# Measure Theory and Lebesgue Integration. Joshua H. Lifton

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Measure Theory and Lebesgue Integration Joshua H. Lifton Originally published 31 March 1999 Revised 5 September 2004

2 bstract This paper originally came out of my 1999 Swarthmore College Mathematics Senior Conference. I ve made minor touch-ups to make it more presentable. This paper begins where the Swarthmore College Mathematics and Statistics course Math 47: Introduction to Real nalysis left off. Namely, basic measure theory is covered with an eye toward exploring the Lebesgue integral and comparing it to the Riemann integral. Knowledge of the notation and techniques used in an introductory analysis course is assumed throughout.

3 Contents cknowledgments ii 1 How to Count Rectangles: Review of Integration Riemann Revisited Shortcomings of Riemann Integration New Way to Count Rectangles: Lebesgue Integration Measure Theory Measure Outer and Inner Measure Shortcomings of Outer and Inner Measure Measurable Sets & Measurable Functions Measurable Sets Measurable Functions Simple Functions The Lebesgue Integral Integrating Bounded Measurable Functions Criteria for Integrability Properties of the Lebesgue Integral Integrating Unbounded Measurable Functions Comparison of Lebesgue and Riemann Integrals Summary of Events Convergence of the Lebesgue Integral Convergence of the Riemann Integral Final Comparison References 24 i

4 cknowledgments Many thanks to all members of the Swarthmore College Mathematics and Statistics Department for their encouragement and tolerance of my mathematics education. ii

5 Chapter 1 How to Count Rectangles: Review of Integration 1.1 Riemann Revisited The development of the integral in most introductory analysis courses is centered almost exclusively on the Riemann integral. This relatively intuitive approach begins by taking a partition P = {x 0,..., x n } of the domain of the real-valued function f in question. Given P, the Riemann sum of f is simply n (x i x i 1 ) f(c i ) where x i 1 < c i < x i. i=1 The integral of f, if it exists, is the limit of the Riemann sum as n. 1.2 Shortcomings of Riemann Integration lthough the Riemann integral suffices in most daily situations, it fails to meet our needs in several important ways. First, the class of Riemann integrable functions is relatively small. Second and related to the first, the Riemann integral does not have satisfactory limit properties. That is, given a sequence of Riemann integrable functions {f n } with a limit function f = lim f n, it does not necessarily follow that the limit function f is Riemann integrable. Third, all L p spaces except for L fail to be complete under the Riemann integral. Examples 1.1 and 1.2 illustrate some of these problems. 1

6 Example 1.1 Consider the sequence of functions {f n } over the interval E = [0, 1]. { 2 n 1 if f n (x) = 2 x 1 n 2 n 1 0 otherwise The limit function of this sequence is simply f = 0. In this example, each function in the sequence is integrable as is the limit function. However, the limit of the sequence of integrals is not equal to the integral of the limit of the sequence. That is, 1 lim 0 f n (x) dx = 1 0 = 1 0 lim f n(x) dx. We will return to this function as an example of how the Lebesgue integral fails as well. Example 1.2 Consider the sequence of functions {d n } over the interval E = [0, 1]. { 1 if x {r n } d n (x) = 0 otherwise where {r n } is the set of the first n elements of some decided upon enumeration of the rational numbers. Each function d n is Riemann integrable since it is discontinuous only at n points. The limit function D = lim d n is given by { 1 if x is rational D(x) = 0 if x is irrational This function, known as the Dirichlet function, is discontinuous everywhere and therefore not Riemann integrable. nother way of showing that D(x) is not Riemann integrable is to take upper and lower sums, which result in 1 and 0, respectively. 1.3 New Way to Count Rectangles: Lebesgue Integration n equally intuitive, but long in coming method of integration, was presented by Lebesgue in Rather than partitioning the domain of the function, as in the Riemann integral, Lebesgue chose to partition the range. Thus, for each interval in the partition, rather than asking for the value of

7 the function between the end points of the interval in the domain, he asked how much of the domain is mapped by the function to some value between two end points in the range. See Figure 1.1. range range domain domain Figure 1.1: Two ways to count rectangles partitioning the range as opposed to partitioning the domain of a function. Partitioning the range of a function and counting the resultant rectangles becomes tricky since we must employ some way of determining (or measuring) how much of the domain is sent to a particular portion of a partition of the range. Measure theory addresses just this problem. s it turns out, the Lebesgue integral solves many of the problems left by the Riemann integral. With this in mind, we now turn to measure theory.

8 Chapter 2 Measure Theory Measure theory is a rich subject in and of itself. However, we present it here expressly for the purpose proposed at the end of 1.3; to define the length of an arbitrary set so as to formalize the idea of the Lebesgue integral. s such, only the very basics of measure theory are presented here and many of the rote proofs are left to the reader. 2.1 Measure Given an interval E = [a, b] and a set S of subsets of E which is closed under countable unions, we define the following. Definition 2.1 set function on S is a function which assigns to each set S a real number. Definition 2.2 set function µ on S is called a measure if the following properties hold. Semi-Positive-Definite: 0 µ() b a for all S. Trivial Case: µ( ) = 0. Monotonicity: µ() µ(b) for all, B S, B Countable dditivity: if = n=1 n, then µ() = n=1 µ( n), where n S for n = 1, 2,... and n m = for n m. s can be seen from the definition, the concept of measure is a general one indeed. 4

9 Example 2.3 Examples of measure abound. To begin at the beginning, consider the trivial measure: µ() = 0 for all S. Example 2.4 Let {x 1,..., x n } E be a finite set of points and E = [a, b]. Let a < x i < x j < b for all i < j. Consider the set = E/{x 1,..., x n }, the interval E without the points x 1,..., x n. Let our measure be such that the measure of any interval with endpoints x < y is y x. Then, m() = m([a, x 1 ) (x 1, x 2 )... (x n 1, x n ) (x n, b]) = m([a, x 1 )) + m((x 1, x 2 )) m((x n 1, x n )) + m((x n, b]) = (x 1 a) + (x 2 x 1 ) (x n x n 1 ) + (b x n ) = b a = m(e). Notice that the only complication which may arise in this example is how we define our set of subsets, S. We will explore this further in Outer and Inner Measure To keep us grounded in reality, we would like to use a measure which puts our intuitive notion of length at ease. Thus we define the following. Definition 2.5 The outer measure of any interval I on the real number line with endpoints a < b is b a and is denoted as m (I). We would like to generalize this definition of outer measure to all sets of real numbers, not just intervals. To do this, we use the following theorem, upon which the remainder of our work depends heavily. Theorem 2.6 Every non-empty open set G R can be uniquely expressed as a finite or countably infinite union of pairwise disjoint open intervals. Proof Idea: Let x G and consider the open interval I x = (a x, b x ) constructed from a x = glb{y (y, x) G} and b x = lub{z (x, z) G}. I x is called the component of x in G. G = x G I x is exactly the decomposition of G we are looking for. It is straight-forward to show that, given

10 x 1, x 2 G, either I x1 and I x2 are disjoint or they are equal. The countability of this collection of open intervals follows easily from their disjointness; simply pick one rational number from each interval and use it to label that interval. Since the intervals are disjoint, no two will have the same label and since the rationals are countable, so will be the intervals which they label. Uniqueness of the decomposition is, as usual, a straight-forward proof by contradiction. With this theorem, it is possible to extend the definition of outer measure to open sets of real numbers. Definition 2.7 The outer measure m (G) of an open set G E is given by i m (I i ) where the I i form the unique decomposition of G into a finite or a countably infinite union of pairwise disjoint open intervals. See Theorem 2.6. From this directly follows a definition of outer measure for any set. Definition 2.8 The outer measure m () of any set R is given by glb{m (G) G and G open in E}. Now that outer measure is well-defined for arbitrary subsets of E, we turn to a closely related measure, inner measure. Definition 2.9 The inner measure of any set E, denoted m (), is defined as m (E) m (E/), where E/ is the compliment of with respect to E. Intuitively, the inner measure is in some ways measuring the same thing as the outer measure, only in a more roundabout way. We cannot take it for granted, however, that the inner and outer measures of any given set are the same, although we are very interested in the cases when they are, as we shall soon see. For now, though, let us state without proof some simple observations regarding inner and outer measure. Lemma 2.10 The measures m and m both exhibit monotonicity. That is, given B E, it follows that, m () m (B) m () m (B)

11 Lemma 2.11 Given a set E, it follows that m () m (). Lemma 2.12 The outer measure m exhibits subadditivity. That is, whenever { n n = 1, 2,...} is a set of subsets of E, then m ( n ) n=1 m ( n ). 2.3 Shortcomings of Outer and Inner Measure Now that we have some concrete measures to work with, we must examine whether they satisfy our needs. The first shortcoming we discover is presented below. Theorem 2.13 The outer measure, m, is not countably additive on the set of all subsets of E. Proof Idea: By construction. It is possible to construct pairwise disjoint sets n such that E = n=1 n and m (E) n=1 m ( n ). This is rather disappointing. It would be nice to have a measure which is defined on all subsets of E and still satisfies our intuition enough to use it as the basis for the development of the Lebesgue integral. However, it turns out that the set of subsets of E on which the outer measure m is countably additive is large enough to make it worth our while to continue using the outer measure. We discuss this point in Chapter 3. n=1

12 Chapter 3 Measurable Sets & Measurable Functions 3.1 Measurable Sets Let s look at a certain subset of all the sets on which inner and outer measure are well-defined the set of Lebesgue measurable sets. Definition 3.1 set E is Lebesgue measurable, or measurable, if m () = m (), in which case the measure of is denoted simply by m() and is given by m() = m () = m (). straight-forward extension of this definition applies to unbounded sets, Definition 3.2 The measure for an unbounded set is defined simply as, m() = lim m( [ n, n]). Lemma 3.3 Measurable sets have the following properties. set E is measurable if and only if m () + m (E/) = b a, where a and b are the endpoints of the interval E. set E is measurable if and only if its complement E/ is measurable. Proof: The lemma follows directly from the definition of measurable (3.1). 8

13 Example 3.4 Referring back to Example 2.4, we see that a set consisting of an interval which is missing a finite number of points has the same outer measure as the interval itself. That is, m (E/{x i,..., x n }) = m (E). Thus, by Lemma 3.3, E/{x i,..., x n } is measurable if and only if m ({x i,..., x n }) = 0. The following theorem takes us back to where we left off at the end of 2.3. Theorem 3.5 The outer measure, m, is countably additive on the set of all measurable subsets of E. That is, whenever { n n = 1, 2,...} is a set of measurable subsets of E, then m ( n ) = n=1 m ( n ) Proof Idea: lthough the proof of this theorem is not difficult, it is somewhat lengthy in its technical detail and is therefore left for the enrichment of the reader. So, knowing that we intend to define the Lebesgue integral in terms of the outer measure, it seems that the size of the class of Lebesgue integrable functions may somehow be limited by the fact that, as far as we know, the outer measure is only defined on measurable sets. We will return to this idea in Chapter 4. n=1 3.2 Measurable Functions Definition 3.6 Let be a bounded measurable subset of R and f : R a function. Then f is said to be measurable on if {x f(x) > r} is measurable (as a set) for every real number r. There are many equivalent definitions of measurable which follow similar lines. In this definition we see the beginnings of being able to count the rectangles created by partitioning the range of a function rather than the domain. See Figure 3.1.

14 rrange domain Figure 3.1: The function f is measurable if the shaded region of the domain is measurable as a set for all choices of the real number r. Example 3.7 Consider again the sequence of functions over E = [0, 1] defined in Example 1.1, { 2 n 1 if f n (x) = 2 x 1 n 2 n 1 0 otherwise We want to show that each f n is a measurable function. There are three cases to consider, corresponding to three possible choices for the real number r. They are, r 2 n : The set {x E f n (x) > r} is the null set and is therefore measurable. 0 r < 2 n : The set {x E f n (x) > r} is the closed interval [ , ] and is therefore measurable (with a measure of n 2 n 1 2 ). n r < 0 : The set {x E f n (x) > r} is the entire interval E and is therefore measurable. Thus, each f n is a measurable function.

15 Example 3.8 Consider again the sequence of functions over E = [0, 1] defined in Example 1.2, { 1 if x {r n } d n (x) = 0 otherwise We want to show that each d n is a measurable function. There are three cases to consider, corresponding to three possible choices for the real number r. They are, r 1 : The set {x E d n (x) > r} is the null set and is therefore measurable. 0 r < 1 : The set {x E d n (x) > r} is the set of the first n rational numbers (in a decided upon enumeration) and is therefore measurable (with a measure of 0). r < 0 : The set {x E d n (x) > r} is the entire interval E and is therefore measurable. Thus, each d n is a measurable function. The above examples naturally bring up the question of whether or not the limit function of a sequence of measurable functions is itself a measurable function. Clearly, the limit function in Example 3.7 is measurable since it is the trivial function. But what of the Dirichlet function in Example 3.8? The following lemma answers this question. Lemma 3.9 If each function in a sequence {f n } is measurable on a set and if f is the pointwise limit function of {f n }, then f is measurable on as well. Proof: The proof of the lemma requires a close examination of the definitions of limit and measurable, as follows. Let x and r R such that f(x) > r. Let p be a natural number such that f(x) > r + 1/p. Then, by definition of limit, there exists a natural number N such that for all n > N, f n (x) > r + 1/p. Thus, This implies that {x f(x) > r} = f(x) = lim f n(x) > r + 1/p > r. p=1 N=1 n=n+1 {x f n (x) > r + 1/p}. Since this set is measurable and r was arbitrary, it follows that f is measurable.

16 3.3 Simple Functions Similar to the way which step functions are put to use in the development the Riemann integral, our development of the Lebesgue integral will make use of a pedestrian class of measurable functions, aptly named simple functions. Definition 3.10 simple function f : R is a measurable function which takes on finitely many values. We ve already seen simple functions in Examples 1.1 and 1.2. The utility of simple functions becomes apparent in the following theorem. Theorem 3.11 function f : R is measurable if and only if it is the pointwise limit of a sequence of simple functions. Proof Idea: ( ) The forward direction of the proof involves constructing a sequence of simple functions whose limit is the given measurable function f. This construction is omitted for brevity. ( ) The reverse direction is given by Lemma 3.9. Example 3.12 Example 3.8 combined with Theorem 3.11 (or even just Lemma 3.9) tells us that the Dirichlet function is indeed measurable.

17 Chapter 4 The Lebesgue Integral 4.1 Integrating Bounded Measurable Functions We introduce the Lebesgue integral by first restricting our attention to bounded measurable functions. The approach here is almost identical to that used in constructing the Riemann integral except that here we partition the range rather than the domain of the function. Let f : R be a bounded measurable function on a bounded measurable subset of R. Let l = glb{f(x) x } and u > lub{f(x) x } where u is arbitrary insofar as it is greater than the least upper bound of f on. s with the Riemann integral, we ll define the Lebesgue integral of f over an interval as the limit of some Lebesgue sum. Definition 4.1 The Lebesgue sum of f with respect to a partition P = {y 0,..., y n } of the interval [l, u] is given as L(f, P ) = n yi m({x y i 1 f(x) < y i }) i=1 where y i [y i 1, y i ] for i = 1,..., n and f is a bounded measurable function over a bounded measurable set R. This is the new way to count rectangles; the yi is the height of the rectangle and the m({x y i 1 f(x) < y i }) serves as the base of the rectangle. The definition of the actual Lebesgue integral is virtually identical to that of the Riemann integral. 13

18 Definition 4.2 bounded measurable function f : R is Lebesgue integrable on if there is a number L R such that, given ɛ > 0, there exists a δ > 0 such that L(f, P ) L < ɛ whenever P < δ. L is known as the Lebesgue integral of f on and is denoted by f dm. 4.2 Criteria for Integrability s with the definition of the Riemann integral, the definition of the Lebesgue integral does not illuminate us as to whether a given function is Lebesgue integrable and, if it is, what the Lebesgue integral is. The following theorem, presented without proof, fills this gap. Theorem 4.3 bounded measurable function f is Lebesgue integrable on a bounded measurable set if and only if, given ɛ > 0, there exist simple functions f and f such that f f f and f dm f dm < ɛ. Corollary 4.4 If f is a bounded measurable function on a bounded measurable set, then f is Lebesgue integrable on. Furthermore, { } f dm =lub f dm f is simple and f f { } =glb f dm f is simple and f f. Proof Idea: This corollary is a result of the proof of Theorem 4.3. Example 4.5 Let s find the Lebesgue integral of the Dirichlet function (see Example 1.2). We know by Example 3.12 that the Dirichlet function is measurable. Thus, by Corollary 4.4, it is Lebesgue integrable. Specifically, we claim that [0,1] D dm = 0. Looking back to the definition of the Lebesgue integral (4.2), we must show that, given ɛ > 0, there exists a δ > 0 such that L(D, P ) < ɛ whenever the width of the partition P is less than δ. Referring to Definition 4.1 for L(D, P ), this means that we must find δ such that L(D, P ) = n yi m({x [0, 1] y i 1 D(x) < y i }) < ɛ, i=1

19 where P = {y 0 = 0,..., y n = 2} is a partition of the interval [0, 2] in the range, y i [y i 1, y i ] for i = 1,..., n, and the absolute value signs have been omitted since all quantities here are inherently positive. We accomplish this by picking δ = ɛ/3. Then L(D, P ) = n yi m({x [0, 1] y i 1 D(x) < y i }) i=1 =y 1 m({x [0, 1] 0 D(x) < ɛ/3}) + y j m({x [0, 1] y j 1 D(x) < y j }), where y j 1 1 < y j. ll other terms in the sum vanish because D(x) only takes on two values, 0 and 1. The first term achieves its maximum when y1 = ɛ/3 and when m({x [0, 1] 0 D(x) < ɛ/3} = 1. Thus, L(D, P ) = n yi m({x [0, 1] y i 1 D(x) < y i }) i=1 [ɛ/3] + [y j m({x [0, 1] y j 1 D(x) < y j })]. lso, we know that y j < 1 + ɛ/3. Finding an upper bound on m({x [0, 1] y j 1 D(x) < y j }) is equivalent to asking for an upper bound on the measure of the rational numbers in the interval [0, 1]. Since any non-empty open interval centered on a rational number contains other rational numbers, an upper bound on the measure of the rational numbers is simply the sum of the measures of all the intervals surrounding the rationals. That is, given an enumeration of the rationals, let the n th rational number be contained in an interval of measure ɛ/3 n+1. In this way, and we finally get that m({x [0, 1] y j 1 D(x) < y j }) < n=1 ɛ 3 n+1 = ɛ 6, L(D, P ) ɛ/3 + y j m({x [0, 1] y j 1 D(x) < y j }) <ɛ/3 + (1 + ɛ/3)(ɛ/6) =ɛ/3 + ɛ/3 + ɛ 2 /18 <ɛ,

20 where we have assumed without loss of generality that ɛ < 1. This completes the proof that [0,1] D dm = 0. Combined with Lemma 3.3, we also know that the irrationals must have measure 1 on the interval [0, 1]. This result makes intuitive sense since the rationals are countably infinite whereas the irrationals are uncountably infinite. 4.3 Properties of the Lebesgue Integral It is comforting to know that the Lebesgue integral shares many properties with the Riemann integral. Theorem 4.6 Let f and g be bounded measurable functions on a bounded measurable set. Then, Monotonicity: If f g, then f dm g dm. Linearity: and (f + g) dm = f dm + g dm cf dm = c f dm for c R. For any numbers l, u R such that l f u, it follows that l m() f dm u m(). f dm f dm. If and B are disjoint bounded measurable sets and f : B R is a bounded measurable function, then f dm = f dm + f dm. B Countable dditivity: If = i=1 i where the i are pairwise disjoint bounded measurable sets, then f dm = i=1 B i f dm. Further comparison of the Riemann and the Lebesgue integral is made in Chapter 5.

21 4.4 Integrating Unbounded Measurable Functions Thus far, we have restricted ourselves to integrating bounded measurable functions. The generalization of the Lebesgue integral to unbounded measurable functions is straight-forward but will not be given in full detail here. In general, though, it involves the introduction of ± as possible values of the integral. s it turns out, all of the properties of the Lebesgue integral listed in 4.3 hold for unbounded measurable functions as well.

22 Chapter 5 Comparison of Lebesgue and Riemann Integrals 5.1 Summary of Events Having achieved what we first set out to do to define an integral based on a new way of counting rectangles, let us look back at the motivation behind this endeavor. s outlined in 1.2, we are striving to define an integral such that the class of integrable functions is large and well-behaved. Ideally, we would like the class of integrable functions to have nice limit properties; that the integral of the limit is the limit of the integral. Examples 1.1 and 1.2 show that this is not in general true of the Riemann integral. side from examining the convergence properties of the Lebesgue integral, we are also interested in how it behaves relative to the Riemann integral. Is a Riemann integrable function Lebesgue integrable and, if so, what are the values of the respective integrals? 5.2 Convergence of the Lebesgue Integral The following lemmas and theorems build upon each other and end with a broad statement which identifies when the limit and Lebesgue integral operations are interchangeable. Lemma 5.1 Let g be a non-negative measurable function on a bounded measurable set. If { n } are measurable subsets of such that 1 2 3, 18

23 and if L R is such that L n g dm for all n, then L n g dm. Proof Idea: (Proof by belief). Given that the Lebesgue integral depends on the measure of subsets of the domain of the function, and that measure follows our intuition, this lemma is easy to believe without proof. Theorem 5.2 (Monotone Convergence Theorem) Let be a bounded measurable subset of R and {f n } be a sequence of measurable functions on such that 0 f 1 f 2. Let f be the pointwise limit of {f n }. That is, lim f n(x) = f(x) for all x E. Then f is integrable and f n dm = lim f dm. Proof: By Lemma 3.9, f is measurable since each f n is measurable. Monotonicity of the Lebesgue integral (Theorem 4.6) implies that { f n dm } is an increasing sequence. So, not discounting as a possibility, this sequence has a limit L. Since f f n implies that f dm f n dm for all n, it follows that f dm L. On the other hand, we can also show that f dm L. Let c (0, 1) and let g : R be a simple function such that 0 g f. Consider the sets n = {x f n (x) cg(x)}, which happen to satisfy the hypotheses of Lemma 5.1. The reason for including c is so that cg < f, which implies that n=1 n =. For any n = 1, 2,, we have L = lim f n dm f n dm c n g dm. n By Lemma 5.1, L c n g dm. Since n=1 n = and c (0, 1) was arbitrary, it follows that L n g dm. By Corollary 4.4, { f dm = lub f dm f is simple and f f we get L f dm. Combining the above two inequalities gives L = f dm, as desired. Lemma 5.3 (Fatou s Lemma) Let R be a bounded measurable set. If {f n } is a sequence of non-negative measurable functions on and f(x) = lim f n (x) lim {glb f k(x) k n} },

24 for every x, then f dm lim Proof: Let f n dm. g n (x) = glb{f k (x) k n} for each n = 1, 2, and for each x. Each g n is a measurable function since the set {x g n (x) > r} = {x f k (x) > r} is measurable. Notice that g n f n for all n and that {g n } is a monotonically increasing sequence of non-negative functions. By definition, k=n lim g n(x) = lim {glb f k(x) k n} lim f n (x) = f(x). Therefore, by the Monotone Convergence Theorem (5.2), g n dm = f dm. lim This, combined with g n (x) dm f n (x) dm for each n and each x due to the monotonicity (Theorem 4.6) of the Lebesgue integral, leads to lim g n dm = f dm lim f n dm, as desired. Theorem 5.4 (Lebesgue Dominated Convergence Theorem) Let R be a bounded measurable set and let {f n } be a sequence of measurable functions on such that lim f n(x) = f(x) for all x. If there exists a function g whose Lebesgue integral is finite such that f n (x) g(x) for all n and all x, then lim f n dm = f dm. Proof: To begin, the functions f n and f have finite Lebesgue integrals since g does (this fact does take some proof, but is omitted here). By our choice of g we have f n + g 0 on the set. So, (f + g) dm lim (f n + g) dm

25 by Fatou s Lemma (5.3) and hence, f dm lim f n dm by the linearity of the Lebesgue integral (Theorem 4.6). The same argument can be used with the function g f n to obtain ( ) f dm lim f n dm. But since lim ( ) f n dm = lim { ( ) glb f k dm { } = lim lub f k dm k n lim f n dm, it follows that f dm lim f n dm. Finally, combining the above two inequalities, f dm lim shows that lim f n dm lim f n dm } k n f dm, f n dm exists and is equal to f dm, as desired. 5.3 Convergence of the Riemann Integral Now that we have a grasp of the convergence properties of the Lebesgue integral, let s compare it with those of the Riemann integral. The Riemann equivalent to the Lebesgue Dominated Convergence Theorem (5.4), is stated below in Theorem 5.5. Theorem 5.5 Let a, b R, a < b, and let {f n } be a uniformly convergent sequence of Riemann integrable functions on [a, b] such that lim f n(x) = f(x) for all x [a, b]. Then b lim a f n (x) dx = b a f(x) dx.

26 Recall that {f n } is uniformly convergent if, given ɛ > 0, there exists a natural number N such that f(x) f n (x) < ɛ whenever n > N, for all x [a, b]. Clearly, the hypotheses placed on {f n } in order for the Lebesgue Dominated Convergence Theorem to hold are much less stringent than requiring {f n } to converge uniformly. Thus, we can at least expect that the class of Lebesgue integrable functions has somewhat better limit properties than those of the class of Riemann integrable functions. In fact, we have already witnessed this in Example 4.5 with the Dirichlet function. 5.4 Final Comparison The following theorem, stated without proof, explicitly relates the Riemann and Lebesgue integrals. Theorem 5.6 If f is Riemann integrable on [a,b], then f is Lebesgue integrable on [a,b], and b f(x) dx = f dm. a What s more, we ve already seen that the converse of this theorem isn t true. Thus, not only does the class of Lebesgue integrable functions have better limit properties, but it is also larger than the class of Riemann integrable functions. s promised in 1.2, we now turn to the Lebesgue integral to re-examine the function given in Example 1.1. Example 5.7 Consider the sequence of functions {f n } over the interval E = [0, 1]. { 2 n 1 if f n (x) = 2 x 1 n 2 n 1 0 otherwise The limit function of this sequence is simply f = 0. In this example, each function in the sequence is Riemann integrable, as is the limit function. However, the limit of the sequence of Riemann integrals is not equal to the Riemann integral of the limit of the sequence. That is, lim 1 0 [a,b] f n (x) dx = 1 0 = By Theorem 5.6, it is also the case that f n dm = 1 0 = lim [0,1] 1 0 [0,1] lim f n(x) dx. lim f n(x) dx.

27 This example goes to show that, although the Lebesgue integral is superior to the Riemann integral insofar as the size of the class of integrable functions and the limit properties of these functions, there are still functions which defy Lebesgue integration. One point we have not yet touched on is the effect of the Lebesgue integral on the L p spaces. It turns out that L 2 is complete under the Lebesgue integral. This is of considerable importance, especially when dealing with the theory behind Fourier series. However interesting that topic may be, though, it will have to wait for another time.

28 References [1] HJ Wilcox, DL Myers, n Introduction to Lebesgue Integration and Fourier Series, Dover Publications, Inc., New York, [2] M Rosenlicht, Introduction to nalysis, Dover Publications, Inc., New York,

### MATHS 730 FC Lecture Notes March 5, Introduction

1 INTRODUCTION MATHS 730 FC Lecture Notes March 5, 2014 1 Introduction Definition. If A, B are sets and there exists a bijection A B, they have the same cardinality, which we write as A, #A. If there exists

### Measure and integration

Chapter 5 Measure and integration In calculus you have learned how to calculate the size of different kinds of sets: the length of a curve, the area of a region or a surface, the volume or mass of a solid.

### Real Analysis Notes. Thomas Goller

Real Analysis Notes Thomas Goller September 4, 2011 Contents 1 Abstract Measure Spaces 2 1.1 Basic Definitions........................... 2 1.2 Measurable Functions........................ 2 1.3 Integration..............................

### Problem set 1, Real Analysis I, Spring, 2015.

Problem set 1, Real Analysis I, Spring, 015. (1) Let f n : D R be a sequence of functions with domain D R n. Recall that f n f uniformly if and only if for all ɛ > 0, there is an N = N(ɛ) so that if n

### Notes on the Lebesgue Integral by Francis J. Narcowich Septemmber, 2014

1 Introduction Notes on the Lebesgue Integral by Francis J. Narcowich Septemmber, 2014 In the definition of the Riemann integral of a function f(x), the x-axis is partitioned and the integral is defined

### Indeed, if we want m to be compatible with taking limits, it should be countably additive, meaning that ( )

Lebesgue Measure The idea of the Lebesgue integral is to first define a measure on subsets of R. That is, we wish to assign a number m(s to each subset S of R, representing the total length that S takes

### MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION. Chapter 2: Countability and Cantor Sets

MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION Chapter 2: Countability and Cantor Sets Countable and Uncountable Sets The concept of countability will be important in this course

### ( f ^ M _ M 0 )dµ (5.1)

47 5. LEBESGUE INTEGRAL: GENERAL CASE Although the Lebesgue integral defined in the previous chapter is in many ways much better behaved than the Riemann integral, it shares its restriction to bounded

### Three hours THE UNIVERSITY OF MANCHESTER. 24th January

Three hours MATH41011 THE UNIVERSITY OF MANCHESTER FOURIER ANALYSIS AND LEBESGUE INTEGRATION 24th January 2013 9.45 12.45 Answer ALL SIX questions in Section A (25 marks in total). Answer THREE of the

### 1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989),

Real Analysis 2, Math 651, Spring 2005 April 26, 2005 1 Real Analysis 2, Math 651, Spring 2005 Krzysztof Chris Ciesielski 1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer

### The Lebesgue Integral

The Lebesgue Integral Having completed our study of Lebesgue measure, we are now ready to consider the Lebesgue integral. Before diving into the details of its construction, though, we would like to give

### Some Background Material

Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary - but important - material as a way of dipping our toes in the water. This chapter also introduces important

### Measure and Integration: Concepts, Examples and Exercises. INDER K. RANA Indian Institute of Technology Bombay India

Measure and Integration: Concepts, Examples and Exercises INDER K. RANA Indian Institute of Technology Bombay India Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai 400076,

### CHAPTER 6. Limits of Functions. 1. Basic Definitions

CHAPTER 6 Limits of Functions 1. Basic Definitions DEFINITION 6.1. Let D Ω R, x 0 be a limit point of D and f : D! R. The limit of f (x) at x 0 is L, if for each " > 0 there is a ± > 0 such that when x

### CHAPTER 5. The Topology of R. 1. Open and Closed Sets

CHAPTER 5 The Topology of R 1. Open and Closed Sets DEFINITION 5.1. A set G Ω R is open if for every x 2 G there is an " > 0 such that (x ", x + ") Ω G. A set F Ω R is closed if F c is open. The idea is

### A Short Journey Through the Riemann Integral

A Short Journey Through the Riemann Integral Jesse Keyton April 23, 2014 Abstract An introductory-level theory of integration was studied, focusing primarily on the well-known Riemann integral and ending

### L p Functions. Given a measure space (X, µ) and a real number p [1, ), recall that the L p -norm of a measurable function f : X R is defined by

L p Functions Given a measure space (, µ) and a real number p [, ), recall that the L p -norm of a measurable function f : R is defined by f p = ( ) /p f p dµ Note that the L p -norm of a function f may

### 6 CARDINALITY OF SETS

6 CARDINALITY OF SETS MATH10111 - Foundations of Pure Mathematics We all have an idea of what it means to count a finite collection of objects, but we must be careful to define rigorously what it means

### MORE ON CONTINUOUS FUNCTIONS AND SETS

Chapter 6 MORE ON CONTINUOUS FUNCTIONS AND SETS This chapter can be considered enrichment material containing also several more advanced topics and may be skipped in its entirety. You can proceed directly

### REVIEW OF ESSENTIAL MATH 346 TOPICS

REVIEW OF ESSENTIAL MATH 346 TOPICS 1. AXIOMATIC STRUCTURE OF R Doğan Çömez The real number system is a complete ordered field, i.e., it is a set R which is endowed with addition and multiplication operations

### Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University

Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University February 7, 2007 2 Contents 1 Metric Spaces 1 1.1 Basic definitions...........................

### We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero

Chapter Limits of Sequences Calculus Student: lim s n = 0 means the s n are getting closer and closer to zero but never gets there. Instructor: ARGHHHHH! Exercise. Think of a better response for the instructor.

### MATH & MATH FUNCTIONS OF A REAL VARIABLE EXERCISES FALL 2015 & SPRING Scientia Imperii Decus et Tutamen 1

MATH 5310.001 & MATH 5320.001 FUNCTIONS OF A REAL VARIABLE EXERCISES FALL 2015 & SPRING 2016 Scientia Imperii Decus et Tutamen 1 Robert R. Kallman University of North Texas Department of Mathematics 1155

### Filters in Analysis and Topology

Filters in Analysis and Topology David MacIver July 1, 2004 Abstract The study of filters is a very natural way to talk about convergence in an arbitrary topological space, and carries over nicely into

### arxiv: v2 [math.ca] 4 Jun 2017

EXCURSIONS ON CANTOR-LIKE SETS ROBERT DIMARTINO AND WILFREDO O. URBINA arxiv:4.70v [math.ca] 4 Jun 07 ABSTRACT. The ternary Cantor set C, constructed by George Cantor in 883, is probably the best known

### Exercise 1. Let f be a nonnegative measurable function. Show that. where ϕ is taken over all simple functions with ϕ f. k 1.

Real Variables, Fall 2014 Problem set 3 Solution suggestions xercise 1. Let f be a nonnegative measurable function. Show that f = sup ϕ, where ϕ is taken over all simple functions with ϕ f. For each n

### Lebesgue Measure. Dung Le 1

Lebesgue Measure Dung Le 1 1 Introduction How do we measure the size of a set in IR? Let s start with the simplest ones: intervals. Obviously, the natural candidate for a measure of an interval is its

### 2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?

MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due 9/5). Prove that every countable set A is measurable and µ(a) = 0. 2 (Bonus). Let A consist of points (x, y) such that either x or y is

### 6.2 Fubini s Theorem. (µ ν)(c) = f C (x) dµ(x). (6.2) Proof. Note that (X Y, A B, µ ν) must be σ-finite as well, so that.

6.2 Fubini s Theorem Theorem 6.2.1. (Fubini s theorem - first form) Let (, A, µ) and (, B, ν) be complete σ-finite measure spaces. Let C = A B. Then for each µ ν- measurable set C C the section x C is

### The Caratheodory Construction of Measures

Chapter 5 The Caratheodory Construction of Measures Recall how our construction of Lebesgue measure in Chapter 2 proceeded from an initial notion of the size of a very restricted class of subsets of R,

### Real Analysis Problems

Real Analysis Problems Cristian E. Gutiérrez September 14, 29 1 1 CONTINUITY 1 Continuity Problem 1.1 Let r n be the sequence of rational numbers and Prove that f(x) = 1. f is continuous on the irrationals.

### 2. The Concept of Convergence: Ultrafilters and Nets

2. The Concept of Convergence: Ultrafilters and Nets NOTE: AS OF 2008, SOME OF THIS STUFF IS A BIT OUT- DATED AND HAS A FEW TYPOS. I WILL REVISE THIS MATE- RIAL SOMETIME. In this lecture we discuss two

### LEBESGUE INTEGRATION. Introduction

LEBESGUE INTEGATION EYE SJAMAA Supplementary notes Math 414, Spring 25 Introduction The following heuristic argument is at the basis of the denition of the Lebesgue integral. This argument will be imprecise,

Lebesgue-Radon-Nikodym Theorem Matt Rosenzweig 1 Lebesgue-Radon-Nikodym Theorem In what follows, (, A) will denote a measurable space. We begin with a review of signed measures. 1.1 Signed Measures Definition

### Nets and filters (are better than sequences)

Nets and filters (are better than sequences) Contents 1 Motivation 2 2 More implications we wish would reverse 2 3 Nets 4 4 Subnets 6 5 Filters 9 6 The connection between nets and filters 12 7 The payoff

### Stanford Mathematics Department Math 205A Lecture Supplement #4 Borel Regular & Radon Measures

2 1 Borel Regular Measures We now state and prove an important regularity property of Borel regular outer measures: Stanford Mathematics Department Math 205A Lecture Supplement #4 Borel Regular & Radon

### Math 117: Topology of the Real Numbers

Math 117: Topology of the Real Numbers John Douglas Moore November 10, 2008 The goal of these notes is to highlight the most important topics presented in Chapter 3 of the text [1] and to provide a few

### Chapter 5. Measurable Functions

Chapter 5. Measurable Functions 1. Measurable Functions Let X be a nonempty set, and let S be a σ-algebra of subsets of X. Then (X, S) is a measurable space. A subset E of X is said to be measurable if

### Hyperreal Calculus MAT2000 Project in Mathematics. Arne Tobias Malkenes Ødegaard Supervisor: Nikolai Bjørnestøl Hansen

Hyperreal Calculus MAT2000 Project in Mathematics Arne Tobias Malkenes Ødegaard Supervisor: Nikolai Bjørnestøl Hansen Abstract This project deals with doing calculus not by using epsilons and deltas, but

### Notes on Integrable Functions and the Riesz Representation Theorem Math 8445, Winter 06, Professor J. Segert. f(x) = f + (x) + f (x).

References: Notes on Integrable Functions and the Riesz Representation Theorem Math 8445, Winter 06, Professor J. Segert Evans, Partial Differential Equations, Appendix 3 Reed and Simon, Functional Analysis,

### Chapter 4. The dominated convergence theorem and applications

Chapter 4. The dominated convergence theorem and applications The Monotone Covergence theorem is one of a number of key theorems alllowing one to exchange limits and [Lebesgue] integrals (or derivatives

### CHAPTER 7. Connectedness

CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set

### Sequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1.

Chapter 3 Sequences Both the main elements of calculus (differentiation and integration) require the notion of a limit. Sequences will play a central role when we work with limits. Definition 3.. A Sequence

### Economics 204 Fall 2011 Problem Set 1 Suggested Solutions

Economics 204 Fall 2011 Problem Set 1 Suggested Solutions 1. Suppose k is a positive integer. Use induction to prove the following two statements. (a) For all n N 0, the inequality (k 2 + n)! k 2n holds.

### CHAPTER I THE RIESZ REPRESENTATION THEOREM

CHAPTER I THE RIESZ REPRESENTATION THEOREM We begin our study by identifying certain special kinds of linear functionals on certain special vector spaces of functions. We describe these linear functionals

### Cauchy Sequences. x n = 1 ( ) 2 1 1, . As you well know, k! n 1. 1 k! = e, = k! k=0. k = k=1

Cauchy Sequences The Definition. I will introduce the main idea by contrasting three sequences of rational numbers. In each case, the universal set of numbers will be the set Q of rational numbers; all

### Limits and Continuity

Chapter Limits and Continuity. Limits of Sequences.. The Concept of Limit and Its Properties A sequence { } is an ordered infinite list x,x,...,,... The n-th term of the sequence is, and n is the index

### Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi

Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.

### A LITTLE REAL ANALYSIS AND TOPOLOGY

A LITTLE REAL ANALYSIS AND TOPOLOGY 1. NOTATION Before we begin some notational definitions are useful. (1) Z = {, 3, 2, 1, 0, 1, 2, 3, }is the set of integers. (2) Q = { a b : aεz, bεz {0}} is the set

### Continuity. Chapter 4

Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of

### REAL AND COMPLEX ANALYSIS

REAL AND COMPLE ANALYSIS Third Edition Walter Rudin Professor of Mathematics University of Wisconsin, Madison Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### INDECOMPOSABILITY IN INVERSE LIMITS WITH SET-VALUED FUNCTIONS

INDECOMPOSABILITY IN INVERSE LIMITS WITH SET-VALUED FUNCTIONS JAMES P. KELLY AND JONATHAN MEDDAUGH Abstract. In this paper, we develop a sufficient condition for the inverse limit of upper semi-continuous

### Analysis I. Classroom Notes. H.-D. Alber

Analysis I Classroom Notes H-D Alber Contents 1 Fundamental notions 1 11 Sets 1 12 Product sets, relations 5 13 Composition of statements 7 14 Quantifiers, negation of statements 9 2 Real numbers 11 21

### van Rooij, Schikhof: A Second Course on Real Functions

vanrooijschikhofproblems.tex December 5, 2017 http://thales.doa.fmph.uniba.sk/sleziak/texty/rozne/pozn/books/ van Rooij, Schikhof: A Second Course on Real Functions Some notes made when reading [vrs].

### 1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3

Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,

### Entrance Exam, Real Analysis September 1, 2017 Solve exactly 6 out of the 8 problems

September, 27 Solve exactly 6 out of the 8 problems. Prove by denition (in ɛ δ language) that f(x) = + x 2 is uniformly continuous in (, ). Is f(x) uniformly continuous in (, )? Prove your conclusion.

### General Power Series

General Power Series James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University March 29, 2018 Outline Power Series Consequences With all these preliminaries

### McGill University Math 354: Honors Analysis 3

Practice problems McGill University Math 354: Honors Analysis 3 not for credit Problem 1. Determine whether the family of F = {f n } functions f n (x) = x n is uniformly equicontinuous. 1st Solution: The

### Finite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product

Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )

### 3 Measurable Functions

3 Measurable Functions Notation A pair (X, F) where F is a σ-field of subsets of X is a measurable space. If µ is a measure on F then (X, F, µ) is a measure space. If µ(x) < then (X, F, µ) is a probability

### Math 117: Continuity of Functions

Math 117: Continuity of Functions John Douglas Moore November 21, 2008 We finally get to the topic of ɛ δ proofs, which in some sense is the goal of the course. It may appear somewhat laborious to use

### INTRODUCTION TO REAL ANALYSIS II MATH 4332 BLECHER NOTES

INTRODUCTION TO REAL ANALYSIS II MATH 4332 BLECHER NOTES You will be expected to reread and digest these typed notes after class, line by line, trying to follow why the line is true, for example how it

### MATH5011 Real Analysis I. Exercise 1 Suggested Solution

MATH5011 Real Analysis I Exercise 1 Suggested Solution Notations in the notes are used. (1) Show that every open set in R can be written as a countable union of mutually disjoint open intervals. Hint:

### MATH NEW HOMEWORK AND SOLUTIONS TO PREVIOUS HOMEWORKS AND EXAMS

MATH. 4433. NEW HOMEWORK AND SOLUTIONS TO PREVIOUS HOMEWORKS AND EXAMS TOMASZ PRZEBINDA. Final project, due 0:00 am, /0/208 via e-mail.. State the Fundamental Theorem of Algebra. Recall that a subset K

### Fundamental Inequalities, Convergence and the Optional Stopping Theorem for Continuous-Time Martingales

Fundamental Inequalities, Convergence and the Optional Stopping Theorem for Continuous-Time Martingales Prakash Balachandran Department of Mathematics Duke University April 2, 2008 1 Review of Discrete-Time

### Lebesgue measure on R is just one of many important measures in mathematics. In these notes we introduce the general framework for measures.

Measures In General Lebesgue measure on R is just one of many important measures in mathematics. In these notes we introduce the general framework for measures. Definition: σ-algebra Let X be a set. A

### Löwenheim-Skolem Theorems, Countable Approximations, and L ω. David W. Kueker (Lecture Notes, Fall 2007)

Löwenheim-Skolem Theorems, Countable Approximations, and L ω 0. Introduction David W. Kueker (Lecture Notes, Fall 2007) In its simplest form the Löwenheim-Skolem Theorem for L ω1 ω states that if σ L ω1

### Chapter 4. Measurable Functions. 4.1 Measurable Functions

Chapter 4 Measurable Functions If X is a set and A P(X) is a σ-field, then (X, A) is called a measurable space. If µ is a countably additive measure defined on A then (X, A, µ) is called a measure space.

### MEASURE AND INTEGRATION. Dietmar A. Salamon ETH Zürich

MEASURE AND INTEGRATION Dietmar A. Salamon ETH Zürich 9 September 2016 ii Preface This book is based on notes for the lecture course Measure and Integration held at ETH Zürich in the spring semester 2014.

### Maths 212: Homework Solutions

Maths 212: Homework Solutions 1. The definition of A ensures that x π for all x A, so π is an upper bound of A. To show it is the least upper bound, suppose x < π and consider two cases. If x < 1, then

### Principle of Mathematical Induction

Advanced Calculus I. Math 451, Fall 2016, Prof. Vershynin Principle of Mathematical Induction 1. Prove that 1 + 2 + + n = 1 n(n + 1) for all n N. 2 2. Prove that 1 2 + 2 2 + + n 2 = 1 n(n + 1)(2n + 1)

### 2. Two binary operations (addition, denoted + and multiplication, denoted

Chapter 2 The Structure of R The purpose of this chapter is to explain to the reader why the set of real numbers is so special. By the end of this chapter, the reader should understand the difference between

### Summer Jump-Start Program for Analysis, 2012 Song-Ying Li. 1 Lecture 7: Equicontinuity and Series of functions

Summer Jump-Start Program for Analysis, 0 Song-Ying Li Lecture 7: Equicontinuity and Series of functions. Equicontinuity Definition. Let (X, d) be a metric space, K X and K is a compact subset of X. C(K)

### ANALYSIS QUALIFYING EXAM FALL 2017: SOLUTIONS. 1 cos(nx) lim. n 2 x 2. g n (x) = 1 cos(nx) n 2 x 2. x 2.

ANALYSIS QUALIFYING EXAM FALL 27: SOLUTIONS Problem. Determine, with justification, the it cos(nx) n 2 x 2 dx. Solution. For an integer n >, define g n : (, ) R by Also define g : (, ) R by g(x) = g n

### THEOREMS, ETC., FOR MATH 515

THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every

### EXISTENCE AND UNIQUENESS OF INFINITE COMPONENTS IN GENERIC RIGIDITY PERCOLATION 1. By Alexander E. Holroyd University of Cambridge

The Annals of Applied Probability 1998, Vol. 8, No. 3, 944 973 EXISTENCE AND UNIQUENESS OF INFINITE COMPONENTS IN GENERIC RIGIDITY PERCOLATION 1 By Alexander E. Holroyd University of Cambridge We consider

### K 4 -free graphs with no odd holes

K 4 -free graphs with no odd holes Maria Chudnovsky 1 Columbia University, New York NY 10027 Neil Robertson 2 Ohio State University, Columbus, Ohio 43210 Paul Seymour 3 Princeton University, Princeton

### Introductory Analysis 2 Spring 2010 Exam 1 February 11, 2015

Introductory Analysis 2 Spring 21 Exam 1 February 11, 215 Instructions: You may use any result from Chapter 2 of Royden s textbook, or from the first four chapters of Pugh s textbook, or anything seen

### MATH 13 SAMPLE FINAL EXAM SOLUTIONS

MATH 13 SAMPLE FINAL EXAM SOLUTIONS WINTER 2014 Problem 1 (15 points). For each statement below, circle T or F according to whether the statement is true or false. You do NOT need to justify your answers.

### CONSTRUCTION OF sequence of rational approximations to sets of rational approximating sequences, all with the same tail behaviour Deﬁnition 1.

CONSTRUCTION OF R 1. MOTIVATION We are used to thinking of real numbers as successive approximations. For example, we write π = 3.14159... to mean that π is a real number which, accurate to 5 decimal places,

### RS Chapter 1 Random Variables 6/5/2017. Chapter 1. Probability Theory: Introduction

Chapter 1 Probability Theory: Introduction Basic Probability General In a probability space (Ω, Σ, P), the set Ω is the set of all possible outcomes of a probability experiment. Mathematically, Ω is just

### Math 396. Bijectivity vs. isomorphism

Math 396. Bijectivity vs. isomorphism 1. Motivation Let f : X Y be a C p map between two C p -premanifolds with corners, with 1 p. Assuming f is bijective, we would like a criterion to tell us that f 1

### Lebesgue Measure and The Cantor Set

Math 0 Final year project Lebesgue Measure and The Cantor Set Jason Baker, Kyle Henke, Michael Sanchez Overview Define a measure Define when a set has measure zero Find the measure of [0, ], I and Q Construct

### MATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions.

MATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions. Continuity Definition. Given a set E R, a function f : E R, and a point c E, the function f is continuous at c if

### Lecture 4 Lebesgue spaces and inequalities

Lecture 4: Lebesgue spaces and inequalities 1 of 10 Course: Theory of Probability I Term: Fall 2013 Instructor: Gordan Zitkovic Lecture 4 Lebesgue spaces and inequalities Lebesgue spaces We have seen how

### Introduction to Topology

Introduction to Topology Randall R. Holmes Auburn University Typeset by AMS-TEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about

### Lemma 15.1 (Sign preservation Lemma). Suppose that f : E R is continuous at some a R.

15. Intermediate Value Theorem and Classification of discontinuities 15.1. Intermediate Value Theorem. Let us begin by recalling the definition of a function continuous at a point of its domain. Definition.

### MH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then

MH 7500 THEOREMS Definition. A topological space is an ordered pair (X, T ), where X is a set and T is a collection of subsets of X such that (i) T and X T ; (ii) U V T whenever U, V T ; (iii) U T whenever

### Real Analysis Chapter 3 Solutions Jonathan Conder. ν(f n ) = lim

. Suppose ( n ) n is an increasing sequence in M. For each n N define F n : n \ n (with 0 : ). Clearly ν( n n ) ν( nf n ) ν(f n ) lim n If ( n ) n is a decreasing sequence in M and ν( )

### Part 2 Continuous functions and their properties

Part 2 Continuous functions and their properties 2.1 Definition Definition A function f is continuous at a R if, and only if, that is lim f (x) = f (a), x a ε > 0, δ > 0, x, x a < δ f (x) f (a) < ε. Notice

### Problem 3. Give an example of a sequence of continuous functions on a compact domain converging pointwise but not uniformly to a continuous function

Problem 3. Give an example of a sequence of continuous functions on a compact domain converging pointwise but not uniformly to a continuous function Solution. If we does not need the pointwise limit of

### Annalee Gomm Math 714: Assignment #2

Annalee Gomm Math 714: Assignment #2 3.32. Verify that if A M, λ(a = 0, and B A, then B M and λ(b = 0. Suppose that A M with λ(a = 0, and let B be any subset of A. By the nonnegativity and monotonicity

### x 0 + f(x), exist as extended real numbers. Show that f is upper semicontinuous This shows ( ɛ, ɛ) B α. Thus

Homework 3 Solutions, Real Analysis I, Fall, 2010. (9) Let f : (, ) [, ] be a function whose restriction to (, 0) (0, ) is continuous. Assume the one-sided limits p = lim x 0 f(x), q = lim x 0 + f(x) exist

Arrow s Paradox Prerna Nadathur January 1, 2010 Abstract In this paper, we examine the problem of a ranked voting system and introduce Kenneth Arrow s impossibility theorem (1951). We provide a proof sketch

### 3 COUNTABILITY AND CONNECTEDNESS AXIOMS

3 COUNTABILITY AND CONNECTEDNESS AXIOMS Definition 3.1 Let X be a topological space. A subset D of X is dense in X iff D = X. X is separable iff it contains a countable dense subset. X satisfies the first

### are Banach algebras. f(x)g(x) max Example 7.4. Similarly, A = L and A = l with the pointwise multiplication

7. Banach algebras Definition 7.1. A is called a Banach algebra (with unit) if: (1) A is a Banach space; (2) There is a multiplication A A A that has the following properties: (xy)z = x(yz), (x + y)z =

### Iowa State University. Instructor: Alex Roitershtein Summer Homework #5. Solutions

Math 50 Iowa State University Introduction to Real Analysis Department of Mathematics Instructor: Alex Roitershtein Summer 205 Homework #5 Solutions. Let α and c be real numbers, c > 0, and f is defined