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International Series of Monographs in Pure and Applied Mathematics, Volume 62: A Course of Higher Mathematics, V: Integration and Functional Analysis focuses on the theory of functions. The book first discusses the Stieltjes integral. Concerns include sets and their powers, Darboux sums, improper Stieltjes integral, jump functions, Helly's theorem, and selection principles. The text then takes a look at set functions and the Lebesgue integral. Operations on sets, measurable sets, properties of closed and open sets, criteria for measurability, and exterior measure and its properties are discuss.
Front Cover; Higher Mathematics; Copyright Page ; Table of Contents; INTRODUCTION; PREFACE; CHAPTER 1. THE STIELTJES INTEGRAL; 1. Sets and their powers; 2. The Stieltjes integral and its basic properties; 3. Darboux sums; 4. The Stieltjes integral of a continuous function; 5· The improper Stieltjes integral; 6. Jump functions; 7. Physical interpretation; 8. Functions of bounded variation; 9· An integrating function of bounded variation; 10. Existence of the Stieltjes integral; 11. Passage to the limit in the Stieltjes integral; 12. Helly's theorem; 13. Selection principle.
14. Space of continuoue fonctione15. General form of the functional in space C; 16. Linear operators in C; 17· Functions of an interval; 18. The general Stieltjes integral; 19. Properties of the (general) Stieltjes integral; 20. The existence of the general Stieltjes integral; 21· Functions of a two-dimensional interval; 22. Passage to point functions; 23. The Stieltjes integral on a plane; 24. Functions of bounded variation on the plane; 25. The space of continuous functions of several variables; 26. The Fourier-Stieltjes integral; 27. Inversion formula; 28. ConvoIution theorem.
44. The limit of a measurable function45. The C properly; 46. Piecewise constant functions; 47. Class B ; 3. The Lebesgue integral; 48. The integral of a bounded function; 49. Properties of the integral; 50· The integral of a non-negative unbounded function; 51. Properties of the integral; 52· Functions of any sign; 53. Complex summable functions; 54. Passage to the limit under the integral sign; 55· The class L2; 56. Convergence in the mean; 57. Hilbert function space; 58· Orthogonal systems of functions; 59. The space l2; 60. Lineals in L2; 61. Examples of closed systems.
62. The Holder and Minhkoskii inequalities63. Integral over a set of infinite measure; 64. The class L2 on a set of infinite measure; 65· An integrating function of bounded variation; 66. The reduction of multiple integrals; 67· The case of the characteristic function; 68· Fubini's theorem; 69. Change of the order of integration; 70. Continuity in the mean; 71. Mean functions; CHAPTER 3. SET FUNCTIONS. ABSOLUTE CONTINUITY GENERALIZATION OF THE INTEGRAL; 72. Additive set functions; 73. Siogular function; 74· The case of one variable; 75. Absolutely continuous set functions; 76. Example.
V.I. Smirnov ; translated [from the Russian] by D.E. Brown ; translation edited by I.N. Sneddon
Translation of: Kurs vysshei matematiki; Moscow, 1939-49.
Course of higher mathematics / Vol. 5, [Integration and functional analysis]
Smirnov, V.I. (Vladimir Ivanovich), 1887-1974.
International series of monographs in pure and applied mathematics ; 62