# SET THEORY AND UNIQUENESS FOR TRIGONOMETRIC SERIES. Alexander S. Kechris* Department of Mathematics Caltech Pasadena, CA 91125

Save this PDF as:

Size: px
Start display at page:

Download "SET THEORY AND UNIQUENESS FOR TRIGONOMETRIC SERIES. Alexander S. Kechris* Department of Mathematics Caltech Pasadena, CA 91125"

## Transcription

1 SET THEORY AND UNIQUENESS FOR TRIGONOMETRIC SERIES Alexander S. Kechris* Department of Mathematics Caltech Pasadena, CA Dedicated to the memory of my friend and colleague Stelios Pichorides Problems concerning the uniqueness of an expansion of a function in a trigonometric series have a long and fascinating history, starting back in the 19th Century with the work of Riemann, Heine and Cantor. The origins of set theory are closely connected with this subject, as it was Cantor s research into the nature of exceptional sets for such uniqueness problems that led him to the creation of set theory. And the earliest application of one of Cantor s fundamental concepts, that of ordinal numbers and transfinite induction, can be glimpsed in his last work on this subject. The purpose of this paper is to give a basic introduction to the application of set theoretic methods to problems concerning uniqueness for trigonometric series. It is written in the style of informal lecture notes for a course or seminar on this subject and, in particular, contains several exercises. The treatment is as elementary as possible and only assumes some familiarity with the most basic results of general topology, measure theory, functional analysis, and descriptive set theory. Standard references to facts that are used without proof are given in the appropriate places. The notes are divided into three parts. The first deals with ordinal numbers and transfinite induction, and gives an exposition of Cantor s work. The second gives an application of Baire category methods, one of the basic set theoretic tools in the arsenal of an analyst. The final part deals with the role of descriptive methods in the study of sets of uniqueness. It closes with an introduction to the modern use of ordinal numbers in descriptive set theory and its applications, and brings us back full circle to the concepts that arose in the beginning of this subject. There is of course much more to this area than what is discussed in these introductory notes. Some references for further study include: Kechris-Louveau [1989, 1992], Cooke [1993], Kahane-Salem [1994], Lyons [1995]. The material in these notes is mostly drawn from these sources. Stelios Pichorides always had a strong interest in the problems of uniqueness for trigonometric series, and, in particular, in the problem of characterizing the sets of uniqueness. I remember distinctly a discussion we had, in the early eighties, during a long ride in the Los Angeles freeways, in which he wondered whether lack of further progress on this * Research partially supported by NSF grant. 1

2 deep problem perhaps had some logical or foundational explanation. (It should be pointed out that mathematical logic was Stelios first love, and he originally went to the University of Chicago to study this field.) His comments are what got me interested in this subject and we have had extensive discussions about this for many years afterwards. Certainly the results proved in the 1980 s (some of which are discussed in Part III below) concerning the Characterization Problem seem to confirm his intuition. Problems concerning sets of uniqueness have fascinated mathematicians for over 100 years now, in part because of the intrinsic nature of the subject and in part because of its intriguing interactions with other areas of classical analysis, measure theory, functional analysis, number theory, and set theory. Once someone asked Paul Erdös, after he gave a talk about one of his favorite number theory problems, somewhat skeptically, why he was so interested in this problem. Erdös replied that if this problem was good enough for Dirichlet and Gauss, it was good enough for him. To paraphrase Erdös, if the problems of uniqueness for trigonometric series were good enough for Riemann, Cantor, Luzin, Menshov, Bari, Salem, Zygmund, and Pichorides, they are certainly good enough for me. 2

3 PART I. ORDINAL NUMBERS AND TRANSFINITE INDUCTION. 1. The problem of uniqueness for trigonometric series. A trigonometric series S is an infinite series of the form S a (a n cos nx + b n sinnx), n=1 where a n C,x R. We view this as a formal expression without any claims about its convergence at a given point x. The Nth partial sum of this series is the trigonometric polynomial S N (x) = a N (a n cos nx + b n sinnx). n=1 If, for some given x, S N (x) s C, we write and call s the sum of the series at x. s = a (a n cos nx + b n sinnx) n=1 A function f :R Cadmits a trigonometric expansion if there is a series S as above so that for every x R f(x) = a (a n cos nx + b n sinnx). n=1 It is clear that any such function is periodic with period 2π. It is a very difficult problem to characterize the functions f which admit a trigonometric expansion, but it is a classical result that any nice enough 2π-periodic function f, for example a continuously differentiable one, admits a trigonometric expansion f(x) = a (a n cos nx + b n sinnx), n=1 there the coefficients a n,b n can be, in fact, computed by the well-known Fourier formulas a n = 1 π 2π 0 f(t)cos nt dt, b n = 1 π 2π 0 f(t)sin nt dt. 3

4 The following question now arises naturally: Is such an expansion unique? If f(x) = a n=1 (a n cos nx + b n sin nx) = a n=1 (a n cos nx + b n sin nx), then, by subtracting, we would have a series c n=1 (c n cos nx + d n sinnx) with 0 = c (c n cos nx + d n sinnx) n=1 for every x. So the problem is equivalent to the following: Uniqueness Problem. If a n=1 (a n cos nx+b n sinnx) is a trigonometric series and a (a n cos nx + b n sin nx) = 0 n=1 for all x R, it is true that a n = b n = 0 for all n? This is the problem (which arose through the work of Riemann and Heine) that Heine proposed, in 1869, to the 24 year old Cantor, who had just accepted a position at the university in Halle, where Heine was a senior colleague. In the next few sections we will give Cantor s solution to the uniqueness problem and see how his search for extensions, allowing exceptional points, led him to the creation of set theory, including the concepts of ordinal numbers and the method of transfinite induction. We will also see how this method can be used to prove the first such major extension. Before we proceed, it would be convenient to also introduce an alternative form for trigonometric series. Every series can be also written as where, letting b 0 = 0, S a (a n cos nx + b n sinnx) n=1 S c n e inx, n= c n = a n ib n,c n = a n + ib n 2 2 (n N). (Thus a n = c n +c n,b n = i(c n c n ).) In this notation, the partial sums S N (x) are given by N S N (x) = c n e inx ; n= N 4

5 and if they converge as N with limit s, we write s = + n= c n e inx. The standard examples of trigonometric series are the Fourier series of integrable functions. Given a 2π-periodic function f :R Cwe say that it is integrable if it is Lebesgue measurable and 1 2π f(t) dt <. 2π 0 In this case we define its Fourier coefficients ˆf(n),n Z, by We call the trigonometric series ˆf(n) = 1 2π f(t)e int dt. 2π 0 S(f) n= ˆf(n)e inx the Fourier series of f, and write S N (f,x) = +N n= N ˆf(n)e inx for its partial sums. Remark. There are trigonometric series, like are not Fourier series. 2. The Riemann Theory. n=2 sin nx log n, which converge everywhere but Riemann was the first mathematician to seriously study general trigonometric series (as opposed to Fourier series), in his Habilitationsschrift (1854). We will prove here two of his main results, that were used by Cantor. These results are beautiful applications of elementary calculus. Let S c n e inx be an arbitrary trigonometric series with bounded coefficients,i.e., c n M for some M and all n Z. Then Riemann had the brilliant idea to consider the function obtained by formally integrating c n e inx twice. This function, called the Riemann function F S of S, is thus defined by F S (x) = c 0x / n= 5 1 n 2c ne inx,x R,

6 where means n = 0 is excluded. Clearly, as c n is bounded, F S is a continuous function onr(but is not periodic), since the above series converges absolutely and uniformly, as 1 n 2 c n e inx M n 2. Now, intuitively, one would hope that F S (x) should be the same as c n e inx, if this sum exists. This may not be quite true, but something close enough to it is. Given a function F :R C, let 2 F(x,h) = F(x + h) + F(x h) 2F(x) and define the second symmetric derivative or second Schwartz derivative of F at x by provided this limit exists. D 2 F(x) = lim h 0 2 F(x,h) h 2, 2.1. Exercise. If F (x) exists, then so does D 2 F(x) and they are equal, but the converse fails Riemann s First Lemma. Let S c n e inx be a trigonometric series with bounded coefficients. If s = c n e inx exists, then D 2 F S (x) exists and D 2 F S (x) = s. Proof. We have by calculating (for n = 0, we let 2.3. Lemma. n=0 sin nh nh Proof. Put A N = N 2 F S (x,2h) 4h 2 = + n= [ ] 2 sinnh c n e inx nh = 1). So it is enough to prove the following: [ [ ] 2 ] a n = a. a n = a lim h 0 n=0 a n. Then ( ) 2 sin nh a n = nh n=0 Let h k 0,h k > 0 and put n=0 sin nh nh [( ) 2 sin nh nh n=0 ( sin(n + 1)h (n + 1)h ( ) 2 ( ) 2 sinnhk sin(n + 1)hk s kn =. nh k (n + 1)h k ) 2 ] A n. Then we have to show that A n n a k A n s kn a. n=0 6

7 We view the infinite matrix (s kn ) as a summability method, i.e., a way of transforming a sequence (x n ) into the sequence i.e., y k = n=0 s kn x n. (y k ) = (s kn ) (x n ), Example. If s kn = 1 k+1 for n k,s kn = 0 for n > k, then y k = x 0+ +x k k+1. A summability method is called regular if x n following result which we leave as an exercise. n x y k k x. Toeplitz proved the 2.4. Exercise. (a) If the matrix (s kn ) satisfies the following conditions, called Toeplitz conditions: (i) s kn (ii) n=0 (iii) k 0, n N, s kn C <, k N, s kn n=0 k 1, then (s kn ) is regular. (b) If (s kn ) satisfies only (i), (ii) and x n 0, then y k 0. So it is enough to check that (i), (ii), (iii) hold for ( ) 2 ( ) 2 sinnhk sin(n + 1)hk s kn =. nh k (n + 1)h k Clearly (i), (iii) hold. For (ii), let u(x) = ( sin x x )2. Then we have 2.5. Exercise. u (x) dx <. 0 Thus s kn = n=0 n=0 0 (n+1)hk nh k u (x)dx u (x) dx = C < Riemann s Second Lemma. Let S c n e inx be a trigonometric series with c n 0. Then 2 F S (x,h) h = F S(x + h) + F S (x h) 2F S (x) h 7 0,

8 as h 0, uniformly on x. Proof. By direct calculation we have again 2 F S (x,2h) 4h = sin 2 (nh) n 2 h c ne inx (where for n = 0, sin2 (nh) n 2 h is defined to be h). Let as before 0 < h k 1, h k 0 and put t kn = sin2 (nh k ) n 2 h k. We have to show that (c n e inx )t kn 0 as k, uniformly on x. Since c n e inx 0, uniformly on x, it is enough to verify that (t kn ) satisfies the first two Toeplitz conditions (i), (ii), of 2.4. Clearly (i) holds. To prove (ii) fix k and choose N > 1 with N 1 h 1 k < N. Then t kn = n=1 N 1 n=1 sin 2 (nh k ) n 2 h k + n=n sin 2 (nh k ) n 2 h k (N 1) h k + 1 h k n=n 1 n h k n=n 1 n(n 1) and we are done. = h k 1 N 1 3, Note that this implies that the graph of F S can have no corners, i.e., if the left- and right-derivatives of F S exist at some point x, then they must be equal. 3. The Cantor Uniqueness Theorem. The following result was proved originally by Cantor with set of positive measure replaced by interval The Cantor-Lebesgue Lemma. If a n cos(nx) + b n sin(nx) 0, for x in a set of positive (Lebesgue) measure, then a n,b n 0. So if c n e inx = 0 for x in a set of positive measure, then c n 0. 8

9 Proof. We can assume that a n,b n R. Let ρ n = a 2 n + b 2 n and ϕ n be such that a n cos(nx) + b n sin(nx) = ρ n cos(nx + ϕ n ). Thus ρ n cos(nx + ϕ n ) 0 on E [0,2π), a set of positive measure. Assume ρ n 0, toward a contradiction. So there is ǫ > 0 and n 0 < n 1 < n 2 < such that ρ nk ǫ. Then cos(n k x + ϕ nk ) 0, so 2cos 2 (n k x + ϕ nk ) 0, i.e., 1 + cos 2(n k x + ϕ nk ) 0 for x E. By Lebesgue Dominated Convergence E (1 + cos 2(n kx + ϕ nk ))dx 0, i.e., letting χ E be the characteristic function of E in the interval [0,2π) extended with period 2π over all of R, and µ(e) be the Lebesgue measure of E: We now have: µ(e) + 2π 0 χ E (x)cos 2(n k x + ϕ nk )dx ] =µ(e) + 2π [Re ˆχ E ( 2n k ) cos 2ϕ nk Im ˆχ E ( 2n k ) sin 2ϕ nk Exercise (Riemann-Lebesgue). If f is an integrable 2π-periodic function, then ˆf(n) 0 as n. [Remark. This is really easy if f is the characteristic function of an interval.] So it follows that µ(e) = 0, a contradiction. We are only one lemma away from the proof of Cantor s Theorem. This lemma was proved by Schwartz in response to a request by Cantor. 3.3 Lemma (Schwartz). Let F : (a,b) R be continuous such that D 2 F(x) 0, x (a,b). Then F is convex on (a,b). In particular, if F : (a,b) Cis continuous and D 2 F(x) = 0, x (a,b), then F is linear on (a,b). Proof. By replacing F by F + ǫx 2,ǫ > 0, and letting ǫ 0, we can assume actually that D 2 F(x) > 0 for all x (a,b). Assume F is not convex, toward a contradiction. Then there is a linear function µx + ν and a < c < d < b such that if G(x) = F(x) (µx + ν), then G(c) = G(d) = 0 and G(x) > 0 for some x (c,d). Let x 0 be a point where G achieves its maximum in [c,d]. Then c < x 0 < d. Now for small enough h, 2 F(x 0,h) 0, so D 2 F(x 0 ) 0, a contradiction. We now have: 3.4. Theorem (Cantor, 1870). If c n e inx = 0 for all x, then c n = 0, n Z. Proof. Let S c n e inx. By the Cantor-Lebesgue Lemma, c n 0, as n, so, in particular, c n is bounded. By Riemann s First Lemma, D 2 F S (x) = 0, x R, and so by Schwartz s Lemma F S is linear, i.e., c 0 x n 2 c ne inx = ax + b. 9

10 Put x = π,x = π and subtract to get a = 0. Put x = 0,x = 2π and subtract to get c 0 = 0. Then 1 n 2 c n e inx = b. Since this series converges uniformly, if m 0 we have 2π 0 = be imx dx 0 = 1 2π n c 2 n e i(n m)x dx 0 so c m = 0 and we are done. = c m m 2, Remark. Kronecker (when he was still on speaking terms with Cantor) pointed out that the use of the fact that c n 0 was unnecessary, i.e., if one can prove that (1) cn e inx = 0, x R&c n 0 c n = 0, n Z, then it follows that (2) c n e inx = 0, x R c n = 0, n Z. To see this assume (1) and take any series c n e inx such that c n e inx = 0, x R (without any other assumption on the c n ). Put for x, x + u and x u, add and divide by 2 to get cn e inx cos nu = 0, x,u R. or equivalently c 0 + (a n cos nx + b n sin nx)cos nu = 0, x,u R. n=1 Considering x fixed now, we note that a n cos nx + b n sinnx 0 (since c 0 + (a n cos nx + b n sinnx) = 0) so we can apply (1) to get that a n cos nx + b n sin nx = 0, for all x,n N, from which easily a n = b n = 0, n N. 4. Sets of uniqueness. Cantor next extended his uniqueness theorem in 1871 by allowing a finite number of exceptional points Theorem (Cantor, 1871). Assume that c n e inx = 0 for all but finitely many x R. Then c n = 0, n Z. Proof. Let S c n e inx. Suppose x 0 = 0 x 1 < x 2 < < x n < 2π = x n+1 are such that for x x i, c n e inx = 0. Then, by Schwartz s Lemma, F S is linear in each interval (x i,x i+1 ). Since by Riemann s Second Lemma the graph of F S has no corners, it follows that F S is linear in [0,2π). The same holds of course for any interval of length 2π, so F S is linear and thus as in the proof of 3.4, c n = 0, n Z. 10

11 In the sequel it will be convenient to identify the unit circlet={e ix : 0 x < 2π} withr/2πz via the map x e ix. Often we can think oftas being [0,2π), or [0,2π] with 0, 2π identified. Functions on T can be also thought of as 2π-periodic functions on R. We denote by λ the Lebesgue measure ont(i.e., the one induced by the above identification with [0, 2π)) normalized so that λ(t) = 1. We now introduce the following basic concept. Definition. Let E T. We say that E is a set of uniqueness if every trigonometric series cn e inx which converges to 0 off E (i.e., c n e inx = 0, for e ix / E, which we will simply write x / E ) is identically 0. Otherwise it is called a set of multiplicity. So 3.4 says that is a set of uniqueness and 4.1 says that every finite set is a set of uniqueness. We denote by U the class of sets of uniqueness and by M the class of sets of multiplicity. Our next goal is to prove the following extension of Cantor s Theorem Theorem (Cantor, Lebesgue see Remarks in 6 below). Every countable closed set is a set of uniqueness. We will give a proof using the method of transfinite induction. 5. The Cantor-Bendixson Theorem. Let E T be a closed set. We define its Cantor-Bendixson derivative E by Note that E E and E is closed as well. E = {x E : x is a limit point of E}. Now define by transfinite induction for each ordinal α a closed set E (α) as follows: E (0) = E, E (α+1) = (E (α) ) E (λ) = E (α), λ a limit ordinal. α<λ The E (α) form a decreasing sequence of closed sets contained in E: E (0) E (1) E (2) E (α) E (β),α β Lemma. If F α, α an ordinal, is a decreasing sequence of closed sets, then for some countable ordinal α 0 we have that F α0 = F α

12 Proof. Fix a basis {U n } for the topology oftand let A α = {n : U n F α = }. Then α β A α A β. If A α A α+1 for all countable ordinals α, let f(α) be the least n with n A α+1 \ A α. Then, if ω 1 is the first uncountable ordinal, which by standard practice we also identify with the set {α : α < ω 1 } of all countable ordinals, we have that f : ω 1 N is injective, which is a contradiction. So for some α 0 < ω 1, A α0 = A α0 +1, so F α0 = F α0 +1. Thus for each closed set E T, there is a least countable α with E (α) = E (α+1) and thus E (α) = E (β) for all α β. We denote this ordinal by rk CB (E) and call it the Cantor-Bendixson rank of E. We also put E ( ) = E (rk CB(E)). Notice that (E ( ) ) = E ( ), so E ( ) is perfect, i.e., every point of it is a limit point (it could be though). We call it the perfect kernel of E Exercise. Show that E ( ) is the largest perfect set contained in E Theorem (Cantor-Bendixson). Let E be closed. Then the set E \E ( ) is countable. In particular, E is countable E ( ) =. Proof. Let x E \ E ( ), so that for some (unique) α < rk CB (E),x E (α) \ E (α+1). Since there are only countably many such α, it is enough to prove the following: 5.4. Lemma. For any closed set F,F \ F is countable. Proof. Fix a countable basis {U n } n N. If x F \F, then there is some n with F U n = {x}. So F \ F = {F U n : F U n is a singleton}, which is clearly countable Exercise. Show that for each closed set E there is a unique decomposition E = P C, P C =, P perfect, C countable Exercise. For each countable successor ordinal α find a countable closed set E with rk CB (E) = α. 6. Sets of uniqueness (cont d). We are now ready to give the Proof of Theorem 4.2. Let E be a countable closed set. Let S c n e inx be such that c n e inx = 0 off E. Since it is clear that any translate (in T) of a set of uniqueness is also a set of uniqueness, we can assume that 0 / E. So we can view E as being a closed set contained in (0,2π). The complement in (0,2π) of any closed subset F of (0,2π) is a disjoint union of open intervals with endpoints in F {0, 2π}, called its contiguous intervals. We will prove by 12

13 transfinite induction on α, that F S is linear on each contiguous interval of E (α). Since E (α 0) = for some α 0, it follows that F S is linear on (0,2π), so, as before, c n = 0, n. α = 0: This is clear since c n e inx = 0 on each contiguous interval of E (0) = E. α α+1: Assume F S is linear in each contiguous interval of E (α). Let then (a,b) be a contiguous interval of E (α+1). Then in each closed subinterval [c,d] (a,b) there are only finitely many points c x 0 < x 1 < < x n d of E (α). Then (c,x 0 ),(x 0,x 1 ),,(x n,d) are contained in contiguous intervals of E (α), so, by induction hypothesis, F S is linear in each one of them, so by the Riemann Second Lemma again, F S is linear on [c,d] and thus on (a,b). α < λ λ (λ a limit ordinal): We use a compactness argument. Fix a contiguous interval (a,b) of E (λ) and a closed subinterval [c,d] (a,b). Then [c,d] (0,2π) \ E (λ) = (0,2π) \ α<λ E (α) = [(0,2π) \ E (α) ]. α<λ Since (0,2π) \ E (α) is open and [c,d] is compact, there are finitely many α 1,,α n < λ with [c,d] [(0,2π) \ E (α) ] (0,2π) \ E (β) α {α 1,,α n } for any α 1,,α n β < λ. Then [c,d] is contained in a contiguous interval of E (β), so, by induction hypothesis, F S is linear on [c,d] and thus on (a,b). This completes the proof. Remark. Cantor published in 1872 the proof of Theorem 4.2 only for the case rk CB (E) < ω, i.e., when the Cantor-Bendixson process terminates in finitely many steps. Apparently at this stage he had, at least at some intuitive level, the idea of extending this process into the transfinite at levels ω, ω + 1,. However this involved conceptual difficulties which led him to re-examine the foundations of the real number system and eventually to create set theory, including, several years later, the rigorous development of the theory of ordinal numbers and transfinite induction. However, after 1872 Cantor never returned to the problem of uniqueness and never published a complete proof of 3.5. This was done, much later, by Lebesgue in Theorem 4.1 was further extended by Bernstein (1908) and W. H. Young (1909) to show that an arbitrary countable set is a set of uniqueness. Finally in 1923 Bari showed that the union of countably many closed sets of uniqueness is a set of uniqueness Exercise (Bernstein, Young). Assume these results and the fact that any uncountable Borel set contains a non- perfect subset. Show that every set which contains no perfect non- set is a set of uniqueness. 13

14 PART II. BAIRE CATEGORY METHODS. 7. Sets of uniqueness and Lebesgue measure. To get some idea about the size of sets of uniqueness, we will first prove the following easy fact Proposition. Let A T be a (Lebesgue) measurable set of uniqueness. Then A is null, i.e., λ(a) = 0. Proof. Assume λ(a) > 0, towards a contradiction. Then, by regularity, λ(f) > 0 for some closed subset F A. Consider the characteristic function χ F of F and its Fourier series S(χ F ). We need the following standard fact Lemma (Localization principle for Fourier series). Let f be an integrable function on T. Then the Fourier series of f converges to 0 in any open interval in which f vanishes. Proof. We have S N (f,x) = N N ˆf(n)e inx = N N ( 1 2π 2π 0 f(t)e int dt)e inx = 1 2π ( N f(t) e )dt in(x t) 2π 0 N where the Dirichlet kernel D n is defined by = 1 2π f(t)d N (x t)dt (= f D n (x)), 2π 0 D n (u) = N N e inu = sin(n )u sin u 2 = cos nu + cot( u )sin nu. 2 So (changing 2π 0 to π π ) S N (f,0) = 1 2π π π f(t)cos ntdt + 1 π f(t)cot( t )sin ntdt. 2π π 2 If now f vanishes in an interval around 0, then clearly f(t)cot( t 2 ) is integrable, so these two integrals converge to 0 by the Riemann-Lebesgue Lemma (3.2). So S N (f,0) 0. 14

15 Thus we have shown that if f vanishes in an interval around 0, then its Fourier series converges to 0 at 0. By translation this is true for any other point. So S(χ F ) converges to 0 in any interval disjoint from F, i.e., S(χ F ) converges to 0 off F. So, since F is a set of uniqueness, ˆχ F (n) = 0 for all n. But ˆχ F (0) = λ(f) > 0, a contradiction Exercise (Bernstein). Show that there is A Tsuch that neither A nort \ A contain a non- perfect set (such a set is called a Bernstein set). Conclude that A is not measurable and therefore that A is a set of uniqueness which is not null. So we have countable U measurable null. In the beginning of the century it was widely believed that U measurable = null, i.e., if a trigonometric series c n e inx converges to 0 almost everywhere (a.e.), then it is identically 0. Recall, for example, here the following standard fact (which we will not use later on) Theorem. (Fejér-Lebesgue). For every integrable function f on T, let σ N (f,x) = S 0(f,x) + S 1 (f,x) + + S N (f,x) N + 1 be the average of the partial sums S N (f,x) of the Fourier series of f. Then σ N (f,x) f(x) a.e. In particular, if a Fourier series converges a.e. to 0, it is identically 0. For example, in Luzin s dissertation Integration and trigonometric series (1915), this problem is discussed and is considered improbable that there is a non-zero trigonometric series that converges to 0 a.e. It thus came as a big surprise when in 1916 Menshov proved that there are indeed trigonometric series which converge to 0 a.e. but are not identically 0. It follows, for example, that any function which admits a trigonometric expansion admits actually more than one, if only convergence a.e. to the function is required. Remark. Menshov also showed that every measurable function admits an a.e. trigonometric expansion, i.e., for any 2π-periodic measurable f there is a trigonometric series cn e inx such that f(x) = c n e inx a.e. By the above this series is not unique. Menshov actually constructed an example of a closed set of multiplicity of measure 0, by an appropriate modification of the standard construction of the Cantor set. The Cantor set in the interval [0, 2π] is constructed by removing the middle 1/3 open interval, then in each of the remaining two closed intervals remaining the middle 1/3 open interval, etc. Now suppose we modify the construction by removing in the first stage the middle 1/2 interval, at the second stage, the middle 1/3 interval, at the third stage the middle 1/4 interval, etc. Denote the resulting perfect set by E M Exercise. Show that λ(e M ) = 0. Now Menshov showed that E M is a set of multiplicity as follows: This set is in a canonical 1-1 correspondence with the set of infinite binary sequences 2 N. Take the usual 15

16 coin-tossing measure on 2 N and transfer it by this correspondence to E M. Call this measure µ M. (Thus µ M gives equal measure 1/2 n to each of the 2 n closed intervals at the nth stage of this construction.) It is a probability Borel measure ont, so we can define its Fourier(- Stieltjes) coefficients ˆµ M (n), as usual, by ˆµ M (n) = e int dµ M (t). Then it turns out that ˆµ M (n)e inx = 0 for x / E, so E is a set of multiplicity (as ˆµ M (0) = µ(e) > 0). This might seem not too surprising, since after all the measure µ lives on E M. However by the same token one should also expect that if we consider the usual Cantor set E C and denote the corresponding measure by µ C we should have ˆµC (n)e inx = 0 off the Cantor set as well, i.e., the Cantor set should also be a set of multiplicity, which is false! This is only one of the many phenomena in this subject which challenge your intuition. What is the difference between µ M,µ C that accounts for this phenomenon? It turns out that it is the following: ˆµ M (n) 0 but ˆµ C (n) 0 as n. In fact, we have the following result Theorem. Let E T, E Tbe a closed set and let µ be a probability Borel measure ontwith µ(e) = 1. Then the following are equivalent: (i) ˆµ(n) 0; (ii) ˆµ(n)e inx = 0, x / E. Clearly (ii) (i) by the Cantor-Lebesgue Lemma. The proof of (i) (ii) requires some further background in the theory of trigonometric series and we will postpone it for a while (see 12,13). Thus Menshov s proof is based on the fact that ˆµ M (n) 0, which is proved by a delicate calculation. We will develop in the sequel a totally different approach to Menshov s Theorem on the existence of null sets of multiplicity, an approach based on the Baire category method. 8. Baire category. A set in a topological space is nowhere dense if its closure has empty interior. A set is first category or meager if it is contained in a countable union of nowhere dense sets. Otherwise it is of the second category or non-meager. It is clear that meager sets form a σ-ideal, i.e., are closed under subsets and countable unions. So this concept determines a notion of topological smallness analogous to that of null sets in measure theory. Of course, for this to be of any interest it better be that it doesn t trivialize, i.e., that the whole space is not meager. This is the case in well-behaved spaces, like complete metric spaces. In fact we have an even stronger statement known as the Baire Category Theorem (for such spaces). We call a set comeager if its complement is meager. Notice that a set is comeager iff it contains a countable intersection of dense open sets. 16

17 8.1 Theorem (The Baire Category Theorem). Let X be a complete metric space. Every comeager set is dense (in particular non- ). This is the basis of a classical method of existence proof in mathematics: Suppose we want to show the existence of a mathematical object x satisfying some property P. The category method consists of finding an appropriate complete metric space X (or other nice topological space satisfying the Baire Category Theorem) and showing that {x X : P(x)} is comeager in X. This not only shows that xp(x), but in fact that in the space X most elements of X have property P or as it is often expressed the generic element of X satisfies property P. (Similarly, if a property holds a.e. we say that the random element satisfies it.) A standard application of the Baire Category Theorem is Banach s proof of the existence of continuous nowhere differentiable functions (originally due to Bolzano and Weierstrass in the 19th century). The argument goes as follows: Let C(T) be the space of real valued continuous 2π-periodic functions with the uniform (or sup) metric d(f,g) = sup{ f(x) g(x) : x T}. It is well-known that this is a complete metric space. We want to show that {f C(T) : x(f (x) does not exist)} is comeager in C(T), so non-. Consider, for each n, the set U n = {f C(T) : x h > 0 f(x + h) f(x) h > n} Exercise. Show that U n is open in C(T). It is not hard to show now that U n is also dense in U(T). Simply approximate any f C(T) by a piecewise linear function g and then approximate g by a piecewise linear function with big slopes, so that it belongs to U n. Thus n U n is a countable intersection of dense open sets, so it is comeager. But clearly if f n U n, f is nowhere differentiable. (So the generic continuous function is nowhere differentiable.) Before we proceed, let us recall that a set A, in a topological space X, is said to have the Baire property (BP) if there is an open set U such that A U is meager. The class of sets with the BP is a σ-algebra (i.e., is closed under countable unions and complements), in fact it is the smallest σ-algebra containing the open sets and the meager sets. The sets with the BP are analogs of the measurable sets. Although there is some analogy between category and measure it should be emphasized that the concepts are orthogonal. This can be expressed by the following important fact: 8.3. Proposition. There is a dense G δ (so comeager) set G T such that λ(g) = 0 (i.e., G is null). 17

18 Proof. Let {d n } be a countable dense set int. For each m let I n,m be an open interval around d n with λ(i n,m ) 1 m 2 n. Let U m = n I n,m, so that U m is dense open with λ(u m ) n λ(i n,m) 2/m. Let G = m U m. 9. Sets of uniqueness and category. We have seen in 7 that a (measurable) set of uniqueness is measure theoretically negligible, i.e., null. The question was raised, already in the 1920 s (see, e.g., the memoir of N. Bari in Fundamenta Mathematica, Bari [1927]) whether they are also topologically negligible, i.e., meager (assuming they have the BP). So we have 9.1. The Category Problem. Is every set of uniqueness with the BP of the first category? This problem was solved affirmatively by Debs and Saint Raymond in Their original proof used the descriptive set theoretic methods that we will discuss in Part III and was quite sophisticated, making use of machinery established in earlier work of Solovay, Kaufman, Kechris-Louveau-Woodin and Kechris-Louveau. In fact Debs and Saint Raymond established the following stronger result Theorem (Debs-Saint Raymond). Let A T be a non-meager set with the BP. Then there is a Borel probability measure µ on T with µ(a) = 1 and ˆµ(n) 0, as n. To see that this implies 9.1 we argue as follows: Let A be a set of uniqueness with the BP. If A is not meager, there is µ, a Borel probability measure, with µ(a) = 1, and ˆµ(n) 0. Since every Borel probability measure is regular, there is closed F A with µ(f) > 0. Let ν = µ F, i.e., ν(x) = µ(x F) Proposition. ˆν(n) 0, as n. Proof. We have for any continuous function ont, fdν = fχ F dµ, so ˆν(n) = χ F (t)e int dµ(t). For each ǫ > 0, there is a trigonometric polynomial P(x) = N N c ke ikx such that χf P dµ < ǫ. Now if d n = P(t)e int dµ(t), we have ( N d n = c k e )e ikx int dµ(t) N N = c kˆµ(n k) 0 k= N 18

19 as n. Moreover ˆν(n) d n = (χ F (t) P(t))e int dµ χ F P dµ < ǫ, so lim n ˆν(n) ǫ, and thus ˆν(n) 0. Then we can apply 7.6 to conclude that ˆν(n)e inx = 0 off E and thus off A. As ˆν(0) 0, this shows that A is a set of multiplicity, a contradiction. Starting with the next section we will give a very different than the original and much simpler proof of 9.2, due to Kechris-Louveau, which is based on the category method and employs only elementary functional analysis. Before we do that though we want to show how 9.2 gives also a much different proof of Menshov s Theorem, which explains this result as an instance of the orthogonality of the concepts of null and meager sets. Indeed, by 8.3 fix a dense G δ set G Twith λ(g) = 0. Then G is comeager, so by 9.2, there is a Borel probability measure µ with µ(g) = 1 and ˆµ(n) 0. As before, find F G closed with µ(f) > 0. Then ˆµ(n)e inx = 0 off F and thus off G and this shows that ˆµ(n)e inx converges to 0 a.e. without being identically 0 (it also shows that F is a null closed set of multiplicity, as Menshov also showed). But this method has also many other applications. For example, in the 1960 s Kahane and Salem raised the following question: Recall that a number x [0, 2π] is called normal in base 2 (say) if for x 2π = 0 x 1x 2, x i {0,1}, and all (a 0,,a k 1 ), a i {0,1}, we have 1 lim n n card {1 m n : x m+i = a m+i,0 i k 1} = 1 2. k Denote by N the set of normal numbers and by N its complement. A famous theorem of Borel asserts that almost every number is normal, i.e., λ(n) = 1, so λ(n ) = 0. Kahane and Salem asked whether N supports any probability Borel measure µ (i.e., µ(n ) = 1) with ˆµ(n) 0. (Measures with ˆµ(n) 0 are somehow considered thick - recall that ˆλ(n) 0, since in fact ˆλ(n) = 0 if n 0. It can be also shown that they are continuous, i.e., give every singleton measure 0. In fact a theorem of Wiener asserts that x T µ({x}) 2 = lim n 1 2N+1 N ˆµ(n) 2.) Lyons, in 1983, answered this affirmatively, by using N delicate analytical tools. However, a totally different proof can be based on 9.2 by noticing that 9.4. Proposition. N is comeager. Proof. We show that N is meager. Note that N n 1F n, 19

20 where with { F k = 2π F n = k n F k, x l 2 l : x l = 0,1 and l=1 x 1 + x k k }. 4 Now F k is closed being the continuous image of the closed (thus compact) subset of 2N { (x 1,x 2, ) 2 N : x x k k }, 4 by the continuous map (x 1,x 2, ) 2π x l 2 l. l=1 So F n is closed Exercise. F n contains no open interval. So each F n is nowhere dense and N is meager. 10. Review of duality in Banach spaces. Good references for the basic results of functional analysis and measure theory that we will use in the sequel are Rudin [1973], [1987]. Let X be a Banach space, i.e., a complete normed linear space, over the complex numbers. Denote by X the dual space of X, i.e., the Banach space of all (bounded or, equivalently, continuous) linear functionals x : X C with the norm { x x (x) = sup x } : x X,x 0 { } = sup x (x) : x X, x 1. It is often convenient to write x (x) = x,x = x,x. A set A X is called convex if for every x,y A and t [0,1],tx + (1 t)y A. There is a fundamental collection of results, collectively known as the Hahn-Banach Theorems, which assert the existence of appropriate linear functionals. We will need here the following separation form of Hahn-Banach. 20

### NOTES ON MEASURE THEORY. M. Papadimitrakis Department of Mathematics University of Crete. Autumn of 2004

NOTES ON MEASURE THEORY M. Papadimitrakis Department of Mathematics University of Crete Autumn of 2004 2 Contents 1 σ-algebras 7 1.1 σ-algebras............................... 7 1.2 Generated σ-algebras.........................

### Chapter 5: Application: Fourier Series

321 28 9 Chapter 5: Application: Fourier Series For lack of time, this chapter is only an outline of some applications of Functional Analysis and some proofs are not complete. 5.1 Definition. If f L 1

### ) + ˆf (n) sin( 2πnt. = 2 u x 2, t > 0, 0 < x < 1. u(0, t) = u(1, t) = 0, t 0. (x, 0) = 0 0 < x < 1.

Introduction to Fourier analysis This semester, we re going to study various aspects of Fourier analysis. In particular, we ll spend some time reviewing and strengthening the results from Math 425 on Fourier

### Problem Set. Problem Set #2. Math 5322, Fall December 3, 2001 ANSWERS

Problem Set Problem Set #2 Math 5322, Fall 2001 December 3, 2001 ANSWERS i Problem 1. [Problem 18, page 32] Let A P(X) be an algebra, A σ the collection of countable unions of sets in A, and A σδ the collection

### Weak topologies. David Lecomte. May 23, 2006

Weak topologies David Lecomte May 3, 006 1 Preliminaries from general topology In this section, we are given a set X, a collection of topological spaces (Y i ) i I and a collection of maps (f i ) i I such

### CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

### REAL ANALYSIS I HOMEWORK 2

REAL ANALYSIS I HOMEWORK 2 CİHAN BAHRAN The questions are from Stein and Shakarchi s text, Chapter 1. 1. Prove that the Cantor set C constructed in the text is totally disconnected and perfect. In other

### 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is:

CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Fourier series. Jan Philip Solovej. English summary of notes for Analysis 1. May 8, 2012

Fourier series Jan Philip Solovej English summary of notes for Analysis 1 May 8, 2012 1 JPS, Fourier series 2 Contents 1 Introduction 2 2 Fourier series 3 2.1 Periodic functions, trigonometric polynomials

### MATH41112/61112 Ergodic Theory Lecture Measure spaces

8. Measure spaces 8.1 Background In Lecture 1 we remarked that ergodic theory is the study of the qualitative distributional properties of typical orbits of a dynamical system and that these properties

### BOREL SETS, WELL-ORDERINGS OF R AND THE CONTINUUM HYPOTHESIS. Proof. We shall define inductively a decreasing sequence of infinite subsets of N

BOREL SETS, WELL-ORDERINGS OF R AND THE CONTINUUM HYPOTHESIS SIMON THOMAS 1. The Finite Basis Problem Definition 1.1. Let C be a class of structures. Then a basis for C is a collection B C such that for

### Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

### CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### Structure of Measurable Sets

Structure of Measurable Sets In these notes we discuss the structure of Lebesgue measurable subsets of R from several different points of view. Along the way, we will see several alternative characterizations

### 9 More on differentiation

Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### A List of Problems in Real Analysis

A List of Problems in Real Analysis W.Yessen & T.Ma June 26, 215 This document was first created by Will Yessen, who now resides at Rice University. Timmy Ma, who is still a student at UC Irvine, now maintains

### Polish spaces and standard Borel spaces

APPENDIX A Polish spaces and standard Borel spaces We present here the basic theory of Polish spaces and standard Borel spaces. Standard references for this material are the books [143, 231]. A.1. Polish

### Compactness in metric spaces

MATHEMATICS 3103 (Functional Analysis) YEAR 2012 2013, TERM 2 HANDOUT #2: COMPACTNESS OF METRIC SPACES Compactness in metric spaces The closed intervals [a, b] of the real line, and more generally the

### SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

### MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,

### INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### Sequences of Functions

Sequences of Functions Uniform convergence 9. Assume that f n f uniformly on S and that each f n is bounded on S. Prove that {f n } is uniformly bounded on S. Proof: Since f n f uniformly on S, then given

### 3 σ-algebras. Solutions to Problems

3 σ-algebras. Solutions to Problems 3.1 3.12 Problem 3.1 (i It is clearly enough to show that A, B A = A B A, because the case of N sets follows from this by induction, the induction step being A 1...

### CHAPTER 1 BASIC TOPOLOGY

CHAPTER 1 BASIC TOPOLOGY Topology, sometimes referred to as the mathematics of continuity, or rubber sheet geometry, or the theory of abstract topological spaces, is all of these, but, above all, it is

### Math 507/420 Homework Assignment #1: Due in class on Friday, September 20. SOLUTIONS

Math 507/420 Homework Assignment #1: Due in class on Friday, September 20. SOLUTIONS 1. Show that a nonempty collection A of subsets is an algebra iff 1) for all A, B A, A B A and 2) for all A A, A c A.

### DISTRIBUTIONS AND FOURIER TRANSFORM

DISTRIBUTIONS AND FOURIER TRANSFORM MIKKO SALO Introduction. The theory of distributions, or generalized functions, provides a unified framework for performing standard calculus operations on nonsmooth

### NONMEASURABLE ALGEBRAIC SUMS OF SETS OF REALS

NONMEASURABLE ALGEBRAIC SUMS OF SETS OF REALS MARCIN KYSIAK Abstract. We present a theorem which generalizes some known theorems on the existence of nonmeasurable (in various senses) sets of the form X+Y.

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.

MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we

### Fourier Series and Fejér s Theorem

wwu@ocf.berkeley.edu June 4 Introduction : Background and Motivation A Fourier series can be understood as the decomposition of a periodic function into its projections onto an orthonormal basis. More

### MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION. Chapter 4: Fourier Series and L 2 ([ π, π], µ) ( 1 π

MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION Chapter 4: Fourier Series and L ([, π], µ) Square Integrable Functions Definition. Let f : [, π] R be measurable. We say that f

### INTRODUCTION TO FOURIER ANALYSIS

NTRODUCTON TO FOURER ANALYSS SAGAR TKOO Abstract. n this paper we introduce basic concepts of Fourier analysis, develop basic theory of it, and provide the solution of Basel problem as its application.

### Chap2: The Real Number System (See Royden pp40)

Chap2: The Real Number System (See Royden pp40) 1 Open and Closed Sets of Real Numbers The simplest sets of real numbers are the intervals. We define the open interval (a, b) to be the set (a, b) = {x

### 1. R In this and the next section we are going to study the properties of sequences of real numbers.

+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

### Fourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks

Chapter 3 Fourier Series 3.1 Some Properties of Functions 3.1.1 Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results

### MATH 461: Fourier Series and Boundary Value Problems

MATH 461: Fourier Series and Boundary Value Problems Chapter III: Fourier Series Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2015 fasshauer@iit.edu MATH 461 Chapter

### Riesz-Fredhölm Theory

Riesz-Fredhölm Theory T. Muthukumar tmk@iitk.ac.in Contents 1 Introduction 1 2 Integral Operators 1 3 Compact Operators 7 4 Fredhölm Alternative 14 Appendices 18 A Ascoli-Arzelá Result 18 B Normed Spaces

### Math 563 Measure Theory Project 1 (Funky Functions Group) Luis Zerón, Sergey Dyachenko

Math 563 Measure Theory Project (Funky Functions Group) Luis Zerón, Sergey Dyachenko 34 Let C and C be any two Cantor sets (constructed in Exercise 3) Show that there exists a function F: [,] [,] with

### 1. Sigma algebras. Let A be a collection of subsets of some fixed set Ω. It is called a σ -algebra with the unit element Ω if. lim sup. j E j = E.

Real Analysis, course outline Denis Labutin 1 Measure theory I 1. Sigma algebras. Let A be a collection of subsets of some fixed set Ω. It is called a σ -algebra with the unit element Ω if (a), Ω A ; (b)

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### 6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

### Lectures 5-6: Taylor Series

Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,

### This chapter describes set theory, a mathematical theory that underlies all of modern mathematics.

Appendix A Set Theory This chapter describes set theory, a mathematical theory that underlies all of modern mathematics. A.1 Basic Definitions Definition A.1.1. A set is an unordered collection of elements.

### Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### Course 421: Algebraic Topology Section 1: Topological Spaces

Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

### Mathematics for Econometrics, Fourth Edition

Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents

### THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

### Applications of Fourier series

Chapter Applications of Fourier series One of the applications of Fourier series is the evaluation of certain infinite sums. For example, n= n,, are computed in Chapter (see for example, Remark.4.). n=

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### If the sets {A i } are pairwise disjoint, then µ( i=1 A i) =

Note: A standing homework assignment for students in MAT1501 is: let me know about any mistakes, misprints, ambiguities etc that you find in these notes. 1. measures and measurable sets If X is a set,

### Measuring sets with translation invariant Borel measures

Measuring sets with translation invariant Borel measures University of Warwick Fractals and Related Fields III, 21 September 2015 Measured sets Definition A Borel set A R is called measured if there is

### 1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

### CHAPTER 5. Product Measures

54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Limits and convergence.

Chapter 2 Limits and convergence. 2.1 Limit points of a set of real numbers 2.1.1 Limit points of a set. DEFINITION: A point x R is a limit point of a set E R if for all ε > 0 the set (x ε,x + ε) E is

### Infinite product spaces

Chapter 7 Infinite product spaces So far we have talked a lot about infinite sequences of independent random variables but we have never shown how to construct such sequences within the framework of probability/measure

### MAA6616 COURSE NOTES FALL 2012

MAA6616 COURSE NOTES FALL 2012 1. σ-algebras Let X be a set, and let 2 X denote the set of all subsets of X. We write E c for the complement of E in X, and for E, F X, write E \ F = E F c. Let E F denote

### Notes on weak convergence (MAT Spring 2006)

Notes on weak convergence (MAT4380 - Spring 2006) Kenneth H. Karlsen (CMA) February 2, 2006 1 Weak convergence In what follows, let denote an open, bounded, smooth subset of R N with N 2. We assume 1 p

### Metric Spaces. Chapter 1

Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### Introduction to Topology

Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................

### Exercise 1. Let E be a given set. Prove that the following statements are equivalent.

Real Variables, Fall 2014 Problem set 1 Solution suggestions Exercise 1. Let E be a given set. Prove that the following statements are equivalent. (i) E is measurable. (ii) Given ε > 0, there exists an

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

### The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line

The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line D. Bongiorno, G. Corrao Dipartimento di Ingegneria lettrica, lettronica e delle Telecomunicazioni,

### Taylor and Maclaurin Series

Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions

### Measure Theory. 1 Measurable Spaces

1 Measurable Spaces Measure Theory A measurable space is a set S, together with a nonempty collection, S, of subsets of S, satisfying the following two conditions: 1. For any A, B in the collection S,

### Vector Spaces II: Finite Dimensional Linear Algebra 1

John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition

### Descriptive Set Theory

Martin Goldstern Institute of Discrete Mathematics and Geometry Vienna University of Technology Ljubljana, August 2011 Polish spaces Polish space = complete metric separable. Examples N = ω = {0, 1, 2,...}.

### Ideal Class Group and Units

Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

### Sequences and Convergence in Metric Spaces

Sequences and Convergence in Metric Spaces Definition: A sequence in a set X (a sequence of elements of X) is a function s : N X. We usually denote s(n) by s n, called the n-th term of s, and write {s

### Set theory, and set operations

1 Set theory, and set operations Sayan Mukherjee Motivation It goes without saying that a Bayesian statistician should know probability theory in depth to model. Measure theory is not needed unless we

### Duality of linear conic problems

Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

### The Ideal Class Group

Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned

### The Banach-Tarski Paradox

University of Oslo MAT2 Project The Banach-Tarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the Banach-Tarski paradox states that for any ball in R, it is possible

### Measure Theory and Probability

Chapter 2 Measure Theory and Probability 2.1 Introduction In advanced probability courses, a random variable (r.v.) is defined rigorously through measure theory, which is an in-depth mathematics discipline

### General theory of stochastic processes

CHAPTER 1 General theory of stochastic processes 1.1. Definition of stochastic process First let us recall the definition of a random variable. A random variable is a random number appearing as a result

### INTRODUCTION TO DESCRIPTIVE SET THEORY

INTRODUCTION TO DESCRIPTIVE SET THEORY ANUSH TSERUNYAN Mathematicians in the early 20 th century discovered that the Axiom of Choice implied the existence of pathological subsets of the real line lacking

### INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem

### Chapter 1. Metric Spaces. Metric Spaces. Examples. Normed linear spaces

Chapter 1. Metric Spaces Metric Spaces MA222 David Preiss d.preiss@warwick.ac.uk Warwick University, Spring 2008/2009 Definitions. A metric on a set M is a function d : M M R such that for all x, y, z

### Chapter 7. Induction and Recursion. Part 1. Mathematical Induction

Chapter 7. Induction and Recursion Part 1. Mathematical Induction The principle of mathematical induction is this: to establish an infinite sequence of propositions P 1, P 2, P 3,..., P n,... (or, simply

### You know from calculus that functions play a fundamental role in mathematics.

CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.

### Analysis I. 1 Measure theory. Piotr Haj lasz. 1.1 σ-algebra.

Analysis I Piotr Haj lasz 1 Measure theory 1.1 σ-algebra. Definition. Let be a set. A collection M of subsets of is σ-algebra if M has the following properties 1. M; 2. A M = \ A M; 3. A 1, A 2, A 3,...

### Ultraproducts and Applications I

Ultraproducts and Applications I Brent Cody Virginia Commonwealth University September 2, 2013 Outline Background of the Hyperreals Filters and Ultrafilters Construction of the Hyperreals The Transfer

### since by using a computer we are limited to the use of elementary arithmetic operations

> 4. Interpolation and Approximation Most functions cannot be evaluated exactly: x, e x, ln x, trigonometric functions since by using a computer we are limited to the use of elementary arithmetic operations

### 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

### Prime Numbers and Irreducible Polynomials

Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

### SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some