Theory of Functions of a Real Variable

Originally published in two volumes, this long out-of-print work by a prominent Soviet mathematician presents a thorough examination of the theory of functions of a real variable. Intended for advanced undergraduates and graduate students of mathematics, the treatment offers a clear account of integration theory and a practical introduction to functional analysis. Prerequisites include a background in the foundations of elementary analysis and some familiarity with the theory of irrational numbers, the theory of limits, continuous functions, Riemann integrals, and infinite series.
Volume I covers infinite and point sets, measurable sets and functions, the Lebesgue integral of a bounded function, square-summable functions, functions of finite variations, the Stieltjes integral, absolutely contiguous functions, and the indefinite Lebesgue integral. Volume II addresses singular integrals, trigonometric series, convex functions, point sets in two-dimensional space, measurable functions of several variables and their integration, set functions and their applications in the theory of integration, transfinite numbers, the Baire classification, certain generalizations of the Lebesgue integral, and some ideas from functional analysis. Many chapters feature challenging exercises.

Soviet mathematician Isidor Pavlovich Natanson (1906–64) was affiliated with St. Petersburg State University. He also wrote a three-volume treatise, Constructive Function Theory. 

Volume II. Infinite SetsII. Point SetsIII. Measurable SetsIV. Measurable FunctionsV. The Lebesgue Integral of a Bounded FunctionVII. Square-Summable FunctionsVIII. Functions of Finite Variations. The Stieltjes IntegralIX.Absolutely Continguous Functions. The Indefinite Lebesgue IntegralIndexVolume IIX. Singular Integrals. Trignometric Series. Convex FunctionsXI. Point Sets in Two-Dimensional SpaceXII. Measurable Functions of Several Variables and their IntegrationXIII. Set functions and their Applications in the Theory of Integration   XIV. Transfinite NumbersXV. The Baire ClassificationXVI. Certain Generalizations of the Lebesgue IntegralXVII. Some Ideas from Functional AnalysisAppendicesIndex.