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CHAPTER Ⅰ．SOME THEOREMS ON REAL-VALUED FUNCTIONS 1

1．Sets and characteristic functions 1

2．Neighborhoods，openness，closure 5

3．Denumerable sets 14

4．Functions and limits 20

5．Bounds 23

6．Upper and lower limits 26

7．Semi-continuous functions 38

8．Functions of bounded variation 44

9．Absolutely continuous functions 47

CHAPTER Ⅱ．THE LEBESGUE INTEGRAL 52

10．Step-functions 52

11．Riemann integrals 57

12．U-functions and L-functions 62

13．Integrals of U-functions and L-functions 66

14．Upper and lower integrals 72

15．The Lebesgue integral 75

16．Consistency of Riemann and Lebesgue integrals 85

17．Integrals over bounded sets 89

18．Integrals over unbounded sets 94

CHAPTER Ⅲ．MEASURABLE SETS AND MEASURABLE FUNCTIONS 101

19．Arithmetic of measurable sets 101

20．Exterior measure and interior measure 109

21．Measurable functions 118

22．Measurable functions and summable functions 125

23．Equivalence of functions 128

24．Summable products．Inequalities 131

CHAPTER Ⅳ．THE INTEGRAL AS A FUNCTION OF SETS；CONVERGENCE THEOREMS 136

25．Multiple integrals and iterated integrals 136

26．Set functions 150

27．The integral as a set function 156

28．Modes of convergence 160

29．Convergence theorems 166

30．Metric spaces；spaces Lp 177

CHAPTER Ⅴ．DIFFERENTIATION 188

31．Dini derivates 188

32．Derivates of monotonic functions 194

33．Derivatives of indefinite Lebesgue integrals 197

34．Derivatives of functions of bounded variation 200

35．Derivatives of absolutely continuous functions 207

36．Integration by parts 209

37．Mean-value theorems 209

38．Substitution theorems 211

39．Differentiation under the integral sign 216

CHAPTER Ⅵ．CONTINUITY PROPERTIES OF MEASURABLE FUNCTIONS 218

40．The classes of Baire 218

41．Metric density and approximate continuity 222

42．Density of continuous functions in Lp．Riemann-Lebesgue theorem 225

43．Lusin＇s theorem 236

44．Non-measurable sets and non-measurable functions 237

CHAPTER Ⅶ．THE LEBESGUE-STIELTJES INTEGRAL 242

45．The difference-function 242

46．Monotonic functions and functions of bounded variation 248

47．Integrals and measure with respect to monotonic functions 251

48．Examples 255

49．Borel sets 261

50．Dependence of integral on integrator 264

51．Integrals with respect to functions of bounded variation 269

52．Properties of the Lebesgue-Stieltjes integral 271

53．Measure functions 277

54．Measure functions and Lebesgue-Stieltjes measure 287

55．Integrals with respect to a measure function 295

56．Measure functions defined by integrals 303

CHAPTER Ⅷ．THE PERRON INTEGRAL 312

57．Definition of the Perron integral 312

58．Elementary properties 316

59．Relation to the Lebesgue integral 322

60．Perron integral of a derivative 323

61．Derivative of the indefinite Perron integral 326

62．Summability of non-negative integrable functions 328

63．Convergence theorems 329

64．Substitution 329

65．Integration by parts 331

66．Second theorem of mean value 335

CHAPTER Ⅸ．DIFFERENTIAL EQUATIONS 336

67．Ascoli＇s theorem 336

68．Existence and uniqueness of solutions 338

69．The solutions as functions of parameters 348

CHAPTER Ⅹ．DIFFERENTIATION OF MULTIPLE INTEGRALS 366

70．Vitali＇s theorem 366

71．Derivates of set functions 372

72．Derivatives of indefinite integrals 374

73．Derivatives of functions of bounded variation 378

APPENDIX 383

List of special symbols and abbreviations 385

INDEX 387