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Lebesgue Integration is usually introduced in either one of the two ways , in most of the books. Either using measure theory OR  skipping measure theory altogether and using approximating functions like step functions, monotone functions etc. The second approach is usually taken so that one can actually understand all the concepts of Lebesgue integral with out going through measure theory. However that approach would make it difficult for a reader to connect Lebesgue Integral and concepts relating to stochastic processes. This is one of rare books which combines both the approaches. It introduces measure theory as well as approximations in the book and uses them in various proofs, wherever the introduction makes the proof easier to follow. Let me attempt to summarize this book.

Chapter 1 : Historical Highlights

The first chapter gives a historical perspective of “Integration”. Right from 408 B.C.E till 1904 , there were lots of brilliant people who contributed to the understanding of the process of integration. The main personalities mentioned are

  • Exodus (408 B.C.E)
  • Hippocrates(430 B.C.E)
  • Archimedes(287- 212 B.C.E),
  • Pierre Fermat (1601 – 1665)
  • Leibnitz (1646 – 1716)
  • Cauchy(1789-1857)
  • Bernhard Riemann(1826-1866),
  • Emile Borel(1871-1956)
  • Camille Jordan(1838-1922)
  • Giuseppe Peano(158-1932)
  • Henri Lebesgue(1875-1941)
  • William Young(1863-1942).

Exodus and Archimedes pioneered the technique called Exhaustion. Fermat, Newton and Leibnitz were masters in providing anti-derivatives for specific functions. But it was Cauchy who laid out a radically different view of analysis, i.e using limits. He described everything from the view of limits. Karl Weierstrass provided the algebraic language needed to express the concepts like limits, continuity. It was followed up by Reimann and Darboux who generalized the work of Cauchy and extended the concepts to integrals for bounded functions. However there were a lot of problems with Riemann Calculus as well as FTC (Fundamental theorem of Calculus) using Riemann Integral. Most of the pathological examples perplexed mathematicians and at the same time raised serious doubts about Riemann Calculus. Peano-Jordan used the concept of inner content and outer content to find integrals. The biggest limitation with Jordan measure was that it was building measure using finite partitions. They seemed to be a ray of hope but their definition missed an important aspect of measurability, i.e content of rationals in [0,1] + content of irrationals in [0,1] > 1 which flew against the usual requirement that measure of disjoint sets needs to be equal to measure of the union of disjoint sets. Finally Lebesgue used the concept of countable covers and introduced outer measure and Lebesgue measure to create a breakthrough in mathematics – Lebesgue Integration. All these developments are summarized in the book with the help of the following visual.

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Chapter 2 : Preliminaries

This chapter mentions all the concepts needed to understand the three core chapters of the book, Lebesgue Measure, Lebesgue measurable function and Lebesgue Integral.

It starts off by introducing basic operations on sets like union, intersection, complements etc. Then it introduces monotone sequence of sets and defines limit superior of a sequence of sets (lim sup) , limit inferior of a sequence of sets ( lim inf). Well, if one is reading this book with absolutely no exposure to set theory, one might wonder the importance of limiting operations of sets. I mean, in any analysis text , one usually comes across lim inf and lim sup for sequences. Lim inf and Lim sup are related to the limit of sequence. However the lim inf and lim sup for a sequence of sets become extremely important in the context of probability. If you have events A1, A2, A3,…An then lim sup (An) means An infinitely often , and lim inf(An) means An almost always.

Functions are then introduced. This is followed up by explaining concepts of cardinality. Terms such as finite set, infinite set, countable, uncountable are explained. For a first time reader of Lebesgue theory, it is difficult to understand the importance of cardinality concepts to measure theory. For all such readers, they would gain immense understanding by going over the historical development of measure theory. Many mathematicians tried to wrestle with the concept of measure considering only finite intersections and finite unions and thus limiting the collection of sets for which measure and integral could be defined. It was Lebesgue/Borel who were the first mathematicians to combine the concepts of cardinality with nowhere dense concepts( Baire’s contribution).Books such as these are good reads after you have gone through the struggles that various mathematicians went through in defining mathematical objects.With out historical background, these books are difficult to follow. But if you have some historical background , Books such as these are very illuminating. Anyway, coming back to the summary, the countability and uncountability are concepts which are key to understanding Lebesgue measure. This is followed up with some concepts relating to the construction of Real line. Nothing too difficult to follow . Just some basic  stuff like least upper bound, greatest lower bound, supremum, infimum definitions are mentioned. Personally I felt Abbot’s book “Understanding Analysis”, does a fantastic job of introducing the Real Line. If you have worked through Abbott, you can quick read this entire chapter.

The highlight of this chapter is the concept relating to sequence of real numbers.lim sup and lim inf of sequences are introduced with pictures. I am always on the lookout for books that explain things using images, for our minds are tuned to visual processing. 80% of our mind is dedicated to visual processing and hence images, more so in math, can make you remember stuff for a LOONG time. Cauchy sequences are then introduced in the book. Well, Cauchy sequences are the life blood to everything in analysis, from as simple as convergence of a series, to understanding metric spaces. Nowadays I tend to reflect on my sheer ignorance of these concepts for a very long time in my life. I had crammed up stuff, learnt to crack a few exams without any good understanding of “Real Analysis”. My first face-off with Real Analysis came out of sheer necessity. I had to teach Calculus to undergraduates during my masters, to take care of my daily expenses. You can’t teach Calculus without a thorough understanding of Analysis. If someone had asked me, whether a sequence can fill up all the numbers between (0,1) back then, in all probability I would have said yes. While in my first semester of my masters, I taught the calculus course to a few students and it was a disaster to say the least. I knew all the formulae, methods, tricks, ways to solve problems relating to limits, differentiation, integration etc. I taught them to crack exams . But my fundamentals were extremely shaky. It was the first time I painfully realized the importance of SOUND fundamentals in Real Analysis. Slowly over the course of time, I improved my teaching skills and brought in the discussion of basic principles in to the class to appreciate the beauty behind all the formulae. Ok..I am deviating from the intent of the post.

Coming back to the summary of this book, the chapter subsequently gives a quick recap of some topological definitions / terms / theorems such as Open sets, Closed Sets, Compact Sets, Limit points, Open Cover, Finite Sub cover, Heine-Borel theorem, Bolzano – Weierstrass theorem. Obviously this chapter provides a good recap ONLY. This is NO GOOD for a beginner. If you want to know thoroughly the topological terms and concepts, you must pick up a book with  enough coverage and problems in it. Math can only be learnt from doing. You cannot read math / listen to math .You understand ONLY when you DO math and probably TEACH math. Books by Abbott / Rudin / Victor Byrant would be the best way to start off on understanding the principles. I found Rudin very tough to go through and chose Abbott’s book instead. For topology, I found Victor Byrant’s book on Metric spaces to be a nice learning resource. I remember the first time when I  came across these terms and really felt clueless. If I want to integrate, why should I know about compactness? But slowly I realized that there was no shortcut to understanding Lebesgue integral or for that matter anything in calculus without actually slogging through Real Analysis and understanding the theorems behind it.

The chapter then talks about Continuous functions and Differentiable functions in a breeze. As I start spending more time on math, I have started to realize that you cannot work on one book at a time. You have got to read at least 2-3 books simultaneously. For example , a proof of theorem might be using Nested Interval property in one book while it might be using Bolzano – Weierstrass theorem to prove the same in another book. A third book might contain the motivation behind the formulation of a theorem. Only when you see things from at least two or three perspectives, you get a decent idea of math principles.

The chapter ends with the discussion of a sequence of functions and uniform convergence of a sequence of functions. Is the limit of derivative same as derivative of the limit? Is the limit of the Sum the same as Sum of limit? Is the limit of Integral same as Integral of limit? These are some of the crucial questions which lead mathematicians to make real analysis more precise. Bressoud’s books Radical Introduction of Real Analysis and Radical Introduction to Lebesgue theory start off with Fourier Series example that perplexed mathematicians for over a century. Fourier series is an example where you can approximate a constant with infinite trigonometric series. You differentiate each term of the series and you get some non sense result. Nobody had a clue the reason for such anomalous behaviour when it was first introduced. Perplexing behaviour of Fourier Series was one of the biggest motivations for mathematicians to bring out the rigor in defining terms like differentiation and integration.

Overall this chapter makes tremendous read for someone who is already familiar with the concepts of real analysis and historical developments of real analysis. My favourite in this genre is “The Calculus Gallery”. I have read it about 3-4 times. Every time I read, I find something that makes me wonder at the immense achievements of the people behind Calculus and Lebesgue Integration.

Chapter 3 : Lebesgue Measure

Chapter 3 introduces Lebesgue measure in a systematic manner. Instead of diving right in to the definition as most books on measure theory do, the approach taken here is very interesting. It lists down a set of 8 desirable attributes of any measure for a subset of R. It then shows the conditions that Lebesgue outer measure satisfies. Out of the laundry list of desirable attributes, there are two attributes which outer measure fails to possess. One is that you cannot have all the subsets of R. Second is that the outer measure does not satisfy countable additivity for a collection of disjoint sets. Outer measure actually carries countable subadditivity property. Hence Lebesgue introduced a new measure which satisfies all the desirable attributes except the attribute that it is applicable to all subsets of R. Such sets for which countable additivity holds good are called Lebesgue measurable sets. There are some sets in R which are not Lebesgue measurable, whose construction is very difficult. In this context, Caratheodory’s condition comes in handy, who came up with a simple condition to check the measurability of the set. One can easily see that null set and R is Lebesgue measurable set. At the same time 2^R , all the subsets of R are not all measurable.

The book then logically introduces sigma algebra, a collection of sets which satisfy certain properties such as countable unions, countable intersections, complements and limits are all present in the collection. One “not to be missed” point is that “Once you are inside a sigma algebra, it is hard to get outside”. All intervals are Lebesgue measurable. However it is desirable to have much more than simple intervals. In any case collection of open intervals is not a sigma algebra. This is where Borel sigma algebra comes in which is nothing but the smallest sigma algebra of all the sigma fields generated by the intervals. Borel Sigma Algebra happens to be a subset of Lebesgue measurable sets. One of the highlights of this book is good visuals. The following visual summarized the structure of all the discussed concepts in this chapter.

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The chapter ends with discussing the topological structure of a measurable set. The discussion helps in understanding that an open set can enclose a measurable set and a closed set can almost exhaust a measurable set. Meaning, Lebesgue measurable sets are “almost open”, “almost closed”

Chapter 4 : Lebesgue Measurable Function

A measurable function is one whose preimages are measurable sets. Functions in the Lebesgue world  are not “measured” but integrated. Besides the definition, various forms of functions are explored in this chapter to check their measurability. With the wide range of applicable functions, the inevitable conclusion is that Lebesgue by defining measurable functions via Measurable sets had cast his net very very wide. The beauty of measurable functions is related to sequence of functions. If one takes a sequence of functions which are measurable, the function to which the sequence converges is also measurable. Basically it means that we cannot escape the world of measurable functions even by taking point wise limits. This is not true about bounded, Riemann-integrable functions, functions in Baire class 1. In those situations, the family of functions was too restrictive to contain all of its pointwise limits. Measurable functions, by contrast, are strikingly inclusive.

The highlight of this chapter is the careful construction of simple functions to approximate a measurable function. The detailed description of the algo for approximation , is by far the best description that I have found amongst all the books on Lebesgue. Here’s a visual to demonstrate the procedure. For example a function like y = x^2  and y = 1/x are approximated using a series of monotonically increasing

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The chapter ends with discussing almost uniform convergence by proving Egoroff theorem. This theorem states that if {fk} is a sequence of measurable functions converging almost everywhere to f, then on a subset of that space where the sequence converges, the convergence is uniform and hence the name almost uniform convergence. I feel this part could have been improved by taking the Bressoud-II approach where the author discusses, convergence in measure, Riesz’s theorem etc. At least by discussing all aspects of convergence like pointwise, almost everywhere, almost uniform convergence, convergence in measure, it is more likely that one gets an nice overall idea. In that sense, this final part of the chapter is a bit let down as it randomly introduces almost uniform convergence concept and ends it abruptly.

Chapter 5 : Lebesgue Integral

The chapter starts off with a basic introduction to Riemann Integral. In most of the real analysis books you find the introduction to Riemann using partitions that eventually concludes with an elegant formulation using Darboux integrals. This book however takes a different approach, the step function approach. The definition of lower Riemann Integral, Upper Riemann Integral, the properties of Riemann Integral like homogeneity, additivity, monotonicity, additivity on the domain and mean value are all derived using step functions. Basically it’s an old wine in new bottle approach but somehow the proofs are much easier to follow. The proofs using step function approach are easier to follow than using let’s say the sup/inf of lower/upper sums that one usually comes across. Even the FTC(Fundamental theorem of Calculus) , the anti-derivative part and the evaluation part are both derived using the step function approach. One basic takeaway from this introduction is that you can formulate Riemann Integrability condition purely based on step functions and that is an intuitively easy way to understand the condition.

The chapter then talks about Lebesgue integral for Bounded functions on sets of finite measure. Homogenity, Additivity, Monotonicity, Additivity on Domain hold good for Simple functions. One of the best proofs in the book is for the theorem, ” Let f be a bounded function on the interval [a,b] If f is Riemann integrable on [a,b] then f is Lebesgue integrable on [a,b]. ” Since step functions are a subset of simple functions, it is proved using the fact that you can sandwich a Lebesgue integrability condition with in Riemann Integrability condition. Subsequently, Homogenity, Additivity, Monotonicity, Additivity on Domain properties are explored for Lebesgue integrals.

How does one decide whether a function is Lebesgue integrable or not ? For functions whose domain is of finite measure, it is easy. If you can sandwich the function between two simple functions such that the difference between their integrals can be made as small as one wants, then the function is Lebesgue integrable. Another obvious way to decide is whether the function one is trying to integrate , is measurable or not. Can there be a Lebesgue integrable function which is not measurable ? No, says the proof of a theorem from this book.

Books such as these should not be missed for a simple reason. You get both intuition and rigorous explanation of results. It is a standard thing that a Lebesgue measurable function can be approximated using Monotone Sequence of simple functions. I always used to wonder, the reason why one should  not , in a dumb fashion generate sequence of simple functions and basically use it for integration purpose. However the book makes the reader realize that it is not always a nice way to do that way. Some of the examples in the book show that it is technically extremely cumbersome to work with sequence of simple functions to approximate the integral. Instead the book says that it is elegant to approximate the lebesgue integral with a sequence of monotone measurable functions. Again examples come before the theory so that reader is well motivated to see the relevance of the theory. Fatou’s lemma is usually used in proving Monotone Convergence theorem. But this book takes a different approach where Monotone Convergence Theorem is used to prove Fatou’s lemma.

Finally the chapter extends it to integration to all measurable functions and not just nonnegative functions. One must be very clear with the difference between the two statements, “Lebesgue integral of f is blahblah“ and “ f is Lebesgue integrable on E and equals blah blah blah”. The finiteness for the integral is crucial to call a function Lebesgue integrable on a set E. So in one sense f is Lebesgue integrable on E only if |f| is integrable. This is very different from Riemann case where there is a chance the negative part of function might cancel some part of positive summands and make the Riemann integral converge. Not in the case of Lebesgue. This means that there would some unbounded indefinite integrals which are Riemann Integrable but not Lebesgue integrable on a measurable set. The chapter ends with a discussion on Lebesgue Dominated Convergence theorem which gives a very broad sufficient (note : it is not a necessary) condition for interchanging limits and integral sign for a sequence of functions. One of the highlights of the book is the detailed derivation of cantor set using ternary expansion, and the construction of devil’s staircase function. 

image Takeaway:

The book is very focused in treatment. Instead of covering a laundry list of topics related to Lebesgue theory , the book focuses on three core concepts i.e Lebesgue Measure, Lebesgue Measurable function and Lebesgue integral and explains them very clearly. The author uses visuals to explain concepts, thus making the content very interesting. The highlight of this book is the conversational tone that the author adopts in explaining stuff, thus enabling the reader connect seemingly disparate theorems/concepts/problems from the book.

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