Some Background Material

 Aubrie Bryan
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1 Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary  but important  material as a way of dipping our toes in the water. This chapter also introduces important notation. But it does not provide a full development of real analysis. So, it is very dry. In fact, we usually can t bring ourselves to make it all the way through when lecturing. We cover some of the material and tell the students to read the rest. Doing a quick read through and looking up any ideas or results that are not familiar would be a useful exercise. 1.1 Sets In several key ways, measure theory and probability are constructed in order to deal with complex sets that arise when describing very practical situations. Definition A set is a collection of objects called elements or points. Some important examples with their notation: Example = {1,2,3, } (natural numbers), = {, 3, 2, 1,0,1,2,3, } = set of rational numbers, = set of real numbers, (integers), + = set of nonnegative real numbers, = set of complex numbers. More complex examples that we use later on include sets of functions. Next some notation and definitions regarding belonging to a set. Definition If A is a set, a A means a belongs to A. We use a / A to indicate that a does not belong to A. If B is a set, then B A (A B) means that every element of B is an element of A, so B is a subset of A. We write A = B if A B and B A. B is a proper subset of A if B A but A has an element not in B. 1
2 2 Chapter 1. Some Background Material Recall the notation that is used to construct subsets. Example The set of odd natural numbers is given by {k : k, k = 2i +1, some i }. There is one special subset that has nothing in it. Definition The empty set is the set that has no elements. We always allow A for any set A, which means is an empty subset. We always have a / where a is any element. Measure theory is built on sets and set operations. The main operations are: Definition Let A and B be sets. A B = {a : a A or a B} A B = {a : a A and a B} A/B = A B = {a : a A and o / B} (Union), (Intersection), (Difference). Note that A/B = is possible. In the case that that there is a largest or master set, so all sets under consideration are subsets of, then we define: Definition For any subset A we denote, /A = A c (Complement of A). Another less familiar operation turns out to be important for measure theory: Definition If A and B are sets, then A B = (A/B) (B/A) (Symmetric Difference). We collect the basic facts about these operations in the theorem below. Theorem Consider subsets A,{A α,α } of a set. Then, α A α c = α A c α, α A α c = α A c α, α A α A β α A α for any β, A A α = α A Aα, A α A α = α A Aα ). Note that we drop the subscript index set in the statements when it is clear which index set is being considered.
3 1.1. Sets Functions Along with sets, measure theory and probability are also built on functions. Definition Let and be sets. A function f from to, f :, is a rule that assigns one element b to each element a. We write b = f (a), a Functions are also called maps, mappings, and transformations. We also consider functions applied to sets. Definition Let f : and A. Then, f (A) = { f (a) : a A}. Note that f (A) may be a proper subset of. There are two important sets associated with a function. Definition The domain of a function is the set of allowed inputs. The range of a function is the set of all outputs of a function for a given domain. In practice, there is some ambiguity in the definitions of domain and range. The natural domain is the set of all inputs for which the function is defined, but we often restrict this to some subset. Likewise, range is often used to refer to a set that contains the actual range of a function, e.g. and + both might be called the range of x 2 for x. It is important to be very precise about the domain and range in measure theory and probability. With this in mind, we define: Definition A map f : is onto if for each b, there is an a with f (a) = b. A map f : is 11 if for any a 1, a 2 with a 1 a 2 ; f (a 1 ) f (a 2 ). The concept of the inverse map to a function is centrally important to measure theory. It is extremely important to pay attention to the domain and range in this situation. Definition Let f : be a map from domain to range. The inverse image of a point b is defined, f 1 (b) = {a : a, f (a) = b}. Note that the inverse image of a point is a set in general. The natural domain of the inverse map to a function f : is the range. The range of the inverse map is a new space whose members consist of sets of points in. Definition Let f : be a map from domain to range. The range of f 1 is the space of equivalence classes on, where a 1 and a 2 are equivalent if f (a 1 ) = f (a 2 ).
4 4 Chapter 1. Some Background Material Cardinality We mentioned above that specifying the size, or cardinality, of an index set is important in certain places. Formalizing that notion, Definition Two sets and are equivalent or have the same cardinality, written, if there is a 1 1 and onto map f :. If = or {1,2,..., n} for some n, we say that is finite. If is finite or, we say that is countable. If is not empty, finite, or countable, we say is is uncountable. Note that there are different cardinalities among the uncountable sets but that is not important for the material below. Example , +, {odd integers}, and are all countable., +,,{x : x, x > 4} are all uncountable and have the same cardinality. Recall that all countable sets are equivalent and, indeed, can be written in the same way. Theorem A countable set can be written as {a 1, a 2, a 3, }, where a 1, a 2, denumerate the points in. This is another way to state the fact that there is a 1 1 and onto map between and {a 1, a 2, a 3,...}. As we said, below we construct complicated sets using unions and intersections. A crucial fact underlying the construction is the following. Theorem The countable union of countable sets is countable. We frequently deal with operations and sums of collections of objects indexed by some set. It is usually important to distinguish the cases of the index set being finite, countable, and uncountable. We use roman letter indices, e.g. i, j, k, l, m, n, for finite and countable collections and greek letter indices, e.g. α,β, for uncountable collections Sequences of sets It turns out that measure theory often deals with countable sequences of sets, and we discuss a few useful ideas. The first notion is convergence of a sequence of sets. Definition Let {A n } be a sequence of subsets of a set. If A 1 A 2 A 3... and A i = A, then we say that {A n } is an increasing sequence of i=1 sets and that A n converges to A. We denote this by A n A.
5 1.2. Real numbers 5 If A 1 A 2 A 3... and A i = A, then we say that {A n } is a decreasing sequence of i=1 sets and A n converges to A. We denote this by A n A. Theorem Let (A n ) be a sequence of subsets of. n 1. If A n A then, A n = A i. i=1 2. If A n A then A c n Ac. If A n A then A c n Ac. The implications of set convergence depends heavily on whether or not the sets in the sequence are nonintersecting. Definition A sequence {A n } of sets in is (pairwise) disjoint if A i A j = for i j. The next set of ideas is based on the observation that given two subsets A, B, we can write the union as a disjoint union: A B = (A) (B A c ). DeMorgan s Law can be used to show the following statements. Theorem Let {A n } be a sequence of subsets of. Then, 1. Set A = i=1 A i. Define the sequence B 1 = A 1 and B n = n i=1 A i for n 2. Then B n A. 2. Define B 1 = A 1 and B n = A n \ n 1 i=1 A i. Then {Bn } is a disjoint sequence of sets with i=1 A i = i=1 B i. References Exercises 1.2 Real numbers For the rest of the book, we work in and use the properties of the real numbers extensively. The one necessary prerequisite for this book is knowledge of the construction and properties of the real numbers. So, it is a good idea to review the reals if these are not familiar. We present a brief overview. Two of the fundamental properties of real numbers are the least upper bound and greatest lower bound properties. Definition A nonempty set A of real numbers is bounded above if there is a number b such that x b for all x A. b is called an upper bound for A. If A is bounded above, then an upper bound c for A is the least upper bound (lub) or supremum for A, if c is less than or equal to any other upper bound of A. We write c = sup A. A nonempty set A of real numbers is bounded below if there is a number b such that b x for all x A. b is called an lower bound for A.
6 6 Chapter 1. Some Background Material If A is bounded below, then a lower bound c for A is the greatest lower bound (glb) or infimum for A, if c is greater than or equal to any other lower bound of A. We write c = inf A. Note that a set may or may not contain its sup or inf if they exist. Definition A bounded set of reals is set that is bounded above and below. A fundamental property of the reals is the existence of the inf and sup. Least Upper Bound (LUB) Property Every nonempty set of real numbers that is bounded above has a least upper bound. Greatest Lower Bound (GLB) Property Every nonempty set of real numbers that is bounded below has a greatest lower bound. Depending on how the real numbers are constructed, the LUB and GLB Properties may be expressed as axioms or as results. In any case, they can be expressed in alternate ways. For the purposes of measure theory, a useful alternative form is completeness, which is a property related to sequences of real numbers. We begin with an important interpretation of a sequence as a function. Definition A sequence {x i } i=1 = {x i } of real numbers is a function from (or + ) to a countable set of real numbers. A sequence may be associated with a particular function or it may be known only by the list of values. Recall that generally we are interested in sequences that have predictable behavior as the index increases. In particular, we usually consider sequences that converge. Definition A sequence {x i } converges to a real number x if for each ε > 0 there is an N such that x i x < ε for all i > N. We call x the limit and we write x = lim i x i = lim i x i and x i x. It is easy to see that limits are unique. Def is very often presented in textbooks, but it is not a practically useful concept, since it requires the limit in order to be verified. If we have the limit in hand, there would seem to be little point in considering the sequence. In practice, we have the sequence and not the limit and we require a condition that guarantees convergence that does not involve the limit. We observe that if a sequence converges, then the terms in the sequence necessarily become close to each other as the indices increase. What about the converse? Definition A sequence {x i } is a Cauchy sequence if for each ε > 0 there is a N such that for all i > N, j > N we have x i x j < ε. Theorem Every convergent sequence of real numbers is a Cauchy sequence and every Cauchy sequence of real numbers converges. Definition Referring to Thm 1.4.1, we say the real numbers are complete. If we are working in a set in which Cauchy sequences converge, then we can compute
7 1.2. Real numbers 7 approximations of the limit of a Cauchy sequence by taking a term in the sequence of sufficiently high index Open, closed, and compact sets The construction of measure theory and probability is based on set operations applied to the most familiar of sets of real numbers. Definition An open interval (a, b) is the set of all real numbers {x : a < x < b}. We also have infinite open intervals, (a, ) = {x : a < x},(, b) = {x : x < b},(, ) = We next generalize the defining property of an open interval. Definition An open set G of real numbers has the property that for each x G, there is an r > 0 such that all y with x y < r belong to G. This is a local property in the sense that r depends on x in general. The following theorem about set operations and open sets underlies the measure theory construction. Theorem The union of any collection of open sets is open. The intersection of any finite number of open sets is open. Example The intersection of an infinite number of open sets may not be open, e.g. 1 n, 1 = {0}. n n=1 The following decomposition result is used for several important proofs in measure theory. Theorem Every open set of real numbers is the union of a countable collection of mutually disjoint open intervals. Now we turn to closed sets. Definition finite. A closed interval is a set [a, b] = {x : a x b}, where a and b are As above, we abstract the important property of being closed. Definition A set F is closed if it is the complement of an open set. By convention, we generally use the notation that G is used for open sets and F is used for closed sets. Of course, there are sets that are neither open or closed. The basic result for set operations and closed sets is: Theorem The intersection of any collection of closed sets is closed. The union of any finite number of closed sets is closed.
8 8 Chapter 1. Some Background Material It is convenient to reformulate the property of being closed in terms of limits. Definition A point x is called a cluster point or an accumulation point of a set A if for every r > 0 there is a y A, y x, such that x y < r. Working from this definition, a cluster point in Acan be obtained as the limit of a sequence of points in A. A cluster point of A may or may not be in A. The following theorem says that cluster points exist under general conditions. Theorem (BolzanoWeierstrass). cluster point. Every bounded set with infinite cardinality has a Finally, the connection to being closed. Theorem A set F is closed if and only if it contains all of its cluster points. A key idea in measure theory is the approximation of a complicated set by a combination of simpler sets. This idea is related to the fundamental property of compactness. Definition A collection of sets covers a set A if A B B. is called a cover of A. If the cover contains only open sets, we call it an open cover. If the cover has a finite number of elements, we call it a finite cover. If is a subset and is also a cover of A, we call a subcover. Definition A set K is called compact if every open cover of K has a finite subcover. The first result is a good exercise. Theorem A closed interval [a, b] is compact. The generalization is a very important property of real numbers. Theorem (HeineBorel). bounded. A subset of is compact if and only if it is closed and Continuous functions We conclude by recalling some facts about real valued functions on sets of real numbers. Let A be an interval in and f :. Definition A function f is continuous at in x A if for any ε > 0 there is a δ > 0 such that f (x) f (y) < ε for all y A with x y < δ. If f is continuous at every point in A, we say f is continuous on A or continuous. This is a local property in the sense that δ and ε depend on x in general. An alternate characterization of continuity involves sequences and limits. Theorem f is continuous at x in A if and only if for every sequence x n A with x n x, f (x n ) f (x).
9 1.2. Real numbers 9 Recall that this definition is applied to a endpoint of a closed interval by considering onesided limits. There is an important connection between continuity and compactness. Theorem The image of a continuous function applied to a compact set is compact. Theorem Every continuous function on a compact domain K has a maximum and minimum value, i.e. there are points x 1 and x 2 in K such that f (x 1 ) f (x) f (x 2 ) for all x K. Next, we consider a stronger version of continuity on a set. Definition A function f on A is uniformly continuous on A if for every ε > 0 there is a δ > 0 such that for all x, y A with x y < δ, f (x) f (y) < ε. The difference between this definition and Def is that δ does not depend on the point in the set under consideration. Uniform continuity on a set implies continuity at each point in the set, but the converse is not true. Example x 1 is continuous but not uniformly continuous on (0,1). Another important way that compactness interacts with continuity is the following: Theorem (Heine). continuous on K. If f is continuous on a compact set K, then f is uniformly There are other notions of continuity that are important. For example, the following is important for measure theory. Definition A function f on A is absolutely continuous if for every ε > 0 there is a δ > 0 such that f (b i ) f (a i ) < ε, i for every finite collection of pairwise disjoint intervals {(a i, b i )}, with (a i, b i ) [a, b] all i and i b i a i < δ. This definition controls how much the function can oscillate. Example The function f (x) on [0,1] that equals 1 n at x = (2n) 2 and equals 0 at x = (2n+1) 2 for n = 1,2, and linearly interpolates between these values for x in between, is uniformly continuous but not absolutely continuous. The issue is that the function "oscillates" too much. A simpler condition to check is Definition that A function f on A is Lipschitz condition if there is a constant L such f (x) f (y) L x y all x, y A.
10 10 Chapter 1. Some Background Material The relation between these properties is Theorem We have: Lipschitz continuity = absolute continuity, absolute continuity = uniform continuity. The proof is a good exercise The extended real number system For convenience, we use a limited version of Definition The extended real number system is = {, } with the rules x, x, < x <, x, x ± = ±, x, + =, =, x (± ) = ±, x > 0, x (± ) =, x < 0, 0 (± ) = 0. Note, that we do not define a value. Also, the convention that 0 (± ) = 0 is only permissible because in measure theory that is usually the correct value that would be assigned with a careful analysis. With these conventions, other structures associated with real numbers are extended in the obvious way. For example, [0,1] = {x : 0 x }. Below, we frequently deal with set functions that take values in the extended reals, e.g. f : [0, ], = a set, References Exercises is an extended real valued nonnegative function. 1.3 Metric spaces Several key concepts and results of metric spaces  one of the main subjects in introductory real analysis  are important to measure theory and probability. Recall that a metric space is an abstraction of a set of points for which the concept of distance of points is defined. A common example is the n dimensional vector space n, but we often have to deal with other spaces in analysis. The notion of distance is important, for example, to limits, area and volume, and the techniques of calculus. We review some relevant concepts and results of metric spaces, without discussing proofs. It is a good idea to keep a real analysis book nearby as a reference. We begin with the definition of distance. Definition A metric on a set is a map d(, ) on to + satisfying:
11 1.3. Metric spaces 11 d(x, y) = d(y, x) for all x, y (Symmetry), d(x, y) 0 for x, y, and d(x, y) = 0 if and only if x = y (Positivity), d(x, y) d(x, z) + d(z, y) for any x, y, z (Triangle Inequality). = (, d) is called a metric space. The following property is a consequence of the triangle inequality. Theorem d(x, y) d(x, z) d(y, z) for all x, y, z. The standard example is Euclidean space: Example On n, we can define a metric using the usual Euclidean distance: n 1/2 d(x, y) = x i y i 2, i=1 where x = (x 1, x 2,..., x n ) and y = (y 1, y 2,..., y n ). Generalizing this example, any norm on a vector space generates a metric: Example Let be any normed vector space with the norm. Define, for any x, y. Then, d is a metric on. d(x, y) = x y, Much of analysis is concerned with metric spaces that are infinite dimensional. An important example is: Definition We use C ([a, b]) to denote the vector space of realvalued continuous functions defined on the interval of real numbers [a, b]. C ([a, b]) is also written as C ([a, b]), C [a, b] and C (a, b). The latter notation is a bit problematic because it is important for the interval to be closed. We define, d( f, g) = sup f (x) g(x) = max f (x) g(x). x [a,b] x [a,b] This is called the sup or max metric. Symmetry is obvious and positivity follows from the continuity of the functions. The triangle inequality follows by observing that for x [a, b], f (x) g(x) f (x) h(x) + h(x) g(x), by the triangle inequality for numbers. We then take the max on both sides. Recall that a continuous function on a closed interval can be written uniquely as a Fourier series. In general, the Fourier series of a continuous function has an infinite number of terms. This is an easy way to understand that C ([a, b]) is infinite dimensional.
12 12 Chapter 1. Some Background Material Sequences The metric opens up the possibility of analysis because it provides a way to talk about convergence of sequences. Definition A sequence {x i } in a metric space (, d) converges if there is an x such that d(x i, x) 0 as i. This means that for any ε > 0, there is a positive integer N such that d(x i, x) < ε for all i > N. We write x i x, lim x i = x, etc. i There is a subtle point in this definition. If {x n } is a sequence in a metric space (, d) and x, then {d(x i, x)} is a sequence of real numbers. In this definition, we define convergence of {x i } using the familiar definition of convergence of a sequence of real numbers. This is a very useful approach. As an example, Theorem A convergent sequence in a metric space has a unique limit. An important issue, or perhaps flaw, with Def is that it requires the limit for verification. We look for an alternative criterion for convergence. Definition A sequence {x i } in a metric space (, d) is a Cauchy sequence if d(x i, x j ) 0 as i, j, which means that for any ε > 0 there is a positive integer N, such that d(x i, x j ) < ε for all i, j > N. We also say that the sequence satisfies the Cauchy criterion. If a sequence converges, then the terms necessarily become close in the limit of large indices. Theorem A convergent sequence in a metric space is also a Cauchy sequence. Does a Cauchy sequence necessarily converge? Note that in the definition of convergence we explicitly assume the limit is in the metric space. That is not so in the definition of a Cauchy sequence. Example Consider = (0,1) with d(x, y) = x y. The sequence {i 1, i, i > 0} is a Cauchy sequence but does not converge because 0. We cannot check the definition of convergence!) It may seem artificial in this example to purposely exclude the limit from. But in practice, we often do not have a limit of a sequence in hand, we may not know if the sequence converges, and even if a sequence converges, we may not know if the limit is in the space in question. Convergence follows in one case: Theorem If a Cauchy sequence has a convergent subsequence, then the entire sequence converges. It may help to think of the convergent subsequence as dragging the rest of the sequence along with it. We give a special name to metric spaces in which all Cauchy sequences converge. Definition A metric space is complete if every Cauchy sequence converges (to an
13 1.3. Metric spaces 13 element in the space). It is important to know the following examples: Example n with the usual metric is complete. Example C ([a, b]) with the sup metric is complete. There are important examples of metric spaces that are not complete. This is often a consequence of the choice of metric. In the following example, we choose a different, but natural, metric on the space of continuous functions. Example Consider the set of continuous realvalued functions on the real interval [a, b], with d( ˆ f, g) = b a f (x) g(x) 2 d x We can verify that d ˆ is a metric, so we get a metric space Ĉ ([a, b]). Now, consider a sequence {g n (x)} in Ĉ ([0,1]), defined 0, 0 x 1/2 1/n, g n (x) = 1 + (x 1/2)/n, 1/2 1/n x 1/2, 1, 1/2 x 1. 1/2. The sequence {g n (x)} Ĉ ([0,1]) is a Cauchy sequence, but g n converges to the discontinuous function given by, 0 x < 1/2, H(x) = 1 x 1/2, with respect to the metric d. ˆ Hence Ĉ ([a, b]) is not complete. Remark 1.1. In general, the choice of a metric is key in the case of infinite dimensional examples Topology The metric also allows the injection of toplogical concepts such as open and closed sets. Note that these concepts can be defined in different ways and the first part of real analysis consists in proving equivalences between various formulations. For example, in Sec we define open sets of real numbers using an open neighborhood condition and list the eventual consequence that a set is closed if and only if it contains its limit points. Below, we start by defining a closed set using a limit condition, then derive a neighborhood condition for open sets. There is no one best formulation, rather the various formulations are most suited for different kinds of proofs. That is, a proof based on one formulation may require fewer lines than other formulations. Definition Let A be a subset of a metric space (, d). A point x is a limit point of A if for every ε > 0 there is y x with d(y, x) < ε. A is closed if it contains all of its limit
14 14 Chapter 1. Some Background Material points. The closure of A, written as A, is the smallest subset of containing A and its limit points. Finally, a set B is open if its complement /B = B c is closed. Example B 1 2 = {(x, y) : d(x, y) < 1} is open. The limit points of B 1 consists of B 1 together with the unit circle {(x, y) : d(x, y) = 1}, so B 1 2 = {(x, y) : d(x, y) 1}. Example [0,1) in 1 with d(x, y) = x y is neither open or closed. Definition If (, d) is a metric space, the open ball of radius r centered at point x 0 is defined, B r (x 0 ) = {x : d(x, x 0 ) < r }. Theorem If G is an open subset of a metric space (, d) if and only if for each point x in G there is an open ball centered at x contained in G. Note that defining the ball to be open by using strict inequality (<) is essential to this result. The proof of this result provides a useful strategy for other arguments. Proof. We show that if G is open, it satisfies open ball condition. G is open if and only if G c is closed. Now x 0 G c, so it is not a limit point of G c, which is closed and contains its limit points. This implies that is an r > 0 such that B r x 0 G c =. Otherwise, we can create a sequence in G c that converges to x 0 by considering a sequence of balls of decreasing radii and choosing a point in each ball. It is a good exercise to show the converse. The notion of compactness is important in understanding sets and functions of real numbers. We extend the notion to metric spaces. Definition A collection {G α } of open sets of a metric space (, d) covers a set K if K G α. {G α } is also called an open cover and a covering of K. A set K is compact if any covering {G α } of K contains a finite subcollection G α1, G α2,...,g αn that covers K. Example Consider (0,1) 1, which is covered by the set, α (0,1) (0,1/2) (1/2 1/4,3/4) (3/4 1/8,7/8). This cover does not contain a finite subcover. Of course, (0,1) is not compact. An equivalent formulation of compactness involves convergence of sequences.
15 1.3. Metric spaces 15 Theorem Let (, d) be a metric space. A set K is compact if and only if every sequence {x i } of points in K contains a subsequence x n1, x n2,... that converges to a limit in K. Thus, being contained in a compact set provides some bounds on how a sequence can behave even when it does not converge. With more assumptions, we get more. Theorem A Cauchy sequence in a metric space that is contained in a compact set converges to a limit in the compact set. Definition A metric space for which every Cauchy sequence converges to a limit in the space is called complete. A related result is Theorem (BolzanoWeierstrass). Every bounded infinitecardinality set in n with the usual metric has a limit point. There is a simple characterization of compactness in Euclidean space. Theorem (HeineBorel). is compact. In n with the usual metric, every closed, bounded subset Remark 1.2. Warning, the HeineBorel result is not true in general metric spaces. Definition Let a = (a 1, a 2,...,a n ) and b = (b 1, b 2,..., b n ) be points in n with the usual metric, such that a i b i for 1 i n. The multiinterval or generalized rectangle is a generalization of a rectangle defined as, Q = {x n : a i x i b i, 1 i n}. Theorem Multiintervals in n are compact Functions We consider a map between two metric spaces (, d ) and (, d ). Definition A function f : is continuous at x 0 if for every ε > 0, there is a δ > 0 such that d ( f (x 0 ), f (y)) < ε for all y with d (x 0, y) < δ. If f is continuous at every point in, we say f is (pointwise) continuous on. f is uniformly continuous on if for ε > 0, there is a δ > 0 such that d ( f (x), f (y)) < ε for all x, y with d (x, y) < δ. Of course, uniform continuity implies pointwise continuity. One important consequence of uniform continuity is inheriting the Cauchy condition. Theorem Let f : be a uniformly continuous function. If {x i } is a Cauchy sequence in, then { f (x i )} is a Cauchy sequence in. We also require the idea of the inverse map.
16 16 Chapter 1. Some Background Material Definition Let f : and B. The set, is called the inverse image of B. f 1 (B) = {x : f (x) B}, In general, the inverse image of even a single point may be a set. Example Consider f (x) = x 2 : [ 1,1] [0,1] with the usual metric on the domain and range. Then f 1 (y) = { y,+ y} for 0 < y 1. Thus, the inverse map to a function f : maps to a new space whose points are sets in in general. We can consider the range of the inverse map to be a space of equivalence classes in. Theorem Suppose f : is a function. 1. If x is a limit point of, the f is continuous at x if and only if f (x i ) f (x) in for any sequence {x i } with x i x in. 2. f is continuous on if and only if f 1 (G) is open in for any open set G in. 3. f is continuous on if and only if f 1 (F ) is closed in for any closed set F in. Conditions (2) and (3) use a notion of behavior induced by the inverse of the map, meaning that we pose a condition on sets in the range of the map and require behavior in the corresponding inverse images. This is a central idea in measure theory. Compactness and continuity relate in a beneficial way. Theorem Suppose f : is a map and is compact. Then, the image of under f, i.e., f () = {y : y = f (x), x } is a compact set in. If = n, then f () is closed and bounded. Moreover, Theorem Let f : be a continuous map where is compact. Then, f is uniformly continuous on. Note that these last two theorems, like others, can be applied to a subset of a metric space, when inherits the metric structure and f is restricted to Denseness and separability In the abstract, we talk about points in a metric space as if they can be produced at will. In many practical situations, we are unable to produce arbitrary points in a metric space. Rather, we have to content ourselves with being able to approximate any given specified point to any desired accuracy using a special collection of simpler points. Here, simple includes generally includes the properties of being in a countable set and being computable, though there can be additional properties.
17 1.3. Metric spaces 17 Example We cannot write down an arbitrary real number in general, e.g. because it has an infinite decimal expansion. However, any real number can be approximated by a sequence of rational numbers, e.g., the sequence of truncated decimal expansions, 3,3.1,3.14,3.141,3.1415, ,... π. So the reals can be approximated by the countable set of rational numbers. Example We cannot write down an arbitrary realvalued function on the real numbers in general. Even in the case of a known function like sin, exp, and log, these labels are placeholders for a function whose values are not specified by a finite number of operations. Rather, we compute approximations to their values in practice. In the next chapter, we discuss the Weierstrass Approximation Theorem, which says roughly that continuous functions can be approximated by polynomials. We take polynomials as being computable in the sense that their values are determined by a finite number of operations. The property of a set being useful for approximation is captured in the following definition. Definition A set A in a metric space (, d) is dense if every point in is a limit point of A or is in A or both. The property of a space containing such an approximation set is given a name. Definition A metric space is separable if it contains a countable dense subset. Compactness is a guarantee. Theorem Every compact metric space is separable. We explore the idea of approximation quite extensively below. As a teaser, we present a result that explains how density can be used to approximate function values. Theorem Assume that (, d ) and (, d ) are metric spaces and that is complete. Assume that we have a map f : A defined on a dense subset A that is uniformly continuous on A. There is a unique continuous map g : such that g(x) = f (x) for x A. We give a name to the new function: Definition The function g is called the extension of f and g extends f from A to. As a consequence, for example, we can approximate the value of a polynomial function on any real number by values of the polynomial computed at numbers with finite decimal expansions, and likewise by the Weierstrass Approximation Theorem mentioned above, we can approximate the values of transcendental functions like exp using polynomials evaluated at numbers with finite decimal expansions.
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