Preface xv

Acknowledgements xvii

**1 Preliminary notions and definitions 1**

1.1 Sets of numbers 1

1.2 Gambles 2

1.3 Subsets and their indicators 5

1.4 Collections of events 5

1.5 Directed sets and Moore–Smith limits 7

1.6 Uniform convergence of bounded gambles 9

1.7 Set functions, charges and measures 10

1.8 Measurability and simple gambles 12

1.9 Real functionals 17

1.10 A useful lemma 19

**PART I LOWER PREVISIONS ON BOUNDED GAMBLES 21**

**2 Introduction 23**

**3 Sets of acceptable bounded gambles 25**

3.1 Random variables 26

3.2 Belief and behaviour 27

3.3 Bounded gambles 28

3.4 Sets of acceptable bounded gambles 29

3.4.1 Rationality criteria 29

3.4.2 Inference 32

**4 Lower previsions** **37**

4.1 Lower and upper previsions 38

4.1.1 From sets of acceptable bounded gambles to lower previsions 38

4.1.2 Lower and upper previsions directly 40

4.2 Consistency for lower previsions 41

4.2.1 Definition and justification 41

4.2.2 A more direct justification for the avoiding sure loss condition 44

4.2.3 Avoiding sure loss and avoiding partial loss 45

4.2.4 Illustrating the avoiding sure loss condition 45

4.2.5 Consequences of avoiding sure loss 46

4.3 Coherence for lower previsions 46

4.3.1 Definition and justification 46

4.3.2 A more direct justification for the coherence condition 50

4.3.3 Illustrating the coherence condition 51

4.3.4 Linear previsions 51

4.4 Properties of coherent lower previsions 53

4.4.1 Interesting consequences of coherence 53

4.4.2 Coherence and conjugacy 56

4.4.3 Easier ways to prove coherence 56

4.4.4 Coherence and monotone convergence 63

4.4.5 Coherence and a seminorm 64

4.5 The natural extension of a lower prevision 65

4.5.1 Natural extension as least-committal extension 65

4.5.2 Natural extension and equivalence 66

4.5.3 Natural extension to a specific domain 66

4.5.4 Transitivity of natural extension 67

4.5.5 Natural extension and avoiding sure loss 67

4.5.6 Simpler ways of calculating the natural extension 69

4.6 Alternative characterisations for avoiding sure loss, coherence, and natural extension 70

4.7 Topological considerations 74

**5 Special coherent lower previsions 76**

5.1 Linear previsions on finite spaces 77

5.2 Coherent lower previsions on finite spaces 78

5.3 Limits as linear previsions 80

5.4 Vacuous lower previsions 81

5.5 {0, 1}-valued lower probabilities 82

5.5.1 Coherence and natural extension 82

5.5.2 The link with classical propositional logic 88

5.5.3 The link with limits inferior 90

5.5.4 Monotone convergence 91

5.5.5 Lower oscillations and neighbourhood filters 93

5.5.6 Extending a lower prevision defined on all continuous bounded gambles 98

**6 n-Monotone lower previsions 101**

6.1 n-Monotonicity 102

6.2 n-Monotonicity and coherence 107

6.2.1 A few observations 107

6.2.2 Results for lower probabilities 109

6.3 Representation results 113

**7 Special n-monotone coherent lower previsions 122**

7.1 Lower and upper mass functions 123

7.2 Minimum preserving lower previsions 127

7.2.1 Definition and properties 127

7.2.2 Vacuous lower previsions 128

7.3 Belief functions 128

7.4 Lower previsions associated with proper filters 129

7.5 Induced lower previsions 131

7.5.1 Motivation 131

7.5.2 Induced lower previsions 133

7.5.3 Properties of induced lower previsions 134

7.6 Special cases of induced lower previsions 138

7.6.1 Belief functions 139

7.6.2 Refining the set of possible values for a random variable 139

7.7 Assessments on chains of sets 142

7.8 Possibility and necessity measures 143

7.9 Distribution functions and probability boxes 147

7.9.1 Distribution functions 147

7.9.2 Probability boxes 149

**8 Linear previsions, integration and duality 151**

8.1 Linear extension and integration 153

8.2 Integration of probability charges 159

8.3 Inner and outer set function, completion and other extensions 163

8.4 Linear previsions and probability charges 166

8.5 The S-integral 168

8.6 The Lebesgue integral 171

8.7 The Dunford integral 172

8.8 Consequences of duality 177

**9 Examples of linear extension 181**

9.1 Distribution functions 181

9.2 Limits inferior 182

9.3 Lower and upper oscillations 183

9.4 Linear extension of a probability measure 183

9.5 Extending a linear prevision from continuous bounded gambles 187

9.6 Induced lower previsions and random sets 188

**10 Lower previsions and symmetry 191**

10.1 Invariance for lower previsions 192

10.1.1 Definition 192

10.1.2 Existence of invariant lower previsions 194

10.1.3 Existence of strongly invariant lower previsions 195

10.2 An important special case 200

10.3 Interesting examples 205

10.3.1 Permutation invariance on finite spaces 205

10.3.2 Shift invariance and Banach limits 208

10.3.3 Stationary random processes 210

**11 Extreme lower previsions 214**

11.1 Preliminary results concerning real functionals 215

11.2 Inequality preserving functionals 217

11.2.1 Definition 217

11.2.2 Linear functionals 217

11.2.3 Monotone functionals 218

11.2.4 n-Monotone functionals 218

11.2.5 Coherent lower previsions 219

11.2.6 Combinations 220

11.3 Properties of inequality preserving functionals 220

11.4 Infinite non-negative linear combinations of inequality preserving functionals 221

11.4.1 Definition 221

11.4.2 Examples 222

11.4.3 Main result 223

11.5 Representation results 224

11.6 Lower previsions associated with proper filters 225

11.6.1 Belief functions 225

11.6.2 Possibility measures 226

11.6.3 Extending a linear prevision defined on all continuous bounded gambles 226

11.6.4 The connection with induced lower previsions 227

11.7 Strongly invariant coherent lower previsions 228

**PART II EXTENDING THE THEORY TO UNBOUNDED GAMBLES 231**

**12 Introduction 233**

**13 Conditional lower previsions 235**

13.1 Gambles 236

13.2 Sets of acceptable gambles 236

13.2.1 Rationality criteria 236

13.2.2 Inference 238

13.3 Conditional lower previsions 240

13.3.1 Going from sets of acceptable gambles to conditional lower previsions 240

13.3.2 Conditional lower previsions directly 252

13.4 Consistency for conditional lower previsions 254

13.4.1 Definition and justification 254

13.4.2 Avoiding sure loss and avoiding partial loss 257

13.4.3 Compatibility with the definition for lower previsions on bounded gambles 258

13.4.4 Comparison with avoiding sure loss for lower previsions on bounded gambles 258

13.5 Coherence for conditional lower previsions 259

13.5.1 Definition and justification 259

13.5.2 Compatibility with the definition for lower previsions on bounded gambles 264

13.5.3 Comparison with coherence for lower previsions on bounded gambles 264

13.5.4 Linear previsions 264

13.6 Properties of coherent conditional lower previsions 266

13.6.1 Interesting consequences of coherence 266

13.6.2 Trivial extension 269

13.6.3 Easier ways to prove coherence 270

13.6.4 Separate coherence 278

13.7 The natural extension of a conditional lower prevision 279

13.7.1 Natural extension as least-committal extension 280

13.7.2 Natural extension and equivalence 281

13.7.3 Natural extension to a specific domain and the transitivity of natural extension 282

13.7.4 Natural extension and avoiding sure loss 283

13.7.5 Simpler ways of calculating the natural extension 285

13.7.6 Compatibility with the definition for lower previsions on bounded gambles 286

13.8 Alternative characterisations for avoiding sure loss, coherence and natural extension 287

13.9 Marginal extension 288

13.10 Extending a lower prevision from bounded gambles to conditional gambles 295

13.10.1 General case 295

13.10.2 Linear previsions and probability charges 297

13.10.3 Vacuous lower previsions 298

13.10.4 Lower previsions associated with proper filters 300

13.10.5 Limits inferior 300

13.11 The need for infinity? 301

**14 Lower previsions for essentially bounded gambles 304**

14.1 Null sets and null gambles 305

14.2 Null bounded gambles 310

14.3 Essentially bounded gambles 311

14.4 Extension of lower and upper previsions to essentially bounded gambles 316

14.5 Examples 322

14.5.1 Linear previsions and probability charges 322

14.5.2 Vacuous lower previsions 323

14.5.3 Lower previsions associated with proper filters 323

14.5.4 Limits inferior 324

14.5.5 Belief functions 325

14.5.6 Possibility measures 325

**15 Lower previsions for previsible gambles 327**

15.1 Convergence in probability 328

15.2 Previsibility 331

15.3 Measurability 340

15.4 Lebesgue’s dominated convergence theorem 343

15.5 Previsibility by cuts 348

15.6 A sufficient condition for previsibility 350

15.7 Previsibility for 2-monotone lower previsions 352

15.8 Convex combinations 355

15.9 Lower envelope theorem 355

15.10 Examples 358

15.10.1 Linear previsions and probability charges 358

15.10.2 Probability density functions: The normal density 359

15.10.3 Vacuous lower previsions 360

15.10.4 Lower previsions associated with proper filters 361

15.10.5 Limits inferior 361

15.10.6 Belief functions 362

15.10.7 Possibility measures 362

15.10.8 Estimation 365

**Appendix A Linear spaces, linear lattices and convexity 368**

**Appendix B Notions and results from topology 371**

B.1 Basic definitions 371

B.2 Metric spaces 372

B.3 Continuity 373

B.4 Topological linear spaces 374

B.5 Extreme points 374

**Appendix C The Choquet integral 376**

C.1 Preliminaries 376

C.1.1 The improper Riemann integral of a non-increasing function 376

C.1.2 Comonotonicity 378

C.2 Definition of the Choquet integral 378

C.3 Basic properties of the Choquet integral 379

C.4 A simple but useful equality 387

C.5 A simplified version of Greco’s representation theorem 389

**Appendix D The extended real calculus 391**

D.1 Definitions 391

D.2 Properties 392

**Appendix E Symbols and notation 396**

References 398

Index 407