria Decision Making

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