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Measure Theory

Donald L. Cohn
Publication Date: 
Number of Pages: 
Birkhäuser Advanced Texts
[Reviewed by
Tushar Das
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In this second edition, Cohn has updated his excellent introduction to measure theory (1st ed. 1980, reprinted in 1993, MathSciNet MR578344) and has made this great textbook even better. Those readers unfamiliar with Cohn’s style will discover that his writing is lucid. It is a treat to behold such a wonderfully put-together book that has certainly been carefully edited and copy-edited, and seems to be (almost incredibly for a book this size) free from typos.

Cohn’s text appears to be not as well-known as, say Folland’s Real Analysis or Rudin’s Real and Complex Analysis — both of which contain introductions to measure theory but then move on to cover distinct terrains. Beyond the basics of measure theory with a dose of functional analysis thrown in for good measure: Folland treats Fourier transforms, distribution theory and probability; while Rudin seamlessly transitions to a full course on complex (and some harmonic) analysis ending with introductions to \(H^p\)-Spaces and Banach algebras. Another famous textbook with comparable content would be Royden’s Real Analysis, but the latter (whose third edition I am familiar with; there is now a fourth edition) is horribly blasé when it comes to statements and proofs of various theorems. For example, there were often, especially towards the latter half of Royden’s text, instances of redundant hypotheses in theorems, and a careful study or (re)discovery of their proofs would often lead to cleaner and tighter material. There are those who believe this “flexibility” to be among Royden’s advantages. In contrast, Cohn is precise in both his statements and proofs throughout the text. All his Is have been dotted and his Ts crossed. This I (currently) believe to be a virtue, especially when exposing the art of analysis to a first-year graduate student, or even a precocious senior. Good habits should be taught early!

Cohn’s textbook, as the title promises, contains a good deal more about measure theory proper — with a view to applications in probability theory and stochastic processes. I can imagine that his attention to detail and mathematical precision may lead more than a few (graduate student) readers to find Cohn somewhat dry. The first edition started right off the bat with abstract measure theory: the definition of a sigma algebra, measures, outer measures, etc. The second edition has some more motivation in the form of an introductory chapter that reviews the Riemann integral, a few classic pitfalls that led to Lebesgue’s theory, and an outline of the main topics and structure of the text. Pedagogically speaking this seems to me an excellent strategy, though my own introduction to measure theory was abstract measure theory first; only much later was there talk of Lebesgue and Riemann. Among other new additions to the second edition, there are concise and useful appendices on the Banach-Tarski paradox and the Henstock-Kurzweil integral.

To echo Mark Kac’s infamous apocrypha, Cohn adds some soul in the form of his entirely new Chapter 10 on Probability Theory. In under 70 pages he covers fundamental results such as the strong law of large numbers, the central limit theorem, the martingale convergence theorem, the construction of Brownian motion and Kolmogorov’s consistency theorem. This is treated as a chapter where a large collection of previously built measure-theoretic tools are (finally?) applied. The reviewer would liked to have seen a section on Ergodic Theory in this new chapter — to provide the budding researcher with another attractive area of application with various strands of active ongoing research interest. A chapter on Geometric Measure Theory would also provide a nice complement to the material developed in this text. But one shouldn’t make one’s wish list, or the text, too long.

Special mention should be made of Cohn’s excellent Chapter 8: “Polish spaces and Analytic Sets’, which covers ground that is often missed in most analysis texts, but is indispensable for analysts/probabilitsts who would like to work on complete separable metric spaces. This material is not altogether absent from the literature. It may be found, e.g., in the first chapter of K. R. Parthasarathy’s Probability Measures on Metric Spaces (1967), reprinted by AMS Chelsea Publishing; or in S. M. Srivastava’s A Course on Borel Sets (Springer, 1998), which has a more leisurely exposition.

To summarize, this is a wonderful text to learn measure theory from and I strongly recommend it.

Tushar Das is an Assistant Professor of Mathematics at the University of Wisconsin – La Crosse.

1. Measures
Algebras and sigma-algebras
Outer measures
Lebesgue measure
Completeness and regularity
Dynkin classes

2. Functions and Integrals
Measurable functions
Properties that hold almost everywhere
The integral
Limit theorems
The Riemann integral
Measurable functions again, complex-valued functions, and image measures

3. Convergence
Modes of Convergence
Normed spaces
Definition of \(\mathcal{L}^p\) and \(L^p\)
Properties of \(\mathcal{L}^p\) and \(L^p\)
Dual spaces

4. Signed and Complex Measures
Signed and complex measures
Absolute continuity
Functions of bounded variation
The duals of the \(L^p\) spaces

5. Product Measures
Fubini’s theorem

6. Differentiation
Change of variable in \(\mathbb{R}^d\)
Differentiation of measures
Differentiation of functions

7. Measures on Locally Compact Spaces
Locally compact spaces
The Riesz representation theorem
Signed and complex measures; duality
Additional properties of regular measures
The \(\mu^*\)-measurable sets and the dual of \(L^1\)
Products of locally compact spaces

8. Polish Spaces and Analytic Sets
Polish spaces
Analytic sets
The separation theorem and its consequences
The measurability of analytic sets
Cross sections
Standard, analytic, Lusin, and Souslin spaces

9. Haar Measure
Topological groups
The existence and uniqueness of Haar measure
The algebras \(L^1(G)\) and \(M (G)\)

10. Probability
Laws of Large Numbers
Convergence in Distribution and the Central Limit Theorem
Conditional Distributions and Martingales
Brownian Motion
Construction of Probability Measures

A. Notation and set theory
B. Algebra
C. Calculus and topology in \(\mathbb{R}^d\)
D. Topological spaces and metric spaces
E. The Bochner integral
F Liftings
G The Banach-Tarski paradox
H The Henstock-Kurzweil and McShane integrals


Index of notation