3.8 Three Types of Convergence

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1 3.8 Three Types of Convergence 3.8 Three Types of Convergence 93 Suppose that we are given a sequence functions {f k } k N on a set X and another function f on X. What does it ean for f k to converge to f? Loosely speaking, convergence eans that f k becoes closer and closer to f as k increases. However, there are any ways to say exactly what we ean by close. Two failiar ways to quantify convergence are pointwise convergence and unifor convergence. These types of convergence were discussed in Sections 0.1 and 0.2, respectively. In this section we will give an alost everywhere version of pointwise and unifor convergence, and then introduce a new notion that we call convergence in easure. Each of these types of convergence (and others that we will encounter later!) have their own iportant role to play in analysis. We need all of these different notions in different circustances; which one to use will depend on the application that we have at hand Pointwise Alost Everywhere Convergence Let {f k } k N be a sequence of functions on a set X, either coplex-valued or extended real-valued. Recalling Definition 0.5, we say that f k converges pointwise to a function f if for each individual eleent x X, the scalar f k (x) converges to f(x) as k. We often write f k f pointwise to denote pointwise convergence. We do not need to have a easure on a doain X in order to discuss pointwise convergence on X. However, if we do have a easure then it is often the case that sets of easure zero are siply not iportant when considering convergence. Therefore, a useful variation on pointwise convergence is pointwise alost everywhere convergence, which is pointwise convergence with the exception of a set of points whose easure is zero. For exaple, this is the type of convergence that is used in the stateent of part (b) of Corollary Here is a precise definition. Definition 3.49 (Pointwise Alost Everywhere Convergence). Let (X, Σ, µ) be a easure space, and let f k, f be easurable functions on E that are either extended real-valued or coplex-valued. We say that f k converges pointwise alost everywhere to f if there exists a easurable set Z X such that µ(z) = 0 and x X\Z, li f k(x) = f(x). k We often denote pointwise alost everywhere convergence by writing f k f pointwise µ-a.e. or siply f k f µ-a.e. (and we ay also oit writing the sybol µ if it is understood). c 2011 by Christopher Heil

2 94 3 Measurable Functions Often, the functions f k not only converge pointwise a.e. to f, but are also onotone increasing except on a set of easure zero, i.e., f 1 (x) f 2 (x) for a.e. x. In this case we say that that f k increases pointwise a.e. to f, and denote this by writing f k ր f µ-a.e., or just f k ր f a.e. A siilar definition is ade for decreasing sequences. Reark Note that if µ is a coplete easure, then Corollary 3.48 iplies that easurability of the liit function f in Definition 3.49 follows autoatically fro the easurability of the functions f k. Exaple The functions χ (0, 1 k ) converge pointwise to the zero function as k. On the other hand, the functions χ [0, 1 k ] do converge pointwise to the zero function, for if x = 0 then the liit is 1, not zero. However, the singleton {0} has Lebesgue easure zero, so χ [0, 1 k ] ց 0 a.e L Convergence Unifor convergence is a stronger requireent than pointwise convergence in that it requires a siultaneity of convergence over all of the doain rather just individual convergence at each x. As discussed in Section 0.2, a convenient way to view unifor convergence is in ters of the unifor nor f u = sup f(x). (3.7) x X Unifor convergence requires that the distance between f k and f, as easured by the unifor nor, shrinks to zero as k increases. That is, f k f uniforly li k f f k u = 0. We can define unifor convergence without needing to have a easure on our doain X. However, if we do have a easure on X then sets with easure zero often don t atter. In keeping with this philosophy, we odify the definition of unifor convergence by siply replacing the supreu that appears in equation (3.7) with an essential supreu. We introduced the essential supreu for functions on R d in Definition 1.47, and the following definition extends this to functions on an arbitrary easure space. Definition 3.52 (Essential Supreu). Let (X, Σ, µ) be a easure space. The essential supreu of a easurable function f : X R is ess sup f(x) = inf { M : f(x) M µ-a.e. }. (3.8) x X

3 3.8 Three Types of Convergence 95 We give the following nae to the essential supreu of f. This nae ay not see very inspired at this point, but it will fit naturally for the terinology for the L p spaces that we will introduce in Chapter 5. Definition 3.53 (L nor). Let (X, Σ, µ) be a easure space, and let f be an easurable function on X, either extended real-valued or coplex-valued. The L nor of f is the essential supreu of f, and it is denoted by f = ess sup f(x). x X Proble 1.27, which was stated specifically for Lebesgue easure, generalizes to abstract easures without change. Consequently, we have f(x) f for a.e. x X. Further, by the definition of supreu we have the inequality f f u. However, the unifor nor and the L nor of f need not be equal. For exaple, if we take X = R and define f(x) = 0 for x 0 and f(0) = 1 then we have f = 0 while f u = 1. On the other hand, Proble 1.29 showed that if f is a continuous function on R d then f = f u. Hence in soe ways we can regard the L nor as being a generalization of the unifor nor. We introduce soe additional terinology associated with the essential supreu and the L nor. Definition 3.54 (Essentially Bounded Function). Let (X, Σ, µ) be a easure space, and let f, f k, g be easurable functions on X, either extended real-valued or coplex-valued. (a) We say that f is essentially bounded if f <. (b) The quantity f g u is called the L distance between f and g. (c) We say that f k converges to f in L nor if li f f k = 0. k The next exercise shows that the L nor alost satisfies the properties given in Definition 0.1 that are necessary in order for to be called a nor. However, there is a subtle difference between stateent (b) in the next exercise and the uniqueness property of a nor given in Definition 0.1. We will discuss this difference in ore detail after the exercise. To avoid technical coplications, we will assue that our easure space is coplete and scalars are coplex in this exercise.

4 96 3 Measurable Functions Exercise Let (X, Σ, µ) be a easure space, and let E b (X) be the set of all easurable, essentially bounded, coplex-valued functions on X, i.e., E b (X) = { f : X C : f is easurable and essentially bounded }. It follows fro Exercise 3.45 that E b (X) is a vector space over the coplex field. Show that unifor nor satisfies the following properties on E b (X): (a) 0 f for all f E b (X), (b) f = 0 if and only if f = 0 a.e., (c) cf = c f for all f E b (X) and scalars c C, (d) f + g f + g for all f, g E b (X). Unfortunately, stateent (b) in Exercise 3.55 is not quite what we need in order to say that is a nor on E b (X). According to Definition 0.1, to be a nor it ust be the case that f = 0 if and only if f is the zero vector in E b (X). The zero vector in E b (X) is the zero function, the function that takes the value 0 for every x X. However, it is not usually true that f = 0 if and only if f is the zero function. Instead we only have the weaker stateent that f = 0 f = 0 a.e. (3.9) The technical terinology for this is that is a seinor on E b (X) rather than a nor. Since our philosophy is that sets of easure zero usually don t atter, this is not really a proble we siply accept the fact that the right-hand side of equation (3.9) says that f is zero alost everywhere, instead of saying that f is zero everywhere. Proble 3.21 shows one way to odify the space on which is defined so that it becoes a true nor on that space. This approach will be useful to us when we define the L p spaces in Chapter 5, but right now it is not iportant. Exaple Consider X = R and f = χ C, the characteristic function of the Cantor set C. Since the Cantor zero has easure zero, its characteristic function satisfies χ C = 0 a.e. Therefore χ C = 0, even though χ C is not the zero function. Whenever we have a nor (or a seinor), it is very iportant to know whether Cauchy sequences ust converge with respect to that nor. Following the usual terinology for nors (Definition 0.2), we say that a sequence {f k } k in E b (X) is Cauchy in L nor or Cauchy with respect to if ε > 0, N > 0, j, k N, f j f k < ε. We will show in the next theore that every Cauchy sequence in E b (X) ust converge in E b (X), at least if our easure µ is a coplete easure on X. If was a true nor on E b (X) then we would describe this by saying that

5 3.8 Three Types of Convergence 97 E b (X) is a coplete nored space (see Definition 0.3). However, since is not a true nor, we will not use that terinology in connection with E b (X). Theore Let (X, Σ, µ) be a coplete easure space. If {f k } k N is a sequence in E b (X) that is Cauchy with respect to, then there exists a function f E b (X) such that f f k 0 as k. Proof. Let {f k } k N be a Cauchy sequence in E b (X). This eans that ε > 0, N > 0, j, k N, f j f k < ε. Let M jk = f j f k. Then f j (x) f k (x) can exceed M jk only for a set of x s that lie in a set of easure zero. That is, if we define Z jk = { f j f k > M jk }, then µ(z jk ) = 0. Since a countable union of zero easure sets has easure zero, it follows that Z = Z jk j,k N has easure zero. For each k N, create a function g k that equals f k alost everywhere but is zero on the set Z: { f k (x), x / Z, g k (x) = 0, x Z. Since µ is a coplete easure, Lea 3.27 iplies that g k is a easurable function. Exercise: Show that the unifor nor of g j g k coincides with its L nor, which also coincides with the L nor of f j g k, i.e., g j g k u = g j g k = f j f k. Consequently, {g k } k N is a uniforly Cauchy in F b (X), the space of bounded, coplex-valued functions on X. Theore 0.12 therefore iplies that there is a bounded function g such that g k g uniforly. This function g is easurable since it is the pointwise liit of easurable functions. Hence g belongs to E b (X). Finally, since g g k differs fro g f k only on a set of easure zero, we have g f k = g g k g g k u 0 as k. Exaple Let Z 1 Z 2 be a nested increasing sequence of subsets of R that each have easure zero. Define Note the two distinct uses of the word coplete in this paragraph! A coplete easure is a easure such that every subset of a null set is easurable, while a coplete nored space is a space such that every Cauchy sequence converges.

6 98 3 Measurable Functions f k (x) = { 1 k, x / Z k, k, x Z k. The functions f k do not converge pointwise to zero since there points x where f k (x) as k increases. On the other hand, f k converges to the zero function with respect to, because 0 f k = f k = 1 k as k. If we like, we could ake analogous versions of Exercise 3.55 and Theore 3.57 for extended real-valued functions that are finite alost everywhere Convergence in Measure Pointwise convergence, pointwise alost everywhere convergence, unifor convergence, and L nor convergence are only soe of the yriad ways in which we ight say that functions f k converge to a given function f. For pointwise convergence we require that f k (x) becoes close to f(x) for each individual x, while for unifor convergence we require that f k (x) and f(x) becoe siultaneously close over all x. Pointwise alost everywhere and L nor convergence are variations where we ignore sets of easure zero. Now we describe another type of convergence criterion that we will call convergence in easure. Suppose that we have functions f k and f on a easure space (X, Σ, µ). The idea of convergence in easure is that we require f k (x) and f(x) to be close except on a set that has saller and saller easure. More precisely, given any fixed ε > 0, the set where f k (x) and f(x) differ by ore than ε should becoe saller and saller in easure as k increases. Here is the explicit definition for coplex-valued functions. Definition 3.59 (Convergence in Measure). Let (X, Σ, µ) be a easure space, and let f k, f be coplex-valued easurable functions on E. Then we say that f k converges in easure to f, and write f k f, if ε > 0, li µ{ f f k > ε } = 0. (3.10) k It is useful to write out equation (3.10) in coplete detail. With all the quantifiers, the requireent for convergence in easure given in (3.10) is that ε, η > 0, N > 0 such that k > N = µ{ f f k ε} < η. (3.11) Proble 3.19 gives an equivalent forulation that requires only the use of a single variable ε instead of two variables ε, η. We can ake a siilar definition for convergence in easure of extended real-valued functions, but there is a technical coplication because f f k ight not be easurable. However, if our easure space is coplete and the functions f k and f are finite a.e. then this is not a proble, so we extend the definition to this case as follows.

7 3.8 Three Types of Convergence 99 Definition 3.60 (Convergence in Measure). Let (X, Σ, µ) be a coplete easure space, and let f k, f be extended real-valued easurable functions on E that are finite a.e. Then we say that f k converges in easure to f, and write f k f, if ε > 0, li µ{ f f k > ε } = 0. k For the reainder of this section we will state results for the case of coplex-valued functions, and assign the reader the task of forulating analogous results for extended real-valued functions. Since { f f k > ε} is the set where f and f k differ by ore than ε, these definitions are saying that the easure of this set ust decrease to zero as k increases. For exaple, if f k = f except on a set E k of easure 1/k, then f k f. This exaple suggests that convergence in easure ight iply pointwise or pointwise alost everywhere convergence, but the following exercises and exaples show that the relationships aong these types of convergence are not quite as straightforward as we ight hope. Exercise Show that L nor convergence iplies both pointwise a.e. convergence and convergence in easure. That is, if (X, Σ, µ) is a easure space and f f k 0, then f k f pointwise a.e. and also f k f. Exaple 3.62 (Shrinking Boxes). Let f k = χ [0, 1 k of the closed interval [0, 1 k ], the characteristic function ]. As we saw in Exaple 3.51, these functions converge pointwise a.e. to the zero function. However, since f k = 1 for every k, the functions f k do not converge in L nor to the zero function. Hence pointwise a.e. convergence does not iply L nor convergence. However, this sequence does converge in easure to the zero function. To see this, fix any 0 < ε < 1. The set where f k differs fro the zero function by ore than ε is precisely the interval [0, 1 k ]. The Lebesgue easure of this set converges to zero as k increases, so f k 0. A variation on the preceding exaple is to take g k = k χ [0, 1 k ]. Even though the heights of these functions grow without bound as k increases, we still have that g k 0 a.e. and g k 0. Exaple 3.63 (Boxes Marching to Infinity). This exaple will show that pointwise a.e. convergence does not iply convergence in easure. Let f k = χ [k,k+1], the characteristic function of the closed interval [k, k + 1]. Then we have that f k 0 pointwise, although we do not have L convergence to the zero function because f k = 1 for every k. To see that we do not have convergence in easure, fix any 0 < ε < 1. The set where f k differs fro 0 by ore than ε is the interval [k, k +1], which has Lebesgue easure 1 for every k. Since this quantity does not shrink to zero, f k does not converge in easure to the zero function.

8 100 3 Measurable Functions Exaple 3.64 (Boxes Marching in Circles). This exaple will show that convergence in easure does not iply pointwise a.e. convergence. Define f 1 = χ [0,1], f 2 = χ [0, 1 2 ], f 3 = χ [ 1 2,1], f 4 = χ [0, 1 3 ], f 5 = χ [ 1 3, 2 3 ], f 6 = χ [ 2 3,1], f 7 = χ [0, 1 4 ], f 8 = χ [ 1 4, 1 2 ], f 9 = χ [ 1 2, 3 4 ], f 10 = χ [ 3 4,1], and so forth. Picturing the graphs of these functions as boxes, the boxes arch fro left to right across the interval [0, 1], then shrink in size and arch across the interval again, and do this over and over. Fix any 0 < ε < 1. For the indices k = 1,..., 10, the Lebesgue easure of { f k > ε} is 1 1, 2, 1 2, 1 3, 1 3, 1 3, 1 4, 1 4, 1 4, 1 4. The Lebesgue easure of { f k > ε} converges to zero as k increases, so f k 0. We do not have pointwise a.e. convergence in this exaple because no atter what point x [0, 1] that we choose, there are infinitely any different values of k such that f k (x) = 0 and infinitely any k such that f k (x) = 1. While f k (x) ay be zero for any particular k, there are always larger k such that f k (x) = 1. Hence f k (x) does not converge to 0 for any x [0, 1]. This sequence of functions does not converge pointwise (or even pointwise a.e.) to the zero function, or to any other function. In suary, convergence in L nor iplies pointwise a.e. convergence and convergence in easure, but pointwise a.e. convergence does not iply convergence in easure, and convergence in easure does not iply pointwise a.e. convergence. Just to coplicate atters, after we prove Egoroff s Theore (Theore 3.68) we will be able to show that, in a finite easure space, pointwise a.e. convergence iplies convergence in easure (see Proble 3.23). In the converse direction, Exaple 3.64 shows us that convergence in easure does not iply pointwise a.e. convergence, even in a finite easure space. However, there is a weaker but still very iportant relation between convergence in easure and pointwise a.e. convergence. Specifically, the next theore will show that if f k converges in easure to f, then there is a subsequence of the f k that converges pointwise alost everywhere to f. For exaple, {χ [0, 1 k ] } k N is a subsequence of the Marching Boxes fro Exaple 3.64, and χ [0, 1 k ] converges pointwise a.e. to the zero function. Theore Let (X, Σ, µ) be a easure space. If f k, f : X C are easurable functions such that f k f, then there exists a subsequence {fnk } k N such that f nk f pointwise a.e.

9 3.8 Three Types of Convergence 101 Proof. Assue f k f. Exercise: Show that there exist indices n1 < n 2 < such that µ { f f nk > 1 k} 2 k, k N. Define E k = { f f nk > 1 k } and H = k= These sets are easurable since f k and f are easurable functions. By construction we have µ(e k ) 2 k, so by subadditivity, E k. µ(h ) µ(e k ) k= k= 1 2 k = Set Z = =1 H. For each N we have µ(z) µ(h ) 2 +1, so Z has easure zero (note that Z = lisupe k in the terinology of Proble 2.18). If x / Z, then x / H for soe. Hence x / E k for all k, which iplies that f(x) f nk (x) 1 for all k. k Thus f nk (x) f(x) when x / Z. Since Z has easure zero, this says that the functions f nk converge pointwise alost everywhere to f. Unlike L convergence, it is not possible to characterize convergence in easure in ters of a nor. Proble 3.22 shows that it is possible to create a etric that corresponds to convergence in easure, but in practice it is usually ore convenient work directly with the definition of convergence in easure instead of trying to use that etric. On the other hand, it is iportant to know whether there is a notion of sequences that are Cauchy with respect to convergence in easure, and whether such Cauchy sequences ust converge in easure. The next definition introduces the Cauchy criterion. Definition 3.66 (Cauchy in Measure). Let (X, Σ, µ) be a easure space. Given easurable functions f k, f : X C, we say that {f k } k N is Cauchy in easure if ε > 0, li µ{ f j f k > ε} = 0. (3.12) j,k Precisely, equation (3.12) eans that ε, η > 0, N > 0 such that j, k > N = µ{ f j f k > ε} < η. Now we prove that if a sequence is Cauchy in easure then it ust converge in easure. The usefulness of the Cauchy criterion is that we can test for it without knowing what the liit function is, or even whether it exists.

10 102 3 Measurable Functions Theore 3.67 (Cauchy Criterion for Convergence in Measure). Let (X, Σ, µ) be a easure space. If f k : X C are easurable functions on X, then the following stateents are equivalent. (a) There exists a easurable function f such that f k f. (b) {f k } k N is Cauchy in easure. Proof. (a) (b). Assue that f k f, and fix ε, η > 0. Then by equation 3.11, there exists an N > 0 such that By the Triangle Inequality, It follows fro this that µ{ f f k > ε} < η for all k > N. f j (x) f k (x) f j (x) f(x) + f(x) f k (x). { f j f k > 2ε} { f j f > ε} { f f k > ε}. Consequently, if k > N then µ{ f j f k > 2ε} µ{ f j f > ε} + µ{ f f k > ε} < η + η = 2η. Therefore {f k } k N is Cauchy in easure. (b) (a). The first part of this proof proceeds uch like the proof of Theore Suppose that {f k } k N is Cauchy in easure. Then there exist indices n 1 < n 2 < such that µ { f nk+1 f nk > 2 k} 2 k, k N. For siplicity of notation, let g k = f nk. Define E k = { g k+1 g k > 2 k} and H = Then µ(e k ) 2 k, so k= E k. Therefore, if we set µ(h ) µ(e k ) k= 2 k = k= Z = =1 H = =1 k= E k = lisupe k, then Z is easurable and µ(z) = 0.

11 3.8 Three Types of Convergence 103 If x / Z then x / H for soe, and therefore x / E k for all k. That is, g k+1 (x) g k (x) 2 k for all k. This iplies that {g k (x)} k N is a Cauchy sequence of coplex scalars (copare Proble 0.3), and therefore ust converge. The function f(x) = { li k(x), k if the liit exists, 0, otherwise, is easurable by Corollary 3.48, and we have by construction that g k f pointwise a.e. Now will show that g k converges in easure to f. Fix ε > 0, and choose large enough that 2 ε. Fix N, and suppose that x / H. If n > k > then we have g n (x) g k (x) n 1 g i+1 (x) g i (x) i=k n 1 2 i 2 k+1 2 ε. i=k Taking the liit as n, this iplies that f(x) g k (x) ε for all x / H and k >. Hence { f gk > ε } H, k >, so li sup k µ { f g k > ε } µ(h ) This is true for every, so we conclude that li k µ{ f g k > ε} = 0. Therefore g k f. This is not quite enough to coplete the proof, becuase {g k } k N is only a subsequence of {f k } k N. However, by Proble 3.18, the fact that {f k } k N is Cauchy in easure and has a subsequence that converges in easure to f iplies that f k f. Additional Probles Let (X, Σ, µ) be a easure space, and let f k, f : X C be easurable functions on X. Show that if {f k } k N is Cauchy in easure and there exists a subsequence such that f nk f, then fk f Let (X, Σ, µ) be a easure space, and let f k, f : X C be easurable functions on X. Prove that f k f if and only if ε > 0, N > 0 such that k > N = µ{ f f k ε} < ε. Forulate and prove an analogous Cauchy criterion.

12 104 3 Measurable Functions Let (X, Σ, µ) be a easure space. Let f k, f, g k, g: X C be easurable functions on X, and prove the following stateents. (a) If f k f and fk g, then f = g a.e. (b) If f k f and gk g, then fk + g k f + g. (c) If µ is a bounded easure, f k f, and gk g, then fk g k fg. (d) The conclusion of part (c) can fail if µ is not bounded Let (X, Σ, µ) be a easure space. (a) Show that f g if f = g a.e. is an equivalence relation on E b (X). (b) Let f denote the equivalence class of f under this relation, i.e., f = { g E b (X) : g = f a.e. }. Define f = f, and show that this quantity is independent of the choice of representative f of f. (c) Let the quotient space L (X) consist of all the distinct equivalence classes of f E b (X), i.e., L (X) = { f : f Eb (X) }. Define operations of addition and scalar ultiplication that ake L (X) into a vector space over the coplex field. What is the zero eleent of L (X)? (d) Show that is a nor on L (X) Let (X, Σ, µ) be a easure space, and let F (X) be the vector space of all coplex-valued easurable functions on X. For each n N, define ρ n (f) = µ { } f > 1 n and for f, g F (X) define d(f, g) = n=1 2 n ρ n (f g) 1 + ρ n (f g). Prove that ρ n (f + g) ρ 2n (f) + ρ 2n (g), and use this to prove the following stateents. (a) d(f, g) 0, (b) d(f, g) = 0 if and only if f = g, (c) d(f, g) = d(g, f), and (d) d(f, h) d(f, g) + d(g, h). Consequently, if we identify functions that are equal alost everywhere then d is a etric on F (x). Prove that convergence with respect to this etric coincides with convergence in easure, i.e., f k f li k d(f, f k) = 0.