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