## Independence, Additivity, UncertaintyThe work on this book started many years ago as an attempt to simplify and unify some results usually taught in courses in mathematical economics. The economic interpretation of the re sults were representations of preferences as sums or integrals and the decomposition of preferences into utilities and probabilities. It later turned out that t.he approach taken in the earlier versions were also the proper approach in generalizing from preferences which were total preorders to preferences which were not total or tran sitive. The same mathematics would even in that situation give representations which were additive. It would also give decomposi tions where concepts of uncertainty appeared. Early versions of some of the results appeared as Working Pa pers No. 135, 140, 150, and 176 from The Center for Research in Management Science, Berkeley. A first version of chapters 2, 4, 6, 7, and 8 appeared 1969 with the title" Mean Groupoids" [177]. They are essentially unchanged -except for some notes especially in chapter 6. Another version appeared 1990 as [178]. Chapter 10 contains results from the same versions and from [181]. Chapter 11 by Birgit Grodal is based on [91] by Grodal and Jp,an-Francois Mertens. Chapters 11 and 12 - also by Birgit Gro dal -contains the results from the earlier versions, but have been extended (by Karl Vind) to take into account the new corollaries of the results in the other chapters. |

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### Contents

I | 1 |

II | 2 |

III | 4 |

IV | 6 |

V | 7 |

VI | 9 |

VII | 14 |

VIII | 16 |

LXXIV | 123 |

LXXV | 124 |

LXXVI | 126 |

LXXVIII | 128 |

LXXIX | 131 |

LXXX | 133 |

LXXXI | 135 |

LXXXII | 136 |

IX | 18 |

X | 20 |

XI | 23 |

XIII | 24 |

XVI | 25 |

XIX | 27 |

XX | 29 |

XXI | 31 |

XXII | 33 |

XXIII | 37 |

XXIV | 38 |

XXV | 39 |

XXVI | 40 |

XXVIII | 43 |

XXIX | 45 |

XXX | 49 |

XXXI | 52 |

XXXII | 54 |

XXXIII | 57 |

XXXIV | 59 |

XXXV | 60 |

XXXVI | 61 |

XXXIX | 62 |

XLII | 63 |

XLIV | 65 |

XLV | 69 |

XLVI | 70 |

XLVII | 73 |

XLVIII | 77 |

L | 78 |

LI | 81 |

LII | 83 |

LIII | 85 |

LIV | 87 |

LV | 88 |

LVI | 91 |

LVII | 95 |

LVIII | 97 |

LIX | 98 |

LXI | 102 |

LXII | 103 |

LXIII | 106 |

LXIV | 107 |

LXV | 108 |

LXVI | 113 |

LXVII | 115 |

LXVIII | 116 |

LXIX | 117 |

LXX | 118 |

LXXI | 119 |

LXXII | 120 |

LXXIII | 121 |

LXXXIII | 137 |

LXXXIV | 138 |

LXXXV | 140 |

LXXXVI | 144 |

LXXXVII | 147 |

LXXXVIII | 149 |

LXXXIX | 152 |

XC | 159 |

XCI | 164 |

XCII | 165 |

XCIII | 169 |

XCIV | 170 |

XCV | 174 |

XCVI | 177 |

XCVII | 183 |

XCVIII | 187 |

XCIX | 189 |

C | 190 |

CI | 192 |

CII | 193 |

CIII | 194 |

CIV | 195 |

CVII | 196 |

CVIII | 197 |

CX | 198 |

CXII | 199 |

CXIII | 201 |

CXIV | 203 |

CXV | 206 |

CXVI | 213 |

CXVII | 215 |

CXVIII | 220 |

CXX | 221 |

CXXI | 222 |

CXXII | 223 |

CXXIII | 225 |

CXXIV | 227 |

CXXV | 228 |

CXXVII | 229 |

CXXVIII | 231 |

CXXIX | 232 |

CXXX | 235 |

CXXXI | 237 |

CXXXII | 240 |

CXXXIII | 243 |

CXXXIV | 244 |

CXXXV | 247 |

CXXXVIII | 248 |

CXXXIX | 252 |

253 | |

269 | |

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### Common terms and phrases

a o b additive affine function algebra arbitrary set assume axiom chapter commutative mean groupoid conditional measures connected mixture constant functions continuous with respect convex sets Corollary countable Definition denoted dense determined equivalence classes example factor independence finite func function f function space functions defined given gives graphV holds homomorphism implies independence assumption independence condition independent with respect lemma Let Q measurable functions measure space metric space mixture with respect Neumann Morgenstern preference notation null set obtained order homomorphism order isomorphism order topology order-complete parameter space probability measures product set product topology properties prove pure preference real functions real numbers relation on Q relative topology Remark representation theorems separable metric space set of measurable space of functions strictly monotonic system of subsets theorem 26 theorem 52 Thomsen tions topological space totally ordered set totally preordered set trivial ug(x utility function

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Page iv - grant from the Ford Foundation to the Graduate School of Business Administration, University of California, Berkeley, and

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