| Preface | p. ix |
| Limit Theorems of Set-Valued and Fuzzy Set-Valued Random Variables | |
| The Space of Set-Valued Random Variables | p. 1 |
| Hyperspaces of a Banach Space | p. 1 |
| The Hausdorff Metric in Hyperspaces and An Embedding Theorem | p. 1 |
| Convergences in Hyperspaces | p. 12 |
| Set-Valued Random Variables | p. 20 |
| The Set of Integrable Selections | p. 26 |
| The Spaces of Integrably Bounded Set-Valued Random Variables | p. 34 |
| The Aumann Integral and the Conditional Expectation of a Set-Valued Random Variable | p. 41 |
| The Aumann Integral and Its Properties | p. 41 |
| Sufficient Conditions for the Aumann Integrals To Be Closed | p. 47 |
| Conditional Expectation and Its Properties | p. 54 |
| Fatou's Lemmas and Lebesgue's Dominated Convergence Theorems | p. 67 |
| Radon-Nikodym Theorems for Set-Valued Measures | p. 73 |
| Set-Valued Measures | p. 74 |
| Radon-Nikodym Theorems for Set-Valued Measures | p. 81 |
| Strong Laws of Large Numbers and Central Limit Theorems for Set-Valued Random Variables | p. 87 |
| Limit Theorems for Set-Valued Random Variables in the Hausdorff Metric | p. 87 |
| Strong Laws of Large Numbers in the Hausdorff Metric | p. 87 |
| Central Limit Theorems | p. 96 |
| Strong Laws of Large Numbers for Set-Valued Random Variables in the Kuratowski-Mosco Sense | p. 100 |
| Gaussian Set-Valued Random Variables | p. 105 |
| Appendix A of Subsection 3.1.2 | p. 108 |
| Convergence Theorems for Set-Valued Martingales | p. 117 |
| Set-Valued Martingales | p. 117 |
| Representation Theorems for Closed Convex Set-Valued Martingales | p. 126 |
| Convergence of Closed Convex Set-Valued Martingales in the Kuratowski-Mosco Sense | p. 134 |
| Convergence of Closed Convex Set-Valued Submartingales and Supermartingales in the Kuratowski-Mosco Sense | p. 138 |
| Convergence of Closed Convex Set-Valued Submartingales in the Kuratowski-Mosco Sense | p. 138 |
| Convergence of Closed Convex Set-Valued Supermartingales in the Kuratowski-Mosco Sense | p. 142 |
| Convergence of Closed Convex Set-Valued Supermartingales (Martingales) Whose Values May Be Unbounded | p. 143 |
| Optional Sampling Theorems for Closed Convex Set-Valued Martingales | p. 150 |
| Doob Decomposition of Set-Valued Submartingales | p. 155 |
| Fuzzy Set-Valued Random Variables | p. 161 |
| Fuzzy Sets | p. 162 |
| The Space of Fuzzy Set-Valued Random Variables | p. 171 |
| Expectations of Fuzzy Set-Valued Random Variables | p. 181 |
| Conditional Expectations of Fuzzy Random Sets | p. 184 |
| The Radon-Nikodym Theorem for Fuzzy Set-Valued Measures | p. 187 |
| Convergence Theorems for Fuzzy Set-Valued Random Variables | p. 191 |
| Embedding Theorems and Gaussian Fuzzy Random Sets | p. 191 |
| Embedding Theorems | p. 191 |
| Gaussian Fuzzy Set-Valued Random Variables | p. 195 |
| Strong Laws of Large Numbers for Fuzzy Set-Valued Random Variables | p. 197 |
| Central Limit Theorems for Fuzzy Set-Valued Random Variables | p. 205 |
| Fuzzy Set-Valued Martingales | p. 214 |
| Convergences in the Graphical Sense for Fuzzy Set-Valued Random Variables | p. 221 |
| Convergences in the Graphical Sense for Fuzzy Sets | p. 221 |
| Separability for the Graphical Convergences and Applications to Strong Laws of Large Numbers | p. 226 |
| Convergence in the Graphical Sense for Fuzzy Set-Valued Martingales and Smartingales | p. 231 |
| References for Part I | p. 235 |
| Practical Applications of Set-Valued Random Variables | |
| Mathematical Foundations for the Applications of Set-Valued Random Variables | p. 253 |
| How Can Limit Theorems Be Applied? | p. 253 |
| Relevant Optimization Techniques | p. 257 |
| Introduction: Optimization of Set Functions Is a Practically Important but Difficult Problem | p. 257 |
| The Existing Methods of Optimizing Set Functions: Their Successes (In Brief) and the Territorial Division Problem as a Challenge | p. 261 |
| A Differential Formalism for Set Functions | p. 264 |
| First Application of the New Formalism: Territorial Division Problem | p. 274 |
| Second Application of the New Formalism: Statistical Example--Excess Mass Method | p. 282 |
| Further Directions, Related Results, and Open Problems | p. 284 |
| Optimization Under Uncertainty and Related Symmetry Techniques | p. 286 |
| Case Study: Selecting Zones in a Plane | p. 286 |
| General Case | p. 290 |
| Applications to Imaging | p. 295 |
| Applications to Astronomy | p. 295 |
| Applications to Agriculture | p. 306 |
| Detecting Trash in Ginned Cotton | p. 306 |
| Classification of Insects in the Cotton Field | p. 313 |
| Applications to Medicine | p. 322 |
| Towards Foundations for Traditional Oriental Medicine | p. 322 |
| Towards Optimal Pain Relief: Acupuncture and Spinal Cord Stimulation | p. 325 |
| Applications to Mechanical Fractures | p. 336 |
| Fault Shapes | p. 336 |
| Best Sensor Locations for Detecting Shapes | p. 337 |
| What Segments are the Best in Representing Contours? | p. 342 |
| Searching For a 'Typical' Image | p. 345 |
| Average Set | p. 345 |
| Average Shape | p. 351 |
| Applications to Data Processing | p. 355 |
| 1-D Case: Why Intervals? A Simple Limit Theorem | p. 355 |
| 2-D Case: Candidate Sets for Complex Interval Arithmetic | p. 360 |
| Multi-D Case: Why Ellipsoids? | p. 362 |
| Conclusions | p. 372 |
| References for Part II | p. 373 |
| Index | p. 387 |
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