CONTENTS 13

46.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .151346.2 The Trace On The Boundary Of An Open Set . . . . . . . . . . . . . . .1515

IV Multifunctions 1519

47 The Yankov von Neumann Aumann theorem 1521

48 Multifunctions and Their Measurability 153348.1 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1533

48.1.1 A Special Case Which Is Easier . . . . . . . . . . . . . . . . . .153748.1.2 Other Measurability Considerations . . . . . . . . . . . . . . . .1538

48.2 Existence of Measurable Fixed Points . . . . . . . . . . . . . . . . . . . .154048.2.1 Simplices And Labeling . . . . . . . . . . . . . . . . . . . . . .154048.2.2 Labeling Vertices . . . . . . . . . . . . . . . . . . . . . . . . .154148.2.3 Measurability Of Brouwer Fixed Points . . . . . . . . . . . . .154348.2.4 Measurability Of Schauder Fixed Points . . . . . . . . . . . . .1551

48.3 A Set Valued Browder Lemma With Measurability . . . . . . . . . . . . .155848.4 A Measurable Kakutani Theorem . . . . . . . . . . . . . . . . . . . . . .156548.5 Some Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . . .156748.6 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .157248.7 Limit Conditions For Nemytskii Operators . . . . . . . . . . . . . . . . .1578

V Complex Analysis 1595

49 The Complex Numbers 159749.1 The Extended Complex Plane . . . . . . . . . . . . . . . . . . . . . . . .159849.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1600

50 Riemann Stieltjes Integrals 160150.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1611

51 Fundamentals Of Complex Analysis 161351.1 Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1613

51.1.1 Cauchy Riemann Equations . . . . . . . . . . . . . . . . . . . .161551.1.2 An Important Example . . . . . . . . . . . . . . . . . . . . . . .1617

51.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .161751.3 Cauchy’s Formula For A Disk . . . . . . . . . . . . . . . . . . . . . . . .161951.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .162651.5 Zeros Of An Analytic Function . . . . . . . . . . . . . . . . . . . . . . .162951.6 Liouville’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .163151.7 The General Cauchy Integral Formula . . . . . . . . . . . . . . . . . . .1632

51.7.1 The Cauchy Goursat Theorem . . . . . . . . . . . . . . . . . . .163251.7.2 A Redundant Assumption . . . . . . . . . . . . . . . . . . . . .163551.7.3 Classification Of Isolated Singularities . . . . . . . . . . . . . .1636