image What’s the connection between probability and real analysis theory ? Why does one need to know at least the basics of Real Analysis to understand probability.Well, that’s because axiomatic probability is built , ground up, using the concepts from real analysis. Though you can pick up any graduate level probability book and go through it by having an intuitive sense of real analysis, there is no substitute to seeing axiomatic probability through the eyes of person who understands real analysis well.

My objective of going through this book was simple : I do not want to prove theorems, even though this book is probably meant to understand proofs better.  All I looking for is to  have  more than an intuitive sense of Real Analysis , but at the same time not to get really deep in to the subject. In that sense, I found this book to be PERFECT.

The book is organized in 8 delightful chapters, taking the reader from Real Line all the way up to Metric spaces. Let me attempt to summarize this book in words.

Chapter 1  : The Real Numbers

This chapter introduces the preliminary tools and concepts of real analysis. It starts off with an example of computing square root of 2 and shows that set of rational numbers Q do not contain such irrational numbers. To prove that square root of 2 exists with out assuming anything is impossible and hence axiom of completeness is introduced which helps us get a grip on such numbers. Axiom of Completeness abbreviated as AoC is the key axiom , based on which further concepts and tools are explored such as nested interval theorem, Cardinality of Sets, Countable and Uncountable sets, Cantor’s theorem. Epilogue mentions about continuum hypothesis, which basically states that there is no set A which follows the inequality : Cardinality of N < Cardinality of A < Cardinality of R.

Chapter 2 : Sequence and Series Infinite series are funny creatures. They cannot be added multiplied subtracted as one does with the finite series. A beautiful introduction to the concept of convergence and divergence of series is given in this chapter. I have not seen a more lucid explanation of convergence in any other book. The illustration showing the epsilon neighbourhood of a function will forever be remember by any reader.

The following logical structure summarizes chapter 2


AoC acts as the crucial step in the proof of the Nested Interval Property (NIP). AoC is also the central step in the Monotone Convergence Theorem(MCT), and NIP is central to proving the Bolzano–Weierstrass Theorem(BW). Finally, one needs BW to prove the Cauchy Criterion (CC) for convergent sequences.One point that cannot be missed from this chapter is that MCT, CC are basically useful tools / concepts which conveniently avoid the term”limit” but still manage to talk about convergence of a series.

Chapter 3 : Basic Topology of R The chapter begins with a fantastic introduction to Cantor Sets. Clear definitions for Open Sets / Closed Sets / Compact Sets / Open Covers / Perfect Sets / Connected Sets / Nowhere Dense sets are given with appropriate examples. Baire’s theorem is introduced in the concluding part of the chapter.

Chapter 4 : Functional Limits and Continuity

The chapter starts off with Dirichlet and Thomae functions where the former is nowhere continuous in R while the latter is continuous at every irrational point of R and discontinuous at all rational points of R. These two functions are good enough to dispel the notion that functions are smooth curved graphs , an image which suits our intuition. However math / real analysis is a subject that is filled with so many examples that stump our intuition and make us go back to understanding the subject rigorously. Epsilon –Delta definition is introduced in this chapter as a method to rigorously verify the existence/non existence of a limit.

Chapter 5 : The Derivative

There are some extremely important questions at the beginning of the chapter which give a flavour of things covered . Some of these questions are as follows: Do all continuous functions have derivatives? If not how nondifferentiable can a continuous function be ? Are all differentiable functions continuous ? If a function f has a derivative at every point in a specific domain, can we say that f’ is continuous ? Can one describe the set of all derivatives ?

Mean value theorem, Rolle’s theorem , L’Hospital’s rule, Darboux’s theorem are the main aspects touched upon in this chapter. One obvious takeaway from this chapter is that differential functions are an exception than a rule. Continuity is a strictly weaker notion than differentiability

Chapter 6 : Sequences and Series of Functions

Polynomial functions are great friends to any applied mathematicians as they are continuous, infinitely differentiable and defined on all of R. They are easy to evaluate and easy to manipulate, both from the  view of algebra (adding, multiplying, factoring) and calculus (integrating, differentiating). It should be no surprise, then, that even in the earliest stages of the development of calculus, mathematicians experimented with the idea of extending the notion of polynomials to functions that are essentially polynomials of infinite degree. Such objects are called power series

One of the first things that comes to mind is “Point wise convergence” where the sequence of functions, i.e, a sequence of real numbers converge point wise to a specific function. But this isn’t enough as a few examples mentioned in the chapter show that, a sequence of continuous functions converge to a function which is not continuous. Also it is possible that a set of sequence of differentiable functions can converge to a function which is not differentiable. Thus a stronger notion of convergence is needed when one uses the limit function as a substitute for the sequence of functions. The concept of “Uniform Convergence” is introduced here which ensures that limit process for a sequence of functions retains the properties of continuity and differentiability of the originating sequence. A few tests for Uniform convergence of a series are stated, notably being Weierstrass M-Test

The all powerful series discovered by Brook Taylor is then introduced at the end of chapter .Taylor series gives a beautiful way to approximate a function to an infinite polynomial series. Lagrange’s remainder theorem is subsequently mentioned which gives an idea about the strength of Taylor series approximation for the function. The chapter then raises the question about the convergence of Taylor series to the function itself. How can we be sure that the Taylor series converges to the original function? Epilogue for the chapter tries to answer the question by analyzing the boundedness of the nth derivative of the generating function.

Chapter 7 : The Riemann Integral

Historically, integration was viewed as the inverse of differentiation. Integral of f was understood to be a function F that satisfies the relation that F’ = f . However this gives one a limited number of functions that can be integrated. What if the function is discontinuous? Thus “Why Riemann” is addressed at the very beginning of the chapter. Instead of using this inverse relationship approach, Cauchy and later Riemann introduced a fresh way to look at things by coming up with “area under the curve” approach. Partitioning the area under the curve and calling it as an integral was a breakthrough in disconnecting the inverse relationship thinking. Splicing the area under the curve to thin slices and adding up the area obtained by the splices, one inevitably stumbles on the continuity definition. Is continuity important for the Riemann sums to converge? Can it handle a function which has countable discontinuities? Can it handle a function which has uncountable discontinuities? These questions make the reader curious to move ahead in the chapter.

The chapter goes on to define Riemann integral in terms of lower sum of partitions and upper sum of partitions of the area under the curve. How does one check whether a function is integrable or not? This aspect is addressed again using the upper sum and lower sum of partitions. Thus the Riemann integral is very well explained for continuous functions, bounded functions in the initial part of the chapter. Cauchy dealt mostly with continuous/ bounded functions and thus the partition approach worked fine. However one immediately starts thinking about functions which are not continuous. The chapter then states that Riemann integral works well for countable discontinuities but could fails for some uncountable discontinuous functions (Dirichlet’s function). What is the criterion for Riemann integrability? This question urges the curious reader to move ahead in the chapter.

Instead of talking about Lebesgue theorem and addressing the criterion for Riemann integrability, the chapter takes a detour to understand a specific aspect of integration applied to sequence of functions. When does a mathematical manipulation such as integration respect the limiting process? This question is precisely addressed in this chapter. It says that if the sequence of functions is uniformly continuous then the one can switch the limit and integral sign.

Lebesque theorem is finally stated to connect the ideas of Riemann integral and continuity. The theorem basically states that “ Let f be a bounded function defined on the interval [a, b]. Then, f is Riemann-integrable if and only if the set of points where f is not continuous has measure zero.” If one goes over the statement carefully, all the confusion relating to the nature of functions which are integrable under Riemann procedure is put to rest. Basically a function with uncountable discontinuity is also Riemann integrable if the measure of those discontinuous sets is 0. The chapter concludes with an interesting example of a differentiable function which is not Riemann integrable specifically because it fails to adhere to Lebesgue theorem.

One inevitable conclusion of this chapter is that one cannot let integrability depend on the continuous nature of functions as there are a ton of derivatives which are not continuous. Hence the logical conclusion for any reader to move to the Lebesque space. Lebesque integral encompasses far more functions than what a Riemann integral can , and the good thing about it is that whenever Riemann integral exists, it converges to Lebesque integral. Example of integrating a Dirichlet function clearly shows the strength of Lebesque integral as compared to the Riemann integral.

image Takeaway :

This book is extremely well written , a feature generally not found for most of the math books. One’s understanding of the Real line, Functions that map on to Real Line would vastly be improved by going over carefully the arguments in various chapters. This book is undoubtedly the most accessible introduction to the world of “Real Analysis”.