PREFACE TO THE FIRST EDITION This text provides an introduction to the Lebesgue integral for advanced undergraduates or beginning graduate students in mathematics. It is also designed to furnish a concise review of the fundamentals for more advanced students who may have forgotten one or two details from their real analysis course and find that more scholarly treatises tell them more than they want to know. The Lebesgue integral has been around for almost a century, and the presentation of the subject has been slicked up con siderably over the years. Most authors prefer to blast through the preliminaries and get quickly to the more interesting results. This very efficient approach puts a great burden on the reader; all the words are there, but none of the music. In this text we deliberately unslick the presentation and grub around in the fundamentals long enough for the reader to develop some in tuition about the subject. For example, the Caratheodory def inition of measurability is slick—even brilliant—but it is not intuitive. In contrast, we stress the importance of additivity for the measure function and so define a set £ e (0,1) to be mea surable if it satisfies the absolutely minimal additivity condi tion: m(E) + m{E) = 1, where E^ = (0,1) — E and m is the outer measure in (0,1). We then show in easy steps that measurability of E is equivalent to the Caratheodory criterion, m(E n T) + m(E^ oT) = m(T) for all T. In this way we remove the magic from the Caratheodory condition, but retain its util ity. After the measure function is defined in (0,1), it is extended to each interval (n, n + 1) in the obvious way and then to the whole line by countable additivity. viii A PRIMER OF LEBESGUE INTEGRATION We define the integral via the famihar upper and lower Darboux sums of the calculus. The only new wrinkle is that now a measurable set is partitioned into a finite number of measurable sets rather than partitioning an interval into a fi nite number of subintervals. The use of upper and lower sums to define the integral is not conceptually different from the usual process of approximating a function by simple functions. How ever, the customary approach to the integral tends to create the impression that the Lebesgue integral differs from the Riemann integral primarily in the fact that the range of the function is par titioned rather than the domain. What is true is that a partition of the range induces an efficient partition of the domain. The real difference between the Riemann and Lebesgue integrals is that the Lebesgue integral uses a more sophisticated concept of length on the line. We take pains to show that both the Riemann-Darboux in tegral and the Lebesgue integral are limits of Riemann sums, for that is the way scientists and engineers tend to think of the integral. This requires that we introduce the concept of a con vergent net. Net convergence also allows us to make sense out of unordered sums and is in any case something every young mathematician should know. After measure and integration have been developed on the line, we define plane outer measure in terms of coverings by rectangles. This early treatment of plane measure serves three purposes. First, it provides a second example of the definition of outer measure, and then measure, starting with a natural geometric concept—here the area of a rectangle. Second, we show that the linear integral really is the area under the curve. Third, plane measure provides the natural concrete example of a product measure and is the prototype for the later development of general product measures. The text is generously interlarded with problems. The prob lems are not intended as an intelligence test, but are calculated to be part of the exposition and to lure the reader away from a passive role. In many cases, the problems provide an essential step in the development. The step may be routine, but the reader is nevertheless encouraged thereby to pause and become actively PREFACE TO THE FIRST EDITION Ix involved in the process. There are also additional exercises at the end of each chapter, and the author earnestly hopes that these will add to the reader's education and enjoyment. The author is pleased to acknowledge the help of Dick Bourgin, Bob Burckel, and Ken Ross, all of whom read the manuscript with great care and suggested many improvements in style and content. PREFACE TO THE SECOND EDITION The principal change from the first edition is the new one-shot definition of the Lebesgue integral. The integral is first defined for bounded functions on sets of finite measure, using upper and lower Darboux sums for finite partitions into measurable sets. This approach is designed to emphasize the similarity of the Lebesgue and Riemann integrals. By introducing countable partitions, we then extend the definition to arbitrary functions (bounded or not) and arbitrary sets (finite measure or not). This elegant touch, like many of my best ideas, was explained to me by A. M. Gleason. Many of the errors and crudities of the first edition have been corrected, and the author is indebted to Robert Burckel, R. K. Getoor, K. R S. Bhaskara Rao, Joel Shapiro, and Nicholas Young for pointing out assorted mistakes. In addition, several anony mous reviewers of the second edition made many helpful sug gestions. I feel confident, however, that there remain enough errors to challenge and reward the conscientious reader. Finally, the author wishes to express his gratitude to Susan Hasegawa and Pat Goldstein for their superb work with the typing and proofreading. H. S. Bear May 2001 1 THE RIEMANN-DARBOUX INTEGRAL We start by recalling the definition of the familiar Riemann- Darboux integral of the calculus, which for brevity we will call the Riemann integral. Our later development of the Lebesgue integral will closely parallel this treatment of the Riemann inte gral. We consider a fixed bounded interval [a, b] and consider only real functions f which are bounded on [a,b]. A partition P of [^, ^] is a set P = {XQ, xi, Xi,..., Xn} of points of [a, b] with a = XQ < Xi < X2 < ' " < Xn = b. Let /" be a given function on [a, b] with w < f(x) < M for all X e [a, b]. For each / = 1, 2,..., w let mi = inf{ f(x) : X/_i < x < Xi}, Mi = sup{ f(x) : Xi-i < X < Xi). In the usual calculus text treatment the infs and sups are taken over the closed intervals [x/_i,X/]. In our treatment of the Lebesgue integral we will partition [a, b] into disjoint sets, so we use the disjoint sets (x/_i, Xi) here. We are effectively ignoring a finite set of function values, f(xo), f(xi), fixi),..., f{Xn), and in so doing we anticipate the important result of Lebesgue that 2 A PRIMER OF LEBESGUE INTEGRATION function values on a set of measure zero (here the set of parti tion points) are not relevant for either the Riemann or Lebesgue integral. The lewder sum L{f, P) and the upper sum U(f, P) for the function f and the partition P are defined as follows: n L{f, P) = ^mi{Xi -x/_i), i=l Clearly m mi and m![ > mi so the sum of the first two terms in L( f, Q) exceeds the first term of L( f, P): m[(x^ — Xo) + m^[(xi — x*) > mi(xi — Xo). The remaining terms of L( /*, Q) and L( /*, P) are the same, so L(f, Q) > L(f, P). We can consider any refinement Q of P as obtained by adding one point at a time, with the lower sum increasing each time we add a point. The argument for the upper sums is similar, ill Proposition 2. Every lower sum is less than or equal to every upper sum, as the geometry demands. Proof. If P and Q are any partitions, and R = P U Q is the common refinement, then L{f, P) < L(f, R) < U(f, R) < U{f, Q). ill 4 A PRIMER OF LEBESGUE INTEGRATION Proposition 3. f is integrable on [a, b] if and only if for each 8>0 there is a partition P of [a,b] such that U{fP) — L{fP)<8. Proof This useful condition is equivalent to sup L{f P) = inf U( /*, P) in view of Proposition 2. ill Proposition 4. If f is integrable on [a, b] and [a, ^] 0 and let P be a partition of [a, b] such that U(f P) — L(f P) < 6. We can assume that a and ^ are points of P, since adding points increases L( /*, P) and decreases U(f P) and makes their difference smaller. If PQ consists of the points of P which are in [a, ^], then PQ is a partition of [a, jS]. Note that U{f P) - Uf P) = X](M, - m,)(x, ~ x,_i) (2) and U( f PQ) — L( f PQ) is the sum of only those terms such that Xi-i and Xi e PQ. Since we omit some non-negative terms from (2)togetU(/',Po)-L(/*,Po), U(f Po) - L{f Po) < U(f P) - Uf P) < 8, ill Problem 1. If f is integrable on [a, b]^ then —f and | f\ are integrable on [a, b], and Xf (-/") = -/.V, |xf/-| < Xf | f\. II Problem 2. li a < c < b then / is integrable on [a, c] and on [c, /?] if and only if f is integrable on [a,b]. In this case rc pb rb f+ f= f. •"•Hil l Ja Jc Ja Problem 3. If /"is integrable on [a, b] and g = /^ except at a fi nite number of points, then g is integrable and f^ g = J^ f. ""HI Problem 4. li a = XQ < xi < - -- < Xn = b and f is defined on [a, b] with /"(x) = yi for x € (x/_i, X/), then f is integrable and - x,_i). (Note that it is immaterial how f is defined on the %/, by the preceding problem.) ""HI 1 THE RIEAAANN-DARBOUX INTEGRAL 5 Problem S. We say that g is a step function on [a, b] if there is a partition a = XQ < Xi < xi < - - < Xn = b such that g is constant on each (x/_i, X/). (By the preceding problem, step functions are integrable with the obvious value for the integral.) Show that if f is integrable on [a, b] there are step functions gn and hn with {gn} increasing and {hn} decreasing and gn < f < hn for all n and all x, and \im J^gn = HmJ^hn = Xf f- (Note: We will be able to show later that gn alid hn approach f except possibly on a set of measure zero.) ""HI Proposition 5. If f is continuous on [a, b], then f is integrable on [a, b\. Proof. If f is continuous on [a,b\ then f is uniformly con tinuous. Hence iis > 0 there is 5 > 0 so that \f{x) — f{x')\ < s whenever \x — x^\ < S.li P is a partition with xi — Xi-\ < 8 for all /, then M/ — mi < s for each /, so U(f P) - L(f P) = J2(M, - m,){x, - x,_i) < 8 Y^{Xi - Xi-i) = s(b-a), ill Problem 6. (i) If f is continuous on [a, b] except at a (or b) and f is bounded on [a,b]^ then f is integrable on [a,b]. (ii) If f is bounded on [a, b] and continuous except at a finite number of points, then f is integrable on [a,b]. (iii) Suppose f is bounded on [a, b] and discontinuous on a (possibly infinite) set £. Assume that for each ^ > 0 there are disjoint intervals (ai, b\),..., {a^, b^) contained in [a, b] such that E c (ai, bi) U ' " U (a^, ^N) and TJiLxih ^ ^i) < ^- Show that f is integrable. ""HI So far we have the integral /^ f defined only when [a, b] is a bounded interval and f is bounded on [a,b]. Now we extend the definition to certain improper cases; i.e., situations where the interval is unbounded, or f is unbounded on a bounded interval. Typical examples of such improper integrals are / —z^dx and / -dx. Jo ^x Jo 1 + x^ A PRIMER OF LEBESGUE INTEGRATION In both these examples the integrand is positive and the defini tion of the integral should give a reasonable value for the area under the curve. The definitions of the integrals above are / —zz: dx = lim / —7= dx, Jo y/x £-^0+ Js ^X coo \ fb 1 / :; :^ dx = lim / dx. Jo 1 + X^ b-^oQ Jo 1 -\- X^ Both these limits are finite, so both functions are said to be (improperly) Riemann integrable on the given interval. The Lebesgue definition of the integral will give the same values. In general, if f is integrable on[a + s.b] for all £ > 0, but f is not bounded on [a, b\ (i.e., not bounded near a)^ we define b f= limj f, J a s—^0+ Ja-\-s provided this limit exists. Similarly, if f is integrable on every interval [a, b] ior b > a we define POO rb J a b—^00 J a when the Hmit exists. Similar definitions are made for Xf f if f is unbounded near /?, and for /^^ /" if /* is integrable on [a, b] for all a < b. These definitions lend themselves to the calculations of ele mentary calculus, but do not coincide with the Lebesgue defi nition if f is not always positive or always negative. For ex ample, if f is {—ly/n on [n,n + 1), n = 1,2,..., then f is improperly Riemann integrable on [1, oc). We will see later that f is Lebesgue integrable if and only if | /*| is Lebesgue in tegrable. Hence the above function is not Lebesgue integrable since ^ - = 00. Problem 7. Show that /o~ '-^dx exists. "Hlll Problem 8. (i) Exhibit a g on [0, oo) such that \g(x)\ = 1 and /o~ g exists.

A Primer of Lebesgue Integration

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