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Theory of Functions of a Real Variable

I. P. Natanson
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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This is a very traditional text in measure and integration on the real line and on \(\mathbb{R}^n\) — traditional partly because it was deliberately written this way, and partly because it was originally written in 1941. It assumes the student has already had a good rigorous course in real analysis up to the Lebesgue integral. This would include the construction of the real numbers, uniform convergence, and the theory of derivatives and of the Riemann integral. The Russian work was aimed at third-year university students, and the English translation at graduate students. The present volume is a Dover unaltered reprint of the two-volume work published by Ungar in 1955 and 1960. This in turn was translated from the 1950 (volume I) and 1957 (volume II) Russian editions. The translation reworks the Russian edition somewhat and adds several appendices by Edwin F. Hewitt.

The work deals primarily with Lebesgue measure on the real line. This is done through outer measure on the real line. It is then re-cast in the language of Borel sets, and the same ideas are used in slightly more generality in \(\mathbb{R}^n\). There’s also a good treatment of several other kinds of integrals, including the Riemann-Stieltjes, Perron, and Denjoy–Khinchin integrals. Beyond integration, it has introductions to absolutely continuous functions, Fourier series, convex functions, transfinite cardinals, Baire category, and functional analysis (mostly about metric spaces, contraction mappings, and fixed-point theorems, rather than functionals).

The exercises are challenging and intended to extend results in the body of the text. Most of them are research results and are credited to well-known mathematicians. Unfortunately these stop after Chapter X except for a few in Chapter XVI. There are no hints or solutions given.

The notation and terminology are antiquated in a few ways, although not confusingly so. The text uses sum and product for the set operations of union and intersection. It refers to summable functions (meaning integrable) and finite variation (meaning bounded variation). It uses the notation \(f_n \Longrightarrow f\) to mean convergence almost everywhere.

Despite its length (at about 550 pages), the book really doesn’t cover very much. It has a very leisurely exposition, everything is written out in detail, and there are lots of worked examples. It’s not a complete course in real analysis, as it assumes you already know derivatives and Riemann integrals and convergence, and it has nothing about partial derivatives or change of variables in multiple integrals, and very little on \(L^p\) spaces except \(L^2\). I think the book is probably not a good fit for present-day curricula, because if we spent this much time on measure and integration we would do it for general measure spaces and not just for \(\mathbb{R}^n\), and we would do it from the context of \(L^p\) spaces. I think it would work fairly well for self-study, although you wouldn’t learn anything about the modern abstract theories. I wouldn’t use it as a reference, because the modern works are more general and more concise references.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.

I. Infinite Sets

II. Point Sets

III. Measurable Sets

IV. Measurable Functions

V. The Lebesgue Integral of a Bounded Function

VI. Summable Functions

VII. Square-Summable Functions

VIII. Functions of Finite Variation. The Stieltjes Integral

IX. Absolutely Continuous Functions. The Indefinite Lebesgue Integral

X. Singular Integrals. Trigonometric Series. Convex functions

XI. Point Sets in Two-Dimensional Space

XII. Measurable Functions of Several Variables and their Integration

XIII. Set Functions and their Applications in the Theory of Integration

XIV. Transfinite Numbers

XV. The Baire Classification

XVI. Certain Generalizations of the Lebesgue Integral

XVII. (omitted)

XVIII. Some Ideas from Functional Analysis