## Finitely Additive Measures and Relaxations of Extremal ProblemsThis monograph constructs correct extensions of extremal problems, including problems of multicriteria optimization as well as more general cone optimization problems. The author obtains common conditions of stability and asymptotic nonsensitivity of extremal problems under perturbation of a part of integral restrictions for finite and infinite systems of restrictions. Features include individual chapters on nonstandard approximation of finitely additive measures by indefinite integrals and constructions of attraction sets. Professor Chentsov illustrates abstract settings by providing examples of problems of impulse control, mathematical programming, and stochastic optimization. |

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### Contents

Relaxations of Extremal Problems cogent discussion and examples | 1 |

12 Relaxations of Integral Constraints | 2 |

13 Constraints Imposed on the Vector Integrand of a Control | 7 |

14 Asymptotic Effects in a Cone Optimization Problem an Example | 13 |

Compactificators in Extremal Problems | 19 |

22 Some Notions from Topology | 20 |

23 Compactificator of an Extremal Problem with Constraints of Asymptotic Character | 26 |

24 Some Notions of Set Theory Products of Topological Spaces | 33 |

55 Strong Boundedness Conditions and Relaxations of Admissible Sets II | 113 |

Relaxations of Attainable Sets | 121 |

62 Limit Representation of Attainable Sets | 122 |

63 Asymptotic Nonsensitivity and Relaxations of Attainable Sets Case of Positive Measures | 125 |

64 Asymptotic Nonsensitivity and Relaxations of Attainable Sets Case of Alternating Measures | 131 |

65 Attainability Domains of Controllable Systems and Their Relaxations | 135 |

Asymptotic Values and Their Generalized Representation | 139 |

72 Example | 140 |

25 The Problem of Asymptotic Optimization in a Preordered Space and Its Extension | 39 |

26 Extension of the Cone Optimization Problem | 45 |

Finitely Additive Measures on a Semialgebra of Sets | 55 |

32 Semialgebras and Algebras of Sets | 57 |

33 Finitely Additive Measures | 59 |

34 Measures and Integral Representation of Linear Functionals | 65 |

35 Compactification in the Class of TwoValued Measures | 71 |

36 Extension of the Cone Optimization Problem in the Class of TwoValued Measures | 72 |

Finitely Additive Measures and Indefinite Integrals | 77 |

42 Weakly Absolutely Continuous Finitely Additive Measures | 79 |

43 Density Properties | 83 |

44 Examples | 89 |

Admissible Sets and Their Relaxations | 97 |

52 Asymptotics of Perturbed Constraints and Its Limit Representation Case of Positive Measures | 100 |

53 Asymptotics of Perturbed Constraints and Its Limit Representation Case of Alternating Measures | 104 |

54 Strong Boundedness Conditions and Relaxations of Admissible Sets I | 108 |

73 Generalized Extremal Problem and Asymptotic Values | 143 |

74 Value Asymptotics for Strongly Bounded Problems | 151 |

75 On an Extremal Problem with InequalityType Constraints | 159 |

Asymptotic Efficiency | 167 |

82 Generalized Problem of Cone Optimization I | 168 |

83 Generalized Problem of Cone Optimization II | 177 |

Issues of Computational Stability | 185 |

92 Constraints on the Vector Integrand | 186 |

93 Computational Stability | 193 |

94 A Problem of the Extremal Choice of Probability Density | 194 |

95 Extension in the Class of Vector Finitely Additive Measures | 198 |

Conclusion | 231 |

References | 235 |

243 | |

### Common terms and phrases

A. G. Chentsov admissible sets algebra algebra of sets analogy arbitrary assertions assume asymptotic optimization asymptotic value attainability domain Attainable Sets bounded bounded variation boundedness Chap chapter coincidence compact space compactification computational stability concerning condition confine ourselves connected continuous control problems conventional solutions convergence corresponding defined definition denote Dirac measures directed set discuss due account easily elements example extremal problems f.a. measures filter base finite intersection property Finitely Additive Measures G add G Step integral integrand introduce Lemma linear linear subspace mapping means measurable space Moreover Namely nonempty family nonempty set nonnegative notation Note obtained obvious operator parameter particular pointwise pointwise ordering preordered proof q G Q realized relation relaxations representation respect semialgebra sense sequel sequence similar specification subspace system of constraints take into account Taking account Theorem Throughout the remainder Throughout this section tions topological space topology vector virtue