{"product_id":"lower-previsions-hardback-9780470723777","title":"Lower Previsions (Hardback) 9780470723777","description":"\u003cfont face=\"Georgia\"\u003e\r\n\u003cp\u003e\u003cfont size=\"6\"\u003eLower Previsions\u003c\/font\u003e\u003cbr\u003e\r\n\r\n\r\n\r\n\r\n\r\n\u003c\/p\u003e\n\u003cp\u003e\u003cfont size=\"4\"\u003eMatthias C. M. Troffaes (Author), Gert de Cooman (Author)\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e9780470723777, Wiley\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003eHardback, published 23 May 2014\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e448 pages\u003cbr\u003e23.6 x 16 x 2.5 cm, 0.694 kg\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\r\n\r\n\r\n\u003cp align=\"justify\"\u003e\u003cstrong\u003e\u003cfont size=\"3\"\u003e\u003cp\u003eThis book has two main purposes. On the one hand, it provides a\u003cbr\u003econcise and systematic development of the theory of lower previsions,\u003cbr\u003ebased on the concept of acceptability, in spirit of the work of\u003cbr\u003eWilliams and Walley. On the other hand, it also extends this theory to\u003cbr\u003edeal with unbounded quantities, which abound in practical\u003cbr\u003eapplications.\u003c\/p\u003e \u003cp\u003eFollowing Williams, we start out with sets of acceptable gambles. From\u003cbr\u003ethose, we derive rationality criteria---avoiding sure loss and\u003cbr\u003ecoherence---and inference methods---natural extension---for\u003cbr\u003e(unconditional) lower previsions. We then proceed to study various\u003cbr\u003easpects of the resulting theory, including the concept of expectation\u003cbr\u003e(linear previsions), limits, vacuous models, classical propositional\u003cbr\u003elogic, lower oscillations, and monotone convergence. We discuss\u003cbr\u003en-monotonicity for lower previsions, and relate lower previsions with\u003cbr\u003eChoquet integration, belief functions, random sets, possibility\u003cbr\u003emeasures, various integrals, symmetry, and representation theorems\u003cbr\u003ebased on the Bishop-De Leeuw theorem.\u003c\/p\u003e \u003cp\u003eNext, we extend the framework of sets of acceptable gambles to consider\u003cbr\u003ealso unbounded quantities. As before, we again derive rationality\u003cbr\u003ecriteria and inference methods for lower previsions, this time also\u003cbr\u003eallowing for conditioning. We apply this theory to construct\u003cbr\u003eextensions of lower previsions from bounded random quantities to a\u003cbr\u003elarger set of random quantities, based on ideas borrowed from the\u003cbr\u003etheory of Dunford integration.\u003c\/p\u003e \u003cp\u003eA first step is to extend a lower prevision to random quantities that\u003cbr\u003eare bounded on the complement of a null set (essentially bounded\u003cbr\u003erandom quantities). This extension is achieved by a natural extension\u003cbr\u003eprocedure that can be motivated by a rationality axiom stating that\u003cbr\u003eadding null random quantities does not affect acceptability.\u003c\/p\u003e \u003cp\u003eIn a further step, we approximate unbounded random quantities by a\u003cbr\u003esequences of bounded ones, and, in essence, we identify those for\u003cbr\u003ewhich the induced lower prevision limit does not depend on the details\u003cbr\u003eof the approximation. We call those random quantities 'previsible'. We\u003cbr\u003estudy previsibility by cut sequences, and arrive at a simple\u003cbr\u003esufficient condition. For the 2-monotone case, we establish a Choquet\u003cbr\u003eintegral representation for the extension. For the general case, we\u003cbr\u003eprove that the extension can always be written as an envelope of\u003cbr\u003eDunford integrals. We end with some examples of the theory.\u003c\/p\u003e\u003c\/font\u003e\u003c\/strong\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003ePreface xv  \u003cp\u003eAcknowledgements xvii\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Preliminary notions and definitions 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Sets of numbers 1\u003c\/p\u003e \u003cp\u003e1.2 Gambles 2\u003c\/p\u003e \u003cp\u003e1.3 Subsets and their indicators 5\u003c\/p\u003e \u003cp\u003e1.4 Collections of events 5\u003c\/p\u003e \u003cp\u003e1.5 Directed sets and Moore–Smith limits 7\u003c\/p\u003e \u003cp\u003e1.6 Uniform convergence of bounded gambles 9\u003c\/p\u003e \u003cp\u003e1.7 Set functions, charges and measures 10\u003c\/p\u003e \u003cp\u003e1.8 Measurability and simple gambles 12\u003c\/p\u003e \u003cp\u003e1.9 Real functionals 17\u003c\/p\u003e \u003cp\u003e1.10 A useful lemma 19\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePART I LOWER PREVISIONS ON BOUNDED GAMBLES 21\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Introduction 23\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Sets of acceptable bounded gambles 25\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Random variables 26\u003c\/p\u003e \u003cp\u003e3.2 Belief and behaviour 27\u003c\/p\u003e \u003cp\u003e3.3 Bounded gambles 28\u003c\/p\u003e \u003cp\u003e3.4 Sets of acceptable bounded gambles 29\u003c\/p\u003e \u003cp\u003e3.4.1 Rationality criteria 29\u003c\/p\u003e \u003cp\u003e3.4.2 Inference 32\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Lower previsions\u003c\/b\u003e \u003cb\u003e37\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Lower and upper previsions 38\u003c\/p\u003e \u003cp\u003e4.1.1 From sets of acceptable bounded gambles to lower previsions 38\u003c\/p\u003e \u003cp\u003e4.1.2 Lower and upper previsions directly 40\u003c\/p\u003e \u003cp\u003e4.2 Consistency for lower previsions 41\u003c\/p\u003e \u003cp\u003e4.2.1 Definition and justification 41\u003c\/p\u003e \u003cp\u003e4.2.2 A more direct justification for the avoiding sure loss condition 44\u003c\/p\u003e \u003cp\u003e4.2.3 Avoiding sure loss and avoiding partial loss 45\u003c\/p\u003e \u003cp\u003e4.2.4 Illustrating the avoiding sure loss condition 45\u003c\/p\u003e \u003cp\u003e4.2.5 Consequences of avoiding sure loss 46\u003c\/p\u003e \u003cp\u003e4.3 Coherence for lower previsions 46\u003c\/p\u003e \u003cp\u003e4.3.1 Definition and justification 46\u003c\/p\u003e \u003cp\u003e4.3.2 A more direct justification for the coherence condition 50\u003c\/p\u003e \u003cp\u003e4.3.3 Illustrating the coherence condition 51\u003c\/p\u003e \u003cp\u003e4.3.4 Linear previsions 51\u003c\/p\u003e \u003cp\u003e4.4 Properties of coherent lower previsions 53\u003c\/p\u003e \u003cp\u003e4.4.1 Interesting consequences of coherence 53\u003c\/p\u003e \u003cp\u003e4.4.2 Coherence and conjugacy 56\u003c\/p\u003e \u003cp\u003e4.4.3 Easier ways to prove coherence 56\u003c\/p\u003e \u003cp\u003e4.4.4 Coherence and monotone convergence 63\u003c\/p\u003e \u003cp\u003e4.4.5 Coherence and a seminorm 64\u003c\/p\u003e \u003cp\u003e4.5 The natural extension of a lower prevision 65\u003c\/p\u003e \u003cp\u003e4.5.1 Natural extension as least-committal extension 65\u003c\/p\u003e \u003cp\u003e4.5.2 Natural extension and equivalence 66\u003c\/p\u003e \u003cp\u003e4.5.3 Natural extension to a specific domain 66\u003c\/p\u003e \u003cp\u003e4.5.4 Transitivity of natural extension 67\u003c\/p\u003e \u003cp\u003e4.5.5 Natural extension and avoiding sure loss 67\u003c\/p\u003e \u003cp\u003e4.5.6 Simpler ways of calculating the natural extension 69\u003c\/p\u003e \u003cp\u003e4.6 Alternative characterisations for avoiding sure loss, coherence, and natural extension 70\u003c\/p\u003e \u003cp\u003e4.7 Topological considerations 74\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Special coherent lower previsions 76\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Linear previsions on finite spaces 77\u003c\/p\u003e \u003cp\u003e5.2 Coherent lower previsions on finite spaces 78\u003c\/p\u003e \u003cp\u003e5.3 Limits as linear previsions 80\u003c\/p\u003e \u003cp\u003e5.4 Vacuous lower previsions 81\u003c\/p\u003e \u003cp\u003e5.5 {0, 1}-valued lower probabilities 82\u003c\/p\u003e \u003cp\u003e5.5.1 Coherence and natural extension 82\u003c\/p\u003e \u003cp\u003e5.5.2 The link with classical propositional logic 88\u003c\/p\u003e \u003cp\u003e5.5.3 The link with limits inferior 90\u003c\/p\u003e \u003cp\u003e5.5.4 Monotone convergence 91\u003c\/p\u003e \u003cp\u003e5.5.5 Lower oscillations and neighbourhood filters 93\u003c\/p\u003e \u003cp\u003e5.5.6 Extending a lower prevision defined on all continuous bounded gambles 98\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 n-Monotone lower previsions 101\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 n-Monotonicity 102\u003c\/p\u003e \u003cp\u003e6.2 n-Monotonicity and coherence 107\u003c\/p\u003e \u003cp\u003e6.2.1 A few observations 107\u003c\/p\u003e \u003cp\u003e6.2.2 Results for lower probabilities 109\u003c\/p\u003e \u003cp\u003e6.3 Representation results 113\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Special n-monotone coherent lower previsions 122\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Lower and upper mass functions 123\u003c\/p\u003e \u003cp\u003e7.2 Minimum preserving lower previsions 127\u003c\/p\u003e \u003cp\u003e7.2.1 Definition and properties 127\u003c\/p\u003e \u003cp\u003e7.2.2 Vacuous lower previsions 128\u003c\/p\u003e \u003cp\u003e7.3 Belief functions 128\u003c\/p\u003e \u003cp\u003e7.4 Lower previsions associated with proper filters 129\u003c\/p\u003e \u003cp\u003e7.5 Induced lower previsions 131\u003c\/p\u003e \u003cp\u003e7.5.1 Motivation 131\u003c\/p\u003e \u003cp\u003e7.5.2 Induced lower previsions 133\u003c\/p\u003e \u003cp\u003e7.5.3 Properties of induced lower previsions 134\u003c\/p\u003e \u003cp\u003e7.6 Special cases of induced lower previsions 138\u003c\/p\u003e \u003cp\u003e7.6.1 Belief functions 139\u003c\/p\u003e \u003cp\u003e7.6.2 Refining the set of possible values for a random variable 139\u003c\/p\u003e \u003cp\u003e7.7 Assessments on chains of sets 142\u003c\/p\u003e \u003cp\u003e7.8 Possibility and necessity measures 143\u003c\/p\u003e \u003cp\u003e7.9 Distribution functions and probability boxes 147\u003c\/p\u003e \u003cp\u003e7.9.1 Distribution functions 147\u003c\/p\u003e \u003cp\u003e7.9.2 Probability boxes 149\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Linear previsions, integration and duality 151\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Linear extension and integration 153\u003c\/p\u003e \u003cp\u003e8.2 Integration of probability charges 159\u003c\/p\u003e \u003cp\u003e8.3 Inner and outer set function, completion and other extensions 163\u003c\/p\u003e \u003cp\u003e8.4 Linear previsions and probability charges 166\u003c\/p\u003e \u003cp\u003e8.5 The S-integral 168\u003c\/p\u003e \u003cp\u003e8.6 The Lebesgue integral 171\u003c\/p\u003e \u003cp\u003e8.7 The Dunford integral 172\u003c\/p\u003e \u003cp\u003e8.8 Consequences of duality 177\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Examples of linear extension 181\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Distribution functions 181\u003c\/p\u003e \u003cp\u003e9.2 Limits inferior 182\u003c\/p\u003e \u003cp\u003e9.3 Lower and upper oscillations 183\u003c\/p\u003e \u003cp\u003e9.4 Linear extension of a probability measure 183\u003c\/p\u003e \u003cp\u003e9.5 Extending a linear prevision from continuous bounded gambles 187\u003c\/p\u003e \u003cp\u003e9.6 Induced lower previsions and random sets 188\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Lower previsions and symmetry 191\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Invariance for lower previsions 192\u003c\/p\u003e \u003cp\u003e10.1.1 Definition 192\u003c\/p\u003e \u003cp\u003e10.1.2 Existence of invariant lower previsions 194\u003c\/p\u003e \u003cp\u003e10.1.3 Existence of strongly invariant lower previsions 195\u003c\/p\u003e \u003cp\u003e10.2 An important special case 200\u003c\/p\u003e \u003cp\u003e10.3 Interesting examples 205\u003c\/p\u003e \u003cp\u003e10.3.1 Permutation invariance on finite spaces 205\u003c\/p\u003e \u003cp\u003e10.3.2 Shift invariance and Banach limits 208\u003c\/p\u003e \u003cp\u003e10.3.3 Stationary random processes 210\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Extreme lower previsions 214\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Preliminary results concerning real functionals 215\u003c\/p\u003e \u003cp\u003e11.2 Inequality preserving functionals 217\u003c\/p\u003e \u003cp\u003e11.2.1 Definition 217\u003c\/p\u003e \u003cp\u003e11.2.2 Linear functionals 217\u003c\/p\u003e \u003cp\u003e11.2.3 Monotone functionals 218\u003c\/p\u003e \u003cp\u003e11.2.4 n-Monotone functionals 218\u003c\/p\u003e \u003cp\u003e11.2.5 Coherent lower previsions 219\u003c\/p\u003e \u003cp\u003e11.2.6 Combinations 220\u003c\/p\u003e \u003cp\u003e11.3 Properties of inequality preserving functionals 220\u003c\/p\u003e \u003cp\u003e11.4 Infinite non-negative linear combinations of inequality preserving functionals 221\u003c\/p\u003e \u003cp\u003e11.4.1 Definition 221\u003c\/p\u003e \u003cp\u003e11.4.2 Examples 222\u003c\/p\u003e \u003cp\u003e11.4.3 Main result 223\u003c\/p\u003e \u003cp\u003e11.5 Representation results 224\u003c\/p\u003e \u003cp\u003e11.6 Lower previsions associated with proper filters 225\u003c\/p\u003e \u003cp\u003e11.6.1 Belief functions 225\u003c\/p\u003e \u003cp\u003e11.6.2 Possibility measures 226\u003c\/p\u003e \u003cp\u003e11.6.3 Extending a linear prevision defined on all continuous bounded gambles 226\u003c\/p\u003e \u003cp\u003e11.6.4 The connection with induced lower previsions 227\u003c\/p\u003e \u003cp\u003e11.7 Strongly invariant coherent lower previsions 228\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePART II EXTENDING THE THEORY TO UNBOUNDED GAMBLES 231\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Introduction 233\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 Conditional lower previsions 235\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 Gambles 236\u003c\/p\u003e \u003cp\u003e13.2 Sets of acceptable gambles 236\u003c\/p\u003e \u003cp\u003e13.2.1 Rationality criteria 236\u003c\/p\u003e \u003cp\u003e13.2.2 Inference 238\u003c\/p\u003e \u003cp\u003e13.3 Conditional lower previsions 240\u003c\/p\u003e \u003cp\u003e13.3.1 Going from sets of acceptable gambles to conditional lower previsions 240\u003c\/p\u003e \u003cp\u003e13.3.2 Conditional lower previsions directly 252\u003c\/p\u003e \u003cp\u003e13.4 Consistency for conditional lower previsions 254\u003c\/p\u003e \u003cp\u003e13.4.1 Definition and justification 254\u003c\/p\u003e \u003cp\u003e13.4.2 Avoiding sure loss and avoiding partial loss 257\u003c\/p\u003e \u003cp\u003e13.4.3 Compatibility with the definition for lower previsions on bounded gambles 258\u003c\/p\u003e \u003cp\u003e13.4.4 Comparison with avoiding sure loss for lower previsions on bounded gambles 258\u003c\/p\u003e \u003cp\u003e13.5 Coherence for conditional lower previsions 259\u003c\/p\u003e \u003cp\u003e13.5.1 Definition and justification 259\u003c\/p\u003e \u003cp\u003e13.5.2 Compatibility with the definition for lower previsions on bounded gambles 264\u003c\/p\u003e \u003cp\u003e13.5.3 Comparison with coherence for lower previsions on bounded gambles 264\u003c\/p\u003e \u003cp\u003e13.5.4 Linear previsions 264\u003c\/p\u003e \u003cp\u003e13.6 Properties of coherent conditional lower previsions 266\u003c\/p\u003e \u003cp\u003e13.6.1 Interesting consequences of coherence 266\u003c\/p\u003e \u003cp\u003e13.6.2 Trivial extension 269\u003c\/p\u003e \u003cp\u003e13.6.3 Easier ways to prove coherence 270\u003c\/p\u003e \u003cp\u003e13.6.4 Separate coherence 278\u003c\/p\u003e \u003cp\u003e13.7 The natural extension of a conditional lower prevision 279\u003c\/p\u003e \u003cp\u003e13.7.1 Natural extension as least-committal extension 280\u003c\/p\u003e \u003cp\u003e13.7.2 Natural extension and equivalence 281\u003c\/p\u003e \u003cp\u003e13.7.3 Natural extension to a specific domain and the transitivity of natural extension 282\u003c\/p\u003e \u003cp\u003e13.7.4 Natural extension and avoiding sure loss 283\u003c\/p\u003e \u003cp\u003e13.7.5 Simpler ways of calculating the natural extension 285\u003c\/p\u003e \u003cp\u003e13.7.6 Compatibility with the definition for lower previsions on bounded gambles 286\u003c\/p\u003e \u003cp\u003e13.8 Alternative characterisations for avoiding sure loss, coherence and natural extension 287\u003c\/p\u003e \u003cp\u003e13.9 Marginal extension 288\u003c\/p\u003e \u003cp\u003e13.10 Extending a lower prevision from bounded gambles to conditional gambles 295\u003c\/p\u003e \u003cp\u003e13.10.1 General case 295\u003c\/p\u003e \u003cp\u003e13.10.2 Linear previsions and probability charges 297\u003c\/p\u003e \u003cp\u003e13.10.3 Vacuous lower previsions 298\u003c\/p\u003e \u003cp\u003e13.10.4 Lower previsions associated with proper filters 300\u003c\/p\u003e \u003cp\u003e13.10.5 Limits inferior 300\u003c\/p\u003e \u003cp\u003e13.11 The need for infinity? 301\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 Lower previsions for essentially bounded gambles 304\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e14.1 Null sets and null gambles 305\u003c\/p\u003e \u003cp\u003e14.2 Null bounded gambles 310\u003c\/p\u003e \u003cp\u003e14.3 Essentially bounded gambles 311\u003c\/p\u003e \u003cp\u003e14.4 Extension of lower and upper previsions to essentially bounded gambles 316\u003c\/p\u003e \u003cp\u003e14.5 Examples 322\u003c\/p\u003e \u003cp\u003e14.5.1 Linear previsions and probability charges 322\u003c\/p\u003e \u003cp\u003e14.5.2 Vacuous lower previsions 323\u003c\/p\u003e \u003cp\u003e14.5.3 Lower previsions associated with proper filters 323\u003c\/p\u003e \u003cp\u003e14.5.4 Limits inferior 324\u003c\/p\u003e \u003cp\u003e14.5.5 Belief functions 325\u003c\/p\u003e \u003cp\u003e14.5.6 Possibility measures 325\u003c\/p\u003e \u003cp\u003e\u003cb\u003e15 Lower previsions for previsible gambles 327\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e15.1 Convergence in probability 328\u003c\/p\u003e \u003cp\u003e15.2 Previsibility 331\u003c\/p\u003e \u003cp\u003e15.3 Measurability 340\u003c\/p\u003e \u003cp\u003e15.4 Lebesgue’s dominated convergence theorem 343\u003c\/p\u003e \u003cp\u003e15.5 Previsibility by cuts 348\u003c\/p\u003e \u003cp\u003e15.6 A sufficient condition for previsibility 350\u003c\/p\u003e \u003cp\u003e15.7 Previsibility for 2-monotone lower previsions 352\u003c\/p\u003e \u003cp\u003e15.8 Convex combinations 355\u003c\/p\u003e \u003cp\u003e15.9 Lower envelope theorem 355\u003c\/p\u003e \u003cp\u003e15.10 Examples 358\u003c\/p\u003e \u003cp\u003e15.10.1 Linear previsions and probability charges 358\u003c\/p\u003e \u003cp\u003e15.10.2 Probability density functions: The normal density 359\u003c\/p\u003e \u003cp\u003e15.10.3 Vacuous lower previsions 360\u003c\/p\u003e \u003cp\u003e15.10.4 Lower previsions associated with proper filters 361\u003c\/p\u003e \u003cp\u003e15.10.5 Limits inferior 361\u003c\/p\u003e \u003cp\u003e15.10.6 Belief functions 362\u003c\/p\u003e \u003cp\u003e15.10.7 Possibility measures 362\u003c\/p\u003e \u003cp\u003e15.10.8 Estimation 365\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix A Linear spaces, linear lattices and convexity 368\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix B Notions and results from topology 371\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eB.1 Basic definitions 371\u003c\/p\u003e \u003cp\u003eB.2 Metric spaces 372\u003c\/p\u003e \u003cp\u003eB.3 Continuity 373\u003c\/p\u003e \u003cp\u003eB.4 Topological linear spaces 374\u003c\/p\u003e \u003cp\u003eB.5 Extreme points 374\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix C The Choquet integral 376\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eC.1 Preliminaries 376\u003c\/p\u003e \u003cp\u003eC.1.1 The improper Riemann integral of a non-increasing function 376\u003c\/p\u003e \u003cp\u003eC.1.2 Comonotonicity 378\u003c\/p\u003e \u003cp\u003eC.2 Definition of the Choquet integral 378\u003c\/p\u003e \u003cp\u003eC.3 Basic properties of the Choquet integral 379\u003c\/p\u003e \u003cp\u003eC.4 A simple but useful equality 387\u003c\/p\u003e \u003cp\u003eC.5 A simplified version of Greco’s representation theorem 389\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix D The extended real calculus 391\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eD.1 Definitions 391\u003c\/p\u003e \u003cp\u003eD.2 Properties 392\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix E Symbols and notation 396\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eReferences 398\u003c\/p\u003e \u003cp\u003eIndex 407\u003c\/p\u003e\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003eSubject Areas: Mathematics [\u003ca title=\"See our other books on Mathematics\" 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