f(x)dx = lim f n (x)dx.

Size: px
Start display at page:

Download "f(x)dx = lim f n (x)dx."

Transcription

1 Chapter 3 Lebesgue Integration 3.1 Introduction The concept of integration as a technique that both acts as an inverse to the operation of differentiation and also computes areas under curves goes back to the origin of the calculus and the work of Isaac Newton ( ) and Gottfried Leibniz ( ). It was Leibniz who introduced the dx notation. The first rigorous attempt to understand integration as a limiting operation within the spirit of analysis was due to Bernhard Riemann ( ). The approach to Riemann integration that is usually taught (as in MA221) was developed by Jean-Gaston Darboux ( ). At the time it was developed, this theory seemed to be all that was needed but as the 19th century drew to a close, some problems appeared: One of the main tasks of integration is to recover a function f from its derivative f. But some functions were discovered for which f was bounded but not Riemann integrable. uppose (f n ) is a sequence of functions converging pointwise to f. The Riemann integral could not be used to find conditions for which f(x)dx = lim f n (x)dx. Riemann integration was limited to computing integrals over R n with respect to Lebesgue measure. Although it was not yet apparent, the emerging theory of probability would require the calculation of expectations of random variables X: E(X) = X(ω)dP (ω). Ω In this chapter, we ll study Lebesgue s powerful techniques which allow us to investigate f(x)dm(x) where f : R is a suitable measurable 23

2 function defined on a measure space (, Σ, m). 1 If we take m to be Lebesgue measure on (R, B(R)) we recover the familiar integral f(x)dx but we will R now be able to integrate many more functions (at least in principle) than Riemann and Darboux. If we take X to be a random variable on a probability space, we get its expectation E(X). Notation. For simplicity we usually write fdm instead of f(x)dm(x). Note that many authors use f(x)m(dx). To simplify even further we ll sometimes write I(f) = fdm. We ll present the construction of the Lebesgue integral in four steps: tep 1: Indicator functions, tep 2: imple Functions, tep 3: Non-negative measurable functions, tep 4: Integrable functions. 3.2 The Lebesgue Integral for imple Functions tep 1. Indicator Functions This is very easy and yet it is very important: If f = 1 A where A Σ 1 A dm = m(a). (3.2.1) e.g. In a probability space we get E(1 A ) = P (A). tep 2. imple Functions Let f = n c i1 Ai be a non-negative simple function so that c i 0 for all 1 i n. We extend (3.2.1) by linearity, i.e. we define and note that fdm [0, ]. fdm = c i m(a i ), (3.2.2) Theorem If f and g are non-negative simple functions and α, β 0 then 1. ( Linearity ) (αf + βg)dm = α fdm + β gdm, 2. (Monotonicity) If f g then fdm gdm. 1 We may also integrate extended measurable functions, but will not develop that here. 24

3 Proof 1. Let f = n c i1 Ai, g = m j=1 d j1 Bj. ince n A i = m j=1 B j =, by the last part of Problem 10 we see that m f = c i 1 Ai = c i 1 Ai m j=1 B = c j i 1 Ai B j. It follows that and so I(αf + βg) = = α = α αf + βg = j=1 m (αc i + βd j )1 Ai B j, j=1 m (αc i + βd j )m(a i B j ) j=1 ( ) m c i m A i B j + β j=1 c i m(a i ) + β = αi(f) + βi(g). ( m n ) d j m A i B j j=1 m d j m(b j ) 2. By (1), I(g) = I(f) + I(g f) but g f is a non-negative simple function and so I(g f) 0. The result follows. Notation. If A Σ, whenever fdm makes sense for some reasonable measurable function f : R we define: I A (f) = fdm = 1 A fdm. A Of course there is no guarantee that I A (f) makes sense and this needs checking at each stage. In Problem 23, you can check that it makes sense when f is non-negative and simple. 3.3 The Lebesgue Integral for Non-negative Measurable Functions We haven t done any analysis yet and at some stage we surely need to take some sort of limit! If f is measurable and non-negative, it may seem attractive to try to take advantage of Theorem and define fdm = 25 j=1

4 lim s ndm. But there are many different choices of simple functions that we could take to make an approximating sequence, and this would make the limiting integral depend on that choice, which is undesirable. Instead Lebesgue used the weaker notion of the supremum to approximate f from below as follows: tep 3. Non-negative measurable functions { } fdm = sup sdm, s simple, 0 s f. (3.3.3) With this definition, fdm [0, ]. The use of the sup makes it harder to prove key properties and we ll have to postpone a full proof of linearity until the next section when we have some more powerful tools. Here are some simple properties that can be proved fairly easily. Theorem If f, g : R are non-negative measurable functions, 1. (Monotonicity) If f g then fdm gdm. 2. I(αf) = αi(f) for all α > 0, 3. If A, B Σ with A B then I A (f) I B (f), 4. If A Σ with m(a) = 0 then I A (f) = 0. Proof. (1) { fdm = sup { sup = gdm } sdm, s simple, 0 s f } sdm, s simple, 0 s g (2) to (4) are Problem 24. Lemma [Markov s inequality] If f : R is a non-negative measurable function and c > 0. m({x ; f(x) c}) 1 fdm c 26

5 Proof. Let E = {x ; f(x) c}. Note that E = f 1 ([c, )) Σ as f is measurable (see Theorem (ii)). By Theorem (3) and (1), fdm fdm E cdm and the result follows. E = cm(e), Definition. Let f, g : R be measurable. We say that f = g almost everywhere, and write this for short as f = g a.e., if m({x ; f(x) g(x)} = 0. In Problem 31 you can show that this gives rise to an equivalence relation on the set of all measurable functions. In probability theory, we use the terminology almost surely for two random variables X and Y that agree almost everywhere, and we write X = Y (a.s.) Corollary If f is a non-negative measurable function and fdm = 0 then f = 0 (a.e.) Proof. Let A = {x ; f(x) 0} and for each n N, A n = {x ; f(x) 1/n}. ince A = n=1 A n, we have m(a) n=1 m(a n) by Theorem 1.5.2, and its sufficient to show that m(a n ) = 0 for all n N. But by Markov s inequality m(a n ) n fdm = 0. In Chapter 1 we indicated that we would be able to use integration to cook up new examples of measures. Let f : R be non-negative and measurable and define I A (f) = fdm for A Σ. We have fdm = 0 by A Theorem (4). To prove that A I A (f) is a measure we then need only prove that it is σ-additive, i.e. that I A (f) = n=1 I A n (f) whenever we have a disjoint union A = n=1 A n. Theorem If f : R is a non-negative measurable function, the mapping from Σ to [0, ] given by A I A (f) is σ-additive. Proof. First assume that f = 1 B for some B Σ. Then by (3.2.1) ( ) I A (f) = m(b A) = m B A n = n=1 m(b A n ) = n=1 27 I An (f), n=1

6 so the result holds in this case. You can then use linearity to show that it is true for non-negative simple functions. Now let f be measurable and non-negative. Then by definition of the supremum, for any ɛ > 0 there exists a simple function s with 0 s f so that I A (f) I A (s) + ɛ. The result holds for simple functions and so by monotonicity we have I A (s) = I An (s) n=1 I An (f). n=1 Combining this with the earlier inequality we find that I A (f) I An (f) + ɛ. n=1 But ɛ was arbitrary and so we conclude that I A (f) I An (f). n=1 The second half of the proof will aim to establish the opposite inequality. First let A 1, A 2 Σ be disjoint. Given any ɛ > 0 we can, as above, find simple functions s 1, s 2 with 0 s j f, so that I Aj (s j ) I Aj (f) ɛ/2 for j = 1, 2. Let s = s 1 s 2 = max{s 1, s 2 }. Then s is simple (check this), 0 s f and s 1 s, s 2 s. o by monotonicity, I Aj (s) I Aj (f) ɛ/2 for j = 1, 2. Add these two inequalities to find that I A1 (s) + I A2 (s) I A1 (f) + I A2 (f) ɛ. But the result is true for simple functions and so we have I A1 A 2 (s) I A1 (f) + I A2 (f) ɛ. By the definition (3.3.3), I A1 A 2 (f) I A1 A 2 (s) and so we have that I A1 A 2 (f) I A1 (f) + I A2 (f) ɛ. But ɛ was arbitrary and so we conclude that I A1 A 2 (f) I A1 (f) + I A2 (f), which is the required inequality for unions of two disjoint sets. By induction we have I A1 A 2 A n (f) I Ai (f), 28

7 for any n 2. But as A 1 A 2 A n A we can use Theorem (3) to find that I A (f) I Ai (f). Now take the limit as n to deduce that I A (f) I Ai (f), as was required. Example. The famous Gaussian measure on R is obtained in this way by taking f(x) = 1 2π e x2 2 for x R, with m being Lebesgue measure. In this case, f(x)dx = 1. To connect R more explicitly with probability theory, let (Ω, F, P ) be a probability space and X : Ω R be a random variable. Equip (R, B(R)) with Lebesgue measure. In Chapter 2, we introduced the probability law p X of X as a probability measure on (R, B(R)). If for all A B(R), p X (A) = I(f X 1 A ) for some non-negative measurable function f X : R R then f X is called the probability density function or pdf of X. o for all A B(R), P (X A) = p X (A) = A f X (x)dx. We say that X is a standard normal if p X is Gaussian measure. We present two useful corollaries to Theorem 3.3.2: Corollary Let f : R be a non-negative measurable function and (E n ) be a sequence of sets in Σ with E n E n+1 for all n N. et E = n=1 E n, Then fdm = lim fdm. E E n Proof. This is in fact an immediate consequence of Theorems and 1.5.1, but it might be helpful to spell out the proof in a little detail, so here goes: We use the disjoint union trick, so write A 1 = E 1, A 2 = E 2 E 1, A 3 = E 3 E 2,.... Then the A n s are mutually disjoint, n=1 A n = E and n A i = 29

8 E n for all n N. Then by Theorem fdm = fdm E A i = lim fdm A i = lim fdm A 1 A 2 A n = lim fdm. E n Corollary If f and g are non-negative measurable functions and f = g (a.e.) then I(f) = I(g). Proof. Let A 1 = {x ; f(x) = g(x)} and A 2 = {x ; f(x) g(x)}. Then A 1, A 2 Σ with A 1 A 2 =, A 1 A 2 = and m(a 2 ) = 0. o by Theorem (4), A 2 fdm = A 2 gdm = 0. But A 1 fdm = A 1 gdm as f = g on A 1 and so by Theorem 3.3.2, fdm = fdm + fdm A 1 A 2 = gdm + gdm = gdm. A 1 A The Monotone Convergence Theorem We haven t yet proved that (f + g)dm = fdm + gdm. Nor have we extended the integral beyond non-negative measurable functions. Before we can do either of these, we need to establish the monotone convergence theorem. This is the first of two important results that address one of the historical problems of integration that we mentioned in section 3.1 Let (f n ) be a sequence of non-negative measurable functions with f n f n+1 for all n N (so the sequence is monotonic increasing). Note that f = lim f n exists and is non-negative and measurable (see Theorem 2.3.5), but f may take values in [0, ]. Theorem [The Monotone Convergence Theorem] If (f n ) is a monotone increasing sequence of non-negative measurable functions from to R then fdm = lim f n dm. 30

9 Proof. As f = sup n N f n, by monotonicity (Theorem 3.3.1(1)), we have f 1 dm f 2 dm fdm. Hence by monotonicity of the integrals, lim f ndm exists (as an extended real number) and lim f n dm fdm. We must now prove the reverse inequality. To simplify notation, let a = lim f ndm. o we need to show that a fdm. Let s be a simple function with 0 s f and choose c R with 0 < c < 1. For each n N, let E n = {x ; f n (x) cs(x)}, and note that E n Σ for all n N by Proposition ince (f n ) is increasing, it follows that E n E n+1 for all n N. Also we have n=1 E n =. To verify this last identity, note that if x with s(x) = 0 then x E n for all n N and if x with s(x) 0 then f(x) s(x) > cs(x) and so for some n, f n (x) cs(x), as f n (x) f(x) as n, i.e. x E n. By Theorem 3.3.1(3) and (1), we have a = lim f n dm f n dm f n dm csdm. E n E n As this is true for all n N, we find that a lim csdm. E n But by Corollary (since (E n ) is increasing), and Theorem 3.3.1(2), lim csdm = csdm = c sdm, E n and so we deduce that a c sdm. But 0 < c < 1 is arbitrary so taking e.g. c = 1 1/k with k = 2, 3, 4,... and letting k, we find that a sdm. But the simple function s for which 0 s f was also arbitrary, so now take the supremum over all such s and apply (3.3.3) to get a fdm, and the proof is complete.. 31

10 Corollary Let f : R be measurable and non-negative. There exists an increasing sequence of simple functions (s n ) converging pointwise to f so that lim s n dm = fdm. Proof. Apply the monotone convergence theorem to the sequence (s n ) constructed in Theorem Theorem Let f, g : R be measurable and non-negative. Then (f + g)dm = fdm + gdm. Proof. By Theorem we can find an increasing sequence of simple functions (s n ) that converges pointwise to f and an increasing sequence of simple functions (t n ) that converges pointwise to g. Hence (s n + t n ) is an increasing sequence of simple functions that converges pointwise to f + g. o by Theorem 3.4.1, Theorem 3.2.1(1) and then Corollary 3.4.1, (f + g)dm = lim (s n + t n )dm = lim s n dm + lim = fdm + gdm. t n dm Another more delicate convergence result can be obtained as a consequence of the monotone convergence theorem. We present this as a theorem, although it is always called Fatou s lemma in honour of the French astronomer (and mathematician) Pierre Fatou ( ). We will find an important use for this result in the next section. Theorem [Fatou s Lemma] If (f n ) is a sequence of non-negative measurable functions from to R then lim inf f n dm lim inf f ndm Proof. Define g n = inf k n f k. Then (g n ) is an increasing sequence which converges to lim inf f n. Now as f l inf k n f k for all l n, by monotonicity (Theorem 3.3.1(1)) we have that for all l n f l dm inf f kdm, k n 32

11 and so inf f l dm inf f kdm. l n k n Now take limits on both sides of this last inequality and apply the monotone convergence theorem to obtain lim inf f n dm lim inf f kdm k n = lim inf f kdm k n = lim inf f ndm Note that we do not require (f n ) to be a bounded sequence, so lim inf f n should be interpreted as an extended measurable function, as discussed at the end of Chapter Lebesgue Integration Completed: Integrability and Dominated Convergence At last we are ready for the final step in the construction of the Lebesgue integral - the extension from non-negative measurable functions to a class of measurable functions that are real-valued. tep 4. For the final step we first take f to be an arbitrary measurable function. We define the positive and negative parts of f, which we denote as f + and f respectively by: f + (x) = max{f(x), 0}, f (x) = max{ f(x), 0}, so both f + and f are measurable (Corollary 2.3.2) and non-negative. We have f = f + f, and using tep 3, we see that we can construct both f +dm and f dm. Provided both of these are not infinite, we define fdm = f + dm f dm. With this definition, fdm [, ]. We say that f is integrable if fdm (, ). Clearly f is integrable if and only if each of f + and f 33

12 are. Define f (x) = f(x) for all x. As f is measurable, it follows from Problem 17(c) that f also is. ince f = f + + f, it is not hard to see that f is integrable if and only if f is. Using this last fact, the condition for integrability of f is often written f dm <. We also have the useful inequality (whose proof is Problem 30(a)): fdm f dm, (3.5.4) for all integrable f. Theorem uppose that f and g are integrable functions from to R. 1. If c R then cf is integrable and cfdm = c fdm, 2. f + g is integrable and (f + g)dm = fdm + gdm, 3. (Monotonicity) If f g then fdm gdm. Proof. (1) and (3) are Problem 29. For (2), we may assume that both f, g are not identically 0. The fact that f + g is integrable if f and g are follows from the triangle inequality (Problem 30 (b)). To show that the integral of the sum is the sum of the integrals, we first need to consider six different cases (writing h = f + g) (i) f 0, g 0, h 0, (ii) f 0, g 0, h 0, (iii) f 0, g 0, h 0, (iv) f 0, g 0, h 0, (v) f 0, g 0, h 0, (vi) f 0, g 0, h 0. Case (i) is Theorem We ll just prove (iii). The others are similar. If h = f + g then f = h + ( g) and this reduces the problem to case (i). Indeed we then have fdm = (f + g)dm + ( g)dm, and so by (1) (f + g)dm = fdm ( g)dm = 34 fdm + gdm.

13 Now write = , where i is the set of all x for which case (i) holds for i = 1, 2,..., 6. These sets are disjoint and measurable and so by a slight extension of Theorem 3.3.2, 2 (f+g)dm = as was required. 6 i (f+g)dm = 6 i fdm+ 6 gdm = i fdm+ gdm, We now present the last of our convergence theorems, the famous Lebesgue dominated convergence theorem - an extremely powerful tool in both the theory and applications of modern analysis: Theorem [Lebesgue dominated convergence theorem] Let (f n ) be a sequence of measurable functions from to R which converges pointwise to a (measurable) function f. uppose there is an integrable function g : R so that f n g for all n N. Then f is integrable and fdm = lim f n dm. [Note that we don t assume that f n is integrable. This follows immediately from the assumptions since by monotonicity (Theorem 3.3.1(1)), f n dm gdm <.] Proof. ince (f n ) converges pointwise to f, ( f n ) converges pointwise to f. By Fatou s lemma (Theorem 3.4.3) and monotonicity (Theorem 3.5.1(3)), we have f dm = lim inf f n dm lim inf f n dm gdm <, and so f is integrable. Also for all n N, g + f n 0 so by Fatou s lemma again, lim inf (g + f n)dm lim inf (g + f n )dm. 2 This works since we only need finite additivity here. 35

14 But lim inf (g+f n ) = g+lim f n = g+f and (using Theorem 3.5.1(2)) lim inf (g + f n)dm = gdm + lim inf f ndm. We then conclude that fdm lim inf f n dm... (i). Repeat this argument with g + f n replaced by g f n which is also nonnegative for all n N. We then find that ( ) fdm lim inf f n dm = lim sup f n dm, and so fdm lim sup f n dm... (ii). Combining (i) and (ii) we see that lim sup f n dm fdm lim inf f n dm... (iii), but we always have lim inf f ndm lim sup f ndm and so lim inf f ndm = lim sup f ndm. Then by Theorem lim f ndm exists and from (iii), we deduce that fdm = lim f ndm. Example. uppose that (, Σ, m) is a finite measure space and (f n ) is a sequence of measurable functions from to R which converge pointwise to f, and are uniformly bounded, i.e. there exists K > 0 so that f n (x) K for all x, n N. Then f is integrable. To see this just take g = K in the dominated convergence theorem and show that it is integrable, which follows from the fact that gdm = Km() <. We will not prove the next theorem, which shows that the Lebesgue integral on R is at least as powerful as the Riemann one (at least when we integrate over finite intervals.) You can find a proof on the module website. Theorem Let f : [a, b] R be bounded and Riemann integrable. Then it is also Lebesgue integrable and the two integrals have the same value. But we can integrate many more functions using Lebesgue integration that are not Riemann integrable, e.g. [a,b] 1 R Q(x)dx = (b a). Note that Theorem only applies to finite closed intervals. We need to be careful on infinite intervals. In the following examples we will freely use the fact (which was established in MA207) that a bounded continuous function on [a, b] is Riemann integrable. Then by Theorem 3.5.3, it is Lebesgue integrable. In particular 36

15 f(x) = x n is integrable on [a, b] for n Z if 0 / [a, b], f(x) = x n is integrable on [a, b] for n Z + if 0 [a, b]. Example 1. how that f(x) = x α is integrable on [1, ) for α > 1. Proof. For each n N define f n (x) = x α 1 [1,n] (x). Then (f n (x)) increases to f(x) as n. We have 1 f n (x)dx = n By the monotone convergence theorem, 1 1 x α dx = 1 α 1 (1 n1 α ). x α dx = 1 α 1 lim (1 n1 α ) = 1 α 1. Example 2. how that f(x) = x α e x is integrable on [0, ) for α > 0. We use the fact that for any M 0, lim x x M e x = 0, so that given any ɛ > 0 there exists R > 0 so that x > R x M e x < ɛ, and choose M so that M α > 1. Now write x α e x = x α e x 1 [0,R] (x) + x α e x 1 (R, ) (x). The first term on the right hand side is clearly integrable. For the second term we use that fact that for all x > R, x α e x = x M e x.x α M < ɛx α M, and the last term on the right hand side is integrable by Example 1. o the result follows by monotonicity (Theorem (1)). As a result of Example 2, we know that the gamma function Γ(α) = x α 1 e x dx exists for all α > 1. It can also be extended to the case α > 0. 0 n!n α Example 3. how that Γ(α) = lim for α > 1. α(α + 1) (α + n) Let Pn α n!n α =. You can check (e.g. by induction and α(α + 1) (α + n) integration by parts) that P α n n α = 1 0 (1 t) n t α 1 dt. 37

16 Make a change of variable x = tn to find that P α n = n 0 ( 1 x n) n x α 1 dx = 0 ( 1 x n) n x α 1 1 [0,n] (x)dx. Now the sequence whose nth term is ( 1 x n) n x α 1 1 [0,n] (x) comprises nonnegative measurable functions, and is monotonic increasing to e x x α 1 as n. The result then follows by the monotone convergence theorem. ummation of series is also an example of Lebesgue integration. uppose for simplicity that we are interested in n=1 a n, where a n 0 for all n N. We consider the sequence (a n ) as a function a : N [0, ). We work with the measure space (N, P(N), m) where m is counting measure. Then every sequence (a n ) gives rise to a non negative measurable function a (why is it measurable?) and a n = a(n)dm(n). n=1 N The integration theory that we ve developed tells us that this either converges, or diverges to +, as we would expect from previous work. 3.6 Further Topics: Fubini s Theorem and Function paces This material is included for interest, it is not examinable. However the first topic (Fubini s theorem) is covered in greater detail for MA451/6352, and that more extensive treatment is examinable Fubini s Theorem Let ( i, Σ i, m i ) be two measure spaces 3 and consider the product space ( 1 2, Σ 1 Σ 2, m 1 m 2 ) as discussed in section 1.6. We can consider integration of measurable functions f : 1 2 R by the procedure that we ve already discussed and there is nothing new to say about the definition and properties of 1 2 fd(m 1 m 2 ) when f is either measurable and non-negative (so the integral may be an extended real number) or when f is integrable (and the integral is a real number.) However from a practical point of view we would always like to calculate a double integral by writing it as a repeated integral 3 Technically speaking, the measures should have an additional property called σ finiteness for the main result below to be valid. 38

17 so that we first integrate with respect to m 1 and then with respect to m 2 (or vice versa). Fubini s theorem, which we will state without proof, tells us that we can do this provided that f is integrable with respect to the product measure. It is named in honour of the Italian mathematician Guido Fubini ( ). Theorem [Fubini s Theorem] Let f be integrable on ( 1 2, Σ 1 Σ 2, m 1 m 2 ) so that 1 2 f(x, y) (m 1 m 2 )(dx, dy) <. Then 1. The mapping f(x, ) is m 2 -integrable, almost everywhere with respect to m 1, 2. The mapping f(, y) is m 1 -integrable, almost everywhere with respect to m 2, 3. The mapping x 2 f(x, y)m 2 (dy) is equal almost everywhere to an integrable function on 1, 4. The mapping y 1 f(x, y)m 1 (dy) is equal almost everywhere to an integrable function on 2, f(x, y)(m 1 m 2 )(dx, dy) = = ( ) f(x, y)m 2 (dy) m 1 (dx) 1 ( 2 ) f(x, y)m 1 (dx) m 2 (dy) Function paces An important application of Lebesgue integration is to the construction of Banach spaces L p (, Σ, m) of equivalence classes of real-valued 4 functions that agree a.e. and which satisfy the requirement ( f p = ) 1 f p p dm <, where 1 p <. In fact p is a norm on L p (, Σ, m). When p = 2 we obtain a Hilbert space with inner product: f, g = fgdm. 4 The complex case also works and is important. 39

18 There is also a Banach space L (, Σ, m) where f = inf{m 0; f(x) M a.e.}. These spaces play important roles in functional analysis and its applications, including partial differential equations, probability theory and quantum mechanics. 40

Math212a1413 The Lebesgue integral.

Math212a1413 The Lebesgue integral. Math212a1413 The Lebesgue integral. October 28, 2014 Simple functions. In what follows, (X, F, m) is a space with a σ-field of sets, and m a measure on F. The purpose of today s lecture is to develop the

More information

Real Analysis Notes. Thomas Goller

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..............................

More information

Measure and integration

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.

More information

2 Lebesgue integration

2 Lebesgue integration 2 Lebesgue integration 1. Let (, A, µ) be a measure space. We will always assume that µ is complete, otherwise we first take its completion. The example to have in mind is the Lebesgue measure on R n,

More information

Lebesgue Integration: A non-rigorous introduction. What is wrong with Riemann integration?

Lebesgue Integration: A non-rigorous introduction. What is wrong with Riemann integration? Lebesgue Integration: A non-rigorous introduction What is wrong with Riemann integration? xample. Let f(x) = { 0 for x Q 1 for x / Q. The upper integral is 1, while the lower integral is 0. Yet, the function

More information

University of Sheffield. School of Mathematics & and Statistics. Measure and Probability MAS350/451/6352

University of Sheffield. School of Mathematics & and Statistics. Measure and Probability MAS350/451/6352 University of Sheffield School of Mathematics & and Statistics Measure and Probability MAS350/451/6352 Spring 2018 Chapter 1 Measure Spaces and Measure 1.1 What is Measure? Measure theory is the abstract

More information

Lebesgue measure and integration

Lebesgue measure and integration Chapter 4 Lebesgue measure and integration If you look back at what you have learned in your earlier mathematics courses, you will definitely recall a lot about area and volume from the simple formulas

More information

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?

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

More information

3 (Due ). 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?

3 (Due ). 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 ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable

More information

Solutions: Problem Set 4 Math 201B, Winter 2007

Solutions: Problem Set 4 Math 201B, Winter 2007 Solutions: Problem Set 4 Math 2B, Winter 27 Problem. (a Define f : by { x /2 if < x

More information

MATHS 730 FC Lecture Notes March 5, Introduction

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

More information

L p Spaces and Convexity

L p Spaces and Convexity L p Spaces and Convexity These notes largely follow the treatments in Royden, Real Analysis, and Rudin, Real & Complex Analysis. 1. Convex functions Let I R be an interval. For I open, we say a function

More information

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

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

More information

Notes on Measure, Probability and Stochastic Processes. João Lopes Dias

Notes on Measure, Probability and Stochastic Processes. João Lopes Dias Notes on Measure, Probability and Stochastic Processes João Lopes Dias Departamento de Matemática, ISEG, Universidade de Lisboa, Rua do Quelhas 6, 1200-781 Lisboa, Portugal E-mail address: jldias@iseg.ulisboa.pt

More information

LEBESGUE MEASURE AND L2 SPACE. Contents 1. Measure Spaces 1 2. Lebesgue Integration 2 3. L 2 Space 4 Acknowledgments 9 References 9

LEBESGUE MEASURE AND L2 SPACE. Contents 1. Measure Spaces 1 2. Lebesgue Integration 2 3. L 2 Space 4 Acknowledgments 9 References 9 LBSGU MASUR AND L2 SPAC. ANNI WANG Abstract. This paper begins with an introduction to measure spaces and the Lebesgue theory of measure and integration. Several important theorems regarding the Lebesgue

More information

Chapter 6. Integration. 1. Integrals of Nonnegative Functions. a j µ(e j ) (ca j )µ(e j ) = c X. and ψ =

Chapter 6. Integration. 1. Integrals of Nonnegative Functions. a j µ(e j ) (ca j )µ(e j ) = c X. and ψ = Chapter 6. Integration 1. Integrals of Nonnegative Functions Let (, S, µ) be a measure space. We denote by L + the set of all measurable functions from to [0, ]. Let φ be a simple function in L +. Suppose

More information

Problem set 5, Real Analysis I, Spring, otherwise. (a) Verify that f is integrable. Solution: Compute since f is even, 1 x (log 1/ x ) 2 dx 1

Problem set 5, Real Analysis I, Spring, otherwise. (a) Verify that f is integrable. Solution: Compute since f is even, 1 x (log 1/ x ) 2 dx 1 Problem set 5, Real Analysis I, Spring, 25. (5) Consider the function on R defined by f(x) { x (log / x ) 2 if x /2, otherwise. (a) Verify that f is integrable. Solution: Compute since f is even, R f /2

More information

Lebesgue Integration on R n

Lebesgue Integration on R n Lebesgue Integration on R n The treatment here is based loosely on that of Jones, Lebesgue Integration on Euclidean Space We give an overview from the perspective of a user of the theory Riemann integration

More information

3 Integration and Expectation

3 Integration and Expectation 3 Integration and Expectation 3.1 Construction of the Lebesgue Integral Let (, F, µ) be a measure space (not necessarily a probability space). Our objective will be to define the Lebesgue integral R fdµ

More information

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

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

More information

THE INVERSE FUNCTION THEOREM

THE INVERSE FUNCTION THEOREM THE INVERSE FUNCTION THEOREM W. PATRICK HOOPER The implicit function theorem is the following result: Theorem 1. Let f be a C 1 function from a neighborhood of a point a R n into R n. Suppose A = Df(a)

More information

(U) =, if 0 U, 1 U, (U) = X, if 0 U, and 1 U. (U) = E, if 0 U, but 1 U. (U) = X \ E if 0 U, but 1 U. n=1 A n, then A M.

(U) =, if 0 U, 1 U, (U) = X, if 0 U, and 1 U. (U) = E, if 0 U, but 1 U. (U) = X \ E if 0 U, but 1 U. n=1 A n, then A M. 1. Abstract Integration The main reference for this section is Rudin s Real and Complex Analysis. The purpose of developing an abstract theory of integration is to emphasize the difference between the

More information

Homework 11. Solutions

Homework 11. Solutions Homework 11. Solutions Problem 2.3.2. Let f n : R R be 1/n times the characteristic function of the interval (0, n). Show that f n 0 uniformly and f n µ L = 1. Why isn t it a counterexample to the Lebesgue

More information

Defining the Integral

Defining the Integral Defining the Integral In these notes we provide a careful definition of the Lebesgue integral and we prove each of the three main convergence theorems. For the duration of these notes, let (, M, µ) be

More information

MATH MEASURE THEORY AND FOURIER ANALYSIS. Contents

MATH MEASURE THEORY AND FOURIER ANALYSIS. Contents MATH 3969 - MEASURE THEORY AND FOURIER ANALYSIS ANDREW TULLOCH Contents 1. Measure Theory 2 1.1. Properties of Measures 3 1.2. Constructing σ-algebras and measures 3 1.3. Properties of the Lebesgue measure

More information

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 (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

More information

Probability and Measure

Probability and Measure Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 84 Paper 4, Section II 26J Let (X, A) be a measurable space. Let T : X X be a measurable map, and µ a probability

More information

( f ^ M _ M 0 )dµ (5.1)

( 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

More information

Chapter 4. The dominated convergence theorem and applications

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

More information

f (x) dx = F (b) F (a), where F is any function whose derivative is

f (x) dx = F (b) F (a), where F is any function whose derivative is Chapter 7 Riemann Integration 7.1 Introduction The notion of integral calculus is closely related to the notion of area. The earliest evidence of integral calculus can be found in the works of Greek geometers

More information

Integration. Darboux Sums. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Darboux Sums Today 1 / 13

Integration. Darboux Sums. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Darboux Sums Today 1 / 13 Integration Darboux Sums Philippe B. Laval KSU Today Philippe B. Laval (KSU) Darboux Sums Today 1 / 13 Introduction The modern approach to integration is due to Cauchy. He was the first to construct a

More information

THEOREMS, ETC., FOR MATH 515

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

More information

Chapter 5. Measurable Functions

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

More information

Lecture 3: Expected Value. These integrals are taken over all of Ω. If we wish to integrate over a measurable subset A Ω, we will write

Lecture 3: Expected Value. These integrals are taken over all of Ω. If we wish to integrate over a measurable subset A Ω, we will write Lecture 3: Expected Value 1.) Definitions. If X 0 is a random variable on (Ω, F, P), then we define its expected value to be EX = XdP. Notice that this quantity may be. For general X, we say that EX exists

More information

Section Integration of Nonnegative Measurable Functions

Section Integration of Nonnegative Measurable Functions 18.2. Integration of Nonnegative Measurable Functions 1 Section 18.2. Integration of Nonnegative Measurable Functions Note. We now define integrals of measurable functions on measure spaces. Though similar

More information

Problem set 4, Real Analysis I, Spring, 2015.

Problem set 4, Real Analysis I, Spring, 2015. Problem set 4, Real Analysis I, Spring, 215. (18) Let f be a measurable finite-valued function on [, 1], and suppose f(x) f(y) is integrable on [, 1] [, 1]. Show that f is integrable on [, 1]. [Hint: Show

More information

Measure and Integration: Solutions of CW2

Measure and Integration: Solutions of CW2 Measure and Integration: s of CW2 Fall 206 [G. Holzegel] December 9, 206 Problem of Sheet 5 a) Left (f n ) and (g n ) be sequences of integrable functions with f n (x) f (x) and g n (x) g (x) for almost

More information

Measure Theory and Lebesgue Integration. Joshua H. Lifton

Measure Theory and Lebesgue Integration. Joshua H. Lifton Measure Theory and Lebesgue Integration Joshua H. Lifton Originally published 31 March 1999 Revised 5 September 2004 bstract This paper originally came out of my 1999 Swarthmore College Mathematics Senior

More information

Outline of Fourier Series: Math 201B

Outline of Fourier Series: Math 201B Outline of Fourier Series: Math 201B February 24, 2011 1 Functions and convolutions 1.1 Periodic functions Periodic functions. Let = R/(2πZ) denote the circle, or onedimensional torus. A function f : C

More information

1.1. MEASURES AND INTEGRALS

1.1. MEASURES AND INTEGRALS CHAPTER 1: MEASURE THEORY In this chapter we define the notion of measure µ on a space, construct integrals on this space, and establish their basic properties under limits. The measure µ(e) will be defined

More information

REAL ANALYSIS I Spring 2016 Product Measures

REAL ANALYSIS I Spring 2016 Product Measures REAL ANALSIS I Spring 216 Product Measures We assume that (, M, µ), (, N, ν) are σ- finite measure spaces. We want to provide the Cartesian product with a measure space structure in which all sets of the

More information

ABSTRACT EXPECTATION

ABSTRACT EXPECTATION ABSTRACT EXPECTATION Abstract. In undergraduate courses, expectation is sometimes defined twice, once for discrete random variables and again for continuous random variables. Here, we will give a definition

More information

Measure Theory on Topological Spaces. Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond

Measure Theory on Topological Spaces. Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond Measure Theory on Topological Spaces Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond May 22, 2011 Contents 1 Introduction 2 1.1 The Riemann Integral........................................ 2 1.2 Measurable..............................................

More information

Integration on Measure Spaces

Integration on Measure Spaces Chapter 3 Integration on Measure Spaces In this chapter we introduce the general notion of a measure on a space X, define the class of measurable functions, and define the integral, first on a class of

More information

SOME PROBLEMS IN REAL ANALYSIS.

SOME PROBLEMS IN REAL ANALYSIS. SOME PROBLEMS IN REAL ANALYSIS. Prepared by SULEYMAN ULUSOY PROBLEM 1. (1 points) Suppose f n : X [, ] is measurable for n = 1, 2, 3,...; f 1 f 2 f 3... ; f n (x) f(x) as n, for every x X. a)give a counterexample

More information

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

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

More information

MATH 202B - Problem Set 5

MATH 202B - Problem Set 5 MATH 202B - Problem Set 5 Walid Krichene (23265217) March 6, 2013 (5.1) Show that there exists a continuous function F : [0, 1] R which is monotonic on no interval of positive length. proof We know there

More information

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9 MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended

More information

Spring 2014 Advanced Probability Overview. Lecture Notes Set 1: Course Overview, σ-fields, and Measures

Spring 2014 Advanced Probability Overview. Lecture Notes Set 1: Course Overview, σ-fields, and Measures 36-752 Spring 2014 Advanced Probability Overview Lecture Notes Set 1: Course Overview, σ-fields, and Measures Instructor: Jing Lei Associated reading: Sec 1.1-1.4 of Ash and Doléans-Dade; Sec 1.1 and A.1

More information

1 Measurable Functions

1 Measurable Functions 36-752 Advanced Probability Overview Spring 2018 2. Measurable Functions, Random Variables, and Integration Instructor: Alessandro Rinaldo Associated reading: Sec 1.5 of Ash and Doléans-Dade; Sec 1.3 and

More information

0.1 Uniform integrability

0.1 Uniform integrability Copyright c 2009 by Karl Sigman 0.1 Uniform integrability Given a sequence of rvs {X n } for which it is known apriori that X n X, n, wp1. for some r.v. X, it is of great importance in many applications

More information

Part II Probability and Measure

Part II Probability and Measure Part II Probability and Measure Theorems Based on lectures by J. Miller Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly)

More information

Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory

Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory Part V 7 Introduction: What are measures and why measurable sets Lebesgue Integration Theory Definition 7. (Preliminary). A measure on a set is a function :2 [ ] such that. () = 2. If { } = is a finite

More information

Chapter IV Integration Theory

Chapter IV Integration Theory Chapter IV Integration Theory This chapter is devoted to the developement of integration theory. The main motivation is to extend the Riemann integral calculus to larger types of functions, thus leading

More information

Notions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy

Notions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy Banach Spaces These notes provide an introduction to Banach spaces, which are complete normed vector spaces. For the purposes of these notes, all vector spaces are assumed to be over the real numbers.

More information

2 Measure Theory. 2.1 Measures

2 Measure Theory. 2.1 Measures 2 Measure Theory 2.1 Measures A lot of this exposition is motivated by Folland s wonderful text, Real Analysis: Modern Techniques and Their Applications. Perhaps the most ubiquitous measure in our lives

More information

II - REAL ANALYSIS. This property gives us a way to extend the notion of content to finite unions of rectangles: we define

II - REAL ANALYSIS. This property gives us a way to extend the notion of content to finite unions of rectangles: we define 1 Measures 1.1 Jordan content in R N II - REAL ANALYSIS Let I be an interval in R. Then its 1-content is defined as c 1 (I) := b a if I is bounded with endpoints a, b. If I is unbounded, we define c 1

More information

Methods of Applied Mathematics

Methods of Applied Mathematics Methods of Applied Mathematics Todd Arbogast and Jerry L. Bona Department of Mathematics, and Institute for Computational Engineering and Sciences The University of Texas at Austin Copyright 1999 2001,

More information

2.1 Convergence of Sequences

2.1 Convergence of Sequences Chapter 2 Sequences 2. Convergence of Sequences A sequence is a function f : N R. We write f) = a, f2) = a 2, and in general fn) = a n. We usually identify the sequence with the range of f, which is written

More information

convergence theorem in abstract set up. Our proof produces a positive integrable function required unlike other known

convergence theorem in abstract set up. Our proof produces a positive integrable function required unlike other known https://sites.google.com/site/anilpedgaonkar/ profanilp@gmail.com 218 Chapter 5 Convergence and Integration In this chapter we obtain convergence theorems. Convergence theorems will apply to various types

More information

Math 328 Course Notes

Math 328 Course Notes Math 328 Course Notes Ian Robertson March 3, 2006 3 Properties of C[0, 1]: Sup-norm and Completeness In this chapter we are going to examine the vector space of all continuous functions defined on the

More information

Problem Set. Problem Set #1. Math 5322, Fall March 4, 2002 ANSWERS

Problem Set. Problem Set #1. Math 5322, Fall March 4, 2002 ANSWERS Problem Set Problem Set #1 Math 5322, Fall 2001 March 4, 2002 ANSWRS i All of the problems are from Chapter 3 of the text. Problem 1. [Problem 2, page 88] If ν is a signed measure, is ν-null iff ν () 0.

More information

It follows from the above inequalities that for c C 1

It follows from the above inequalities that for c C 1 3 Spaces L p 1. Appearance of normed spaces. In this part we fix a measure space (, A, µ) (we may assume that µ is complete), and consider the A - measurable functions on it. 2. For f L 1 (, µ) set f 1

More information

The Lebesgue Integral

The Lebesgue Integral The Lebesgue Integral Brent Nelson In these notes we give an introduction to the Lebesgue integral, assuming only a knowledge of metric spaces and the iemann integral. For more details see [1, Chapters

More information

MEASURE AND INTEGRATION. Dietmar A. Salamon ETH Zürich

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.

More information

5 Measure theory II. (or. lim. Prove the proposition. 5. For fixed F A and φ M define the restriction of φ on F by writing.

5 Measure theory II. (or. lim. Prove the proposition. 5. For fixed F A and φ M define the restriction of φ on F by writing. 5 Measure theory II 1. Charges (signed measures). Let (Ω, A) be a σ -algebra. A map φ: A R is called a charge, (or signed measure or σ -additive set function) if φ = φ(a j ) (5.1) A j for any disjoint

More information

4 Integration 4.1 Integration of non-negative simple functions

4 Integration 4.1 Integration of non-negative simple functions 4 Integration 4.1 Integration of non-negative simple functions Throughout we are in a measure space (X, F, µ). Definition Let s be a non-negative F-measurable simple function so that s a i χ Ai, with disjoint

More information

Notes on the Lebesgue Integral by Francis J. Narcowich November, 2013

Notes on the Lebesgue Integral by Francis J. Narcowich November, 2013 Notes on the Lebesgue Integral by Francis J. Narcowich November, 203 Introduction In the definition of the Riemann integral of a function f(x), the x-axis is partitioned and the integral is defined in

More information

(1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define

(1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define Homework, Real Analysis I, Fall, 2010. (1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define ρ(f, g) = 1 0 f(x) g(x) dx. Show that

More information

Tools from Lebesgue integration

Tools from Lebesgue integration Tools from Lebesgue integration E.P. van den Ban Fall 2005 Introduction In these notes we describe some of the basic tools from the theory of Lebesgue integration. Definitions and results will be given

More information

i. v = 0 if and only if v 0. iii. v + w v + w. (This is the Triangle Inequality.)

i. v = 0 if and only if v 0. iii. v + w v + w. (This is the Triangle Inequality.) Definition 5.5.1. A (real) normed vector space is a real vector space V, equipped with a function called a norm, denoted by, provided that for all v and w in V and for all α R the real number v 0, and

More information

3. (a) What is a simple function? What is an integrable function? How is f dµ defined? Define it first

3. (a) What is a simple function? What is an integrable function? How is f dµ defined? Define it first Math 632/6321: Theory of Functions of a Real Variable Sample Preinary Exam Questions 1. Let (, M, µ) be a measure space. (a) Prove that if µ() < and if 1 p < q

More information

REAL ANALYSIS I HOMEWORK 4

REAL ANALYSIS I HOMEWORK 4 REAL ANALYSIS I HOMEWORK 4 CİHAN BAHRAN The questions are from Stein and Shakarchi s text, Chapter 2.. Given a collection of sets E, E 2,..., E n, construct another collection E, E 2,..., E N, with N =

More information

Math 118B Solutions. Charles Martin. March 6, d i (x i, y i ) + d i (y i, z i ) = d(x, y) + d(y, z). i=1

Math 118B Solutions. Charles Martin. March 6, d i (x i, y i ) + d i (y i, z i ) = d(x, y) + d(y, z). i=1 Math 8B Solutions Charles Martin March 6, Homework Problems. Let (X i, d i ), i n, be finitely many metric spaces. Construct a metric on the product space X = X X n. Proof. Denote points in X as x = (x,

More information

7: FOURIER SERIES STEVEN HEILMAN

7: FOURIER SERIES STEVEN HEILMAN 7: FOURIER SERIES STEVE HEILMA Contents 1. Review 1 2. Introduction 1 3. Periodic Functions 2 4. Inner Products on Periodic Functions 3 5. Trigonometric Polynomials 5 6. Periodic Convolutions 7 7. Fourier

More information

Metric Spaces and Topology

Metric Spaces and Topology Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies

More information

Probability and Measure

Probability and Measure Chapter 4 Probability and Measure 4.1 Introduction In this chapter we will examine probability theory from the measure theoretic perspective. The realisation that measure theory is the foundation of probability

More information

Riemann integral and volume are generalized to unbounded functions and sets. is an admissible set, and its volume is a Riemann integral, 1l E,

Riemann integral and volume are generalized to unbounded functions and sets. is an admissible set, and its volume is a Riemann integral, 1l E, Tel Aviv University, 26 Analysis-III 9 9 Improper integral 9a Introduction....................... 9 9b Positive integrands................... 9c Special functions gamma and beta......... 4 9d Change of

More information

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

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

More information

Summary of Real Analysis by Royden

Summary of Real Analysis by Royden Summary of Real Analysis by Royden Dan Hathaway May 2010 This document is a summary of the theorems and definitions and theorems from Part 1 of the book Real Analysis by Royden. In some areas, such as

More information

Advanced Analysis Qualifying Examination Department of Mathematics and Statistics University of Massachusetts. Tuesday, January 16th, 2018

Advanced Analysis Qualifying Examination Department of Mathematics and Statistics University of Massachusetts. Tuesday, January 16th, 2018 NAME: Advanced Analysis Qualifying Examination Department of Mathematics and Statistics University of Massachusetts Tuesday, January 16th, 2018 Instructions 1. This exam consists of eight (8) problems

More information

HARMONIC ANALYSIS. Date:

HARMONIC ANALYSIS. Date: HARMONIC ANALYSIS Contents. Introduction 2. Hardy-Littlewood maximal function 3. Approximation by convolution 4. Muckenhaupt weights 4.. Calderón-Zygmund decomposition 5. Fourier transform 6. BMO (bounded

More information

CHAPTER 6. Differentiation

CHAPTER 6. Differentiation CHPTER 6 Differentiation The generalization from elementary calculus of differentiation in measure theory is less obvious than that of integration, and the methods of treating it are somewhat involved.

More information

Lebesgue-Radon-Nikodym Theorem

Lebesgue-Radon-Nikodym Theorem 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

More information

MTH 404: Measure and Integration

MTH 404: Measure and Integration MTH 404: Measure and Integration Semester 2, 2012-2013 Dr. Prahlad Vaidyanathan Contents I. Introduction....................................... 3 1. Motivation................................... 3 2. The

More information

+ 2x sin x. f(b i ) f(a i ) < ɛ. i=1. i=1

+ 2x sin x. f(b i ) f(a i ) < ɛ. i=1. i=1 Appendix To understand weak derivatives and distributional derivatives in the simplest context of functions of a single variable, we describe without proof some results from real analysis (see [7] and

More information

h(x) lim H(x) = lim Since h is nondecreasing then h(x) 0 for all x, and if h is discontinuous at a point x then H(x) > 0. Denote

h(x) lim H(x) = lim Since h is nondecreasing then h(x) 0 for all x, and if h is discontinuous at a point x then H(x) > 0. Denote Real Variables, Fall 4 Problem set 4 Solution suggestions Exercise. Let f be of bounded variation on [a, b]. Show that for each c (a, b), lim x c f(x) and lim x c f(x) exist. Prove that a monotone function

More information

THE STONE-WEIERSTRASS THEOREM AND ITS APPLICATIONS TO L 2 SPACES

THE STONE-WEIERSTRASS THEOREM AND ITS APPLICATIONS TO L 2 SPACES THE STONE-WEIERSTRASS THEOREM AND ITS APPLICATIONS TO L 2 SPACES PHILIP GADDY Abstract. Throughout the course of this paper, we will first prove the Stone- Weierstrass Theroem, after providing some initial

More information

Analysis Finite and Infinite Sets The Real Numbers The Cantor Set

Analysis Finite and Infinite Sets The Real Numbers The Cantor Set Analysis Finite and Infinite Sets Definition. An initial segment is {n N n n 0 }. Definition. A finite set can be put into one-to-one correspondence with an initial segment. The empty set is also considered

More information

0.1 Pointwise Convergence

0.1 Pointwise Convergence 2 General Notation 0.1 Pointwise Convergence Let {f k } k N be a sequence of functions on a set X, either complex-valued or extended real-valued. We say that f k converges pointwise to a function f if

More information

d(x n, x) d(x n, x nk ) + d(x nk, x) where we chose any fixed k > N

d(x n, x) d(x n, x nk ) + d(x nk, x) where we chose any fixed k > N Problem 1. Let f : A R R have the property that for every x A, there exists ɛ > 0 such that f(t) > ɛ if t (x ɛ, x + ɛ) A. If the set A is compact, prove there exists c > 0 such that f(x) > c for all x

More information

A Brief Introduction to the Theory of Lebesgue Integration

A Brief Introduction to the Theory of Lebesgue Integration A Brief Introduction to the Theory of Lebesgue Integration Kevin Sigler June 8, 25 Introduction Gonzalez-Velasco s paper The Lebesgue Integral as a Riemann Integral provides a non-standard, direct construction

More information

MAS221 Analysis, Semester 1,

MAS221 Analysis, Semester 1, MAS221 Analysis, Semester 1, 2018-19 Sarah Whitehouse Contents About these notes 2 1 Numbers, inequalities, bounds and completeness 2 1.1 What is analysis?.......................... 2 1.2 Irrational numbers.........................

More information

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

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 (, )

More information

REAL AND COMPLEX ANALYSIS

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

More information

Introduction to Real Analysis

Introduction to Real Analysis Christopher Heil Introduction to Real Analysis Chapter 0 Online Expanded Chapter on Notation and Preliminaries Last Updated: January 9, 2018 c 2018 by Christopher Heil Chapter 0 Notation and Preliminaries:

More information

Reminder Notes for the Course on Measures on Topological Spaces

Reminder Notes for the Course on Measures on Topological Spaces Reminder Notes for the Course on Measures on Topological Spaces T. C. Dorlas Dublin Institute for Advanced Studies School of Theoretical Physics 10 Burlington Road, Dublin 4, Ireland. Email: dorlas@stp.dias.ie

More information

36-752: Lecture 1. We will use measures to say how large sets are. First, we have to decide which sets we will measure.

36-752: Lecture 1. We will use measures to say how large sets are. First, we have to decide which sets we will measure. 0 0 0 -: Lecture How is this course different from your earlier probability courses? There are some problems that simply can t be handled with finite-dimensional sample spaces and random variables that

More information

Introduction to Real Analysis Alternative Chapter 1

Introduction to Real Analysis Alternative Chapter 1 Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces

More information

The Lebesgue Integral

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

More information