REAL AND COMPLEX ANALYSIS REAL AND COMPLEX ANALYSIS Third Edition

INTEGRA nON 7 If no two members of {All} have an element in common, then {All} is a disjoint collection of sets: We write A B = {x: x E A, x ~ B}, and denote the complement of A by AC whenever it is clear from the context with respect to which larger set the complement is taken. The cartesian product Al x ... x An of the sets AI' ... , An is the set of all ordered n-tuples (aI' ... , an) where ai E Ai for i = 1, ... , n. The real line (or real number system) is Rl, and (k factors). The extended real number system is Rl with two symbols, 00 and 00, adjoined, and with the obvious ordering. If 00 :s; a :s; b :s; 00, the interval [a, b] and the segment (a, b) are defined to be [a, b] = {x: a :s; x :s; b}, (a, b) = {x: a < x < b}. We also write [a, b) = {x: a :s; x < b}, (a, b] = {x: a < x :s; b}. If E c [ 00, 00] and E i= 0, the least upper bound (supremum) and greatest lower bound (infimum) of E exist in [ 00, 00] and are denoted by sup E and inf E. Sometimes (but only when sup E E E) we write max E for sup E. The symbol f:X-Y means thatfis afunction (or mapping or transformation) of the set X into the set Y; i.e., f assigns to each x E X an element f(x) E Y. If A c X and BeY, the image of A and the inverse image (or pre-image) of Bare f(A) = {y: y = f(x) for some x E A}, f1(B) = {x:f(x) E B}. Note thatf-l(B) may be empty even when B i= 0. The domain offis X. The range offisf(X). Iff(X) = Y,fis said to map X onto Y. We writef1(y), instead off1({y}), for every y E Y. Iff-l(y) consists of at most one point, for each y E Y,f is said to be one-to-one. Iff is one-to-one, then f1 is a function with domainf(X) and range X. If f: X [ 00, 00] and E c X, it is customary to write supx e E f(x) rather than sup f(E). If f: X Y and g: Y Z, the composite function g 0 f: X Z is defined by the formula (g 0 f)(x) = g(f(x» (x EX). 8 REAL AND COMPLEX ANALYSIS If the range of J lies in the real line (or in the com plex plane), then J is said to be a real Junction (or a complex Junction). For a complex function/, the statement "J;;::. 0" means that all valuesJ(x) ofJare nonnegative real numbers. The Con~ept of Measurability The class of measurable functions plays a fundamental role in integration theory. It has some basic properties in common with another most important class of functions, namely, the continuous ones. It is helpful to keep these similarities in mind. Our presentation is therefore organized in such a way that the analogies between the concepts topological space, open set, and continuous Junction, on the one hand, and measurable space, measurable set, and measurable Junction, on the other, are strongly emphasized. It seems that the relations between these concepts emerge most clearly when the setting is quite abstract, and this (rather than a desire for mere generality) motivates our approach to the subject. 1.2 Definition (a) A collection 't of subsets of a set X is said to be a topology in X if 't has the following three properties: (i) 0 E 't and X E 'to (ii) If V; E dor i = 1, ... , n, then VI n V2 n ... n v.. E 'to (iii) If {~} is an arbitrary collection of members of't (finite, countable, or uncountable), then Ua ~ E 'to (b) If 't is a topology in X, then X is called a topological space, and the members of't are called the open sets in X. (c) If X and Yare topological spaces and ifJis a mapping of X into Y, then J is said to be continuous provided that J -l( V) is an open set in X for every open set V in Y. 1.3 Definition (a) A collection IDl of subsets of a set X is said to be a a-algebra in X if IDl has the following properties: (i) X E IDl. (ii) If A E IDl, then AC E IDl, where AC is the complement of A relative to X. (iii) If A = U:,= I An and if An E IDl for n = 1,2, 3, ... , then A E IDl. (b) If IDl is a a-algebra in X, then X is called a measurable space, and the members of IDl are called the measurable sets in X. . (c) If X is a measurable space, Y is a topological space, andJis a mapping of X into Y, then J is said to be measurable provided that J -1(V) is a measurable set in X for every open set V in Y. ABSTRACT INTEGRA nON 9INTEGRA nON 9 It would perhaps be more satisfactory to apply the term" measurable space" to the ordered pair (X, rol), rather than to X. After all, X is a set, and X has not been changed in any way by the fact that we now also have a u-algebra of its subsets in mind. Similarly, a topological space is an ordered pair (X, 'tf But if this sort of thing were systematically done in all mathematics, the terminology would become awfully cumbersome. We shall discuss this again at somewhat greater length in Sec. 1.21. 1.4 Comments on Definition 1.2 The most familiar topological spaces are the metric spaces. We shall assume some familiarity with metric spaces but shall give the basic definitions, for the sake of completeness. A metric space is a set X in which a distance function (or metric) p is defined, with the following properties: (a) 0::;; p(x, y) < 00 for all x and y E X. (b) p(x, y) = 0 if and only if x = y. (c) p(x, y) = p(y, x) for all x and y E X. (d) p(x, y) ::;; p(x, z) + p(z, y) for all x, y, and z E X. Property (d) is called the triangle inequality. If x E X and r ~ 0, the open ball with center at x and radius r is the set {y E X: p(x, y) < r}. If X is a metric space and if 't is the collection of all sets E c X which are arbitrary unions of open balls, then 't is a topology in X. This is not hard to verify; the intersection property depends on the fact that if x E B 1 1\ B 2 , where Bl and B2 are open balls, then x is the center of an open ball B c Bl 1\ B 2. We lea ve this as an exercise. For instance, in the real line Rl a set is open if and only if it is a union of open segments (a, b). In the plane R2, the open sets are those which are unions of open circular discs. Another topological space, which we shall encounter frequently, is the extended real line [00, 00]; its topology is defined by declaring the following sets to be open: (a, b), [ 00, a), (a, 00], and any union of segments of this type. The definition of continuity given in Sec. 1.2(c) is a global one. Frequently it is desirable to define continuity locally: A mapping f of X into Y is said to be continuous at the point Xo E X if to every neighborhood V of f(x o) there corresponds a neighborhood W of Xo such thatf(W) c V. (A neighborhood of a point x i~, by definition, an open set which contains x.) When X and Yare metric spaces, this local definition is of course the same as the usual epsilon-delta definition, and is equivalent to the requirement that limf(xn) = f(x o) in Y whenever lim Xn = Xo in X. The following easy proposition relates the local and global definitions of continuity in the expected manner: 1.5 Proposition Let X and Y be topological spaces. A mapping f of X into Y is continuous if and only iffis continuous at every point of x. 10 REAL AND COMPLEX ANALYSIS PROOF Iff is continuous and Xo E X, then f -l(V) is a neighborhood of xo, for every neighborhood V of f(xo). Since f(f -l(V» c V, it follows that f is continuous at Xo. If f is continuous at every point of X and if V is open in Y, every point x Ef-l(V) has a neighborhood w" such that f(Wx) c V. Therefore w" c f-1(V). It follows thatf-1(V) is the union of the open sets w", SOf-1(V) is itself open. Thusfis continuous. IIII 1.6 Comments on Definition 1.3 Let Wl be a o--algebra in a set X. Referring to Properties (i) to (iii) of Definition 1.3(a), we immediately derive the following facts. (a) Since 0 = XC, (i) and (ii) imply that 0 E Wl. (b) Taking An+l = An+2 = ... = 0 in (iii), we see that A1 u A2 U ... U An E Wl if Ai E Wl for i = 1, ... , n. (c) Since Wl is closed under the formation of countable (and also finite) intersections. (d) Since A B = JJ< n A, we have A B E Wl if A E Wl and BE Wl. The prefix 0refers to the fact that (iii) is required to hold for all countable unions of members of Wl. If (iii) is required for finite unions only, then Wl is called an algebra of sets. 1.7 Theorem Let Y and Z be topological spaces, and let g: Y -+ Z be continuous. (a) If X is a topological space, iff: X -+ Y is continuous, and if h = g 0 J, then h: X -+ Z is continuous. (b) If X is a measurable space, iff: X -+ Y is measurable, and if h = g 0 J, then h: X -+ Z is measurable. Stated informally, continuous functions of continuous functions are continuous; continuous functions of measurable functions are measurable. PROOF If V is open in Z, then g-l(V) is open in Y, and Iffis continuous, it follows that h1(V) is open, proving (a). Iffis measurable, it follows that h1(V) is measurable, proving (b). IIII ABSTRACT INTEGRATION 11INTEGRATION 11 1.8 Theorem Let u and v be real measurable functions on a measurable space X, let CI> be a continuous mapping of the plane into a topological space Y, and define h(x) = CI>(u(x), v(x» for x E X. Then h: X -4 Y is measurable. PROOF Put f(x) = (u(x), v(x». Then f maps X into the plane. Since h = CI> 0 f, Theorem 1.7 shows that it is enough to prove the measurability off If R is any open rectangle in, the plane, with sides parallel to the axes, then R is the cartesian product of two segments 11 and 12 , and which is measurable, by our assumption on u and v. Every open set V in the plane is a countable union of such rectangles R;, and since f -1( V) is measurable. IIII 1.9 Let X be a measurable space. The following propositions are corollaries of Theorems 1.7 and 1.8: (a) Iff = u + iv, where u and v are real measurable functions on X, then f Is a complex measurable function on X. This follows from Theorem 1.8, with CI>(z) = z. (b) Iff = u + iv is a complex measurable function on X, then u, v, and 1 flare real measurable functions on X. This follows from Theorem 1.7, with g(z) = Re (z), 1m (z), and 1 z I. (c) Iff and 9 are complex measurable functions on X, then so aref + 9 andfg. For realfand 9 this foll~ws from Theorem 1.8, with Cl>(s, t) = s + t and CI>(s, t) = st. The complex case then follows from (a) and (b). (d) If E is a measurable set in X and if then XE is a measurable function. if x E E if x if E This is obvious. We call XE the characteristic function of the set E. The letter X will be reserved for characteristic functions throughout this book. (e) Iff is a complex measurable function on X; there is a complex measurable function IX on X such that IIX 1 = 1 and f = IX 1 fl. 12 REAL AND COMPLEX ANALYSIS PROOF Let E = {x:f(x) = O}, let Y be the complex plane with the origin removed, define <p(z) = z/l z 1 for z E Y, and put IX(X) = <p(f(x) + X~x» (x EX). If x E E, IX(X) = 1; if x ¢ E, IX(X) = f(x)1 1 f(x) I. Since <p is continuous on Y and since E is measurable (why?), the measurability of IX follows from (c), (d), and Theorem 1.7. IIII We now show that a-algebras exist in great profusion. 1.10 Theorem If $' is any collection of subsets of X, there exists a smallest a-algebra Wi* in X such that$' c Wi*. This Wi* is sometimes called the a-algebra generated by $'. PROOF Let n be the family of all a-algebras Wi in X which contain $'. Since the collection of all subsets of X is such a a-algebra, n is not empty. Let Wi* be the intersection of all Wi E n. It is clear that$' c Wi* and that Wi* lies in every a-algebra in X which contains $'. To complete the proof, we have to show that Wi* is itself a a-algebra. If An E Wi* for n = 1,2,3, ... , and if Wi En, then An E Wi, so U An E Wi, since Wi is a a-algebra. Since U An E Wi for every Wi E n, we conclude that U An E Wi*. The other two defining properties of a a-algebra are verified in the same manner. I I I I 1.11 Borel Sets Let X be a topological space. By Theorem 1.10, there exists a smallest a-algebra fJI in X such that every open set in X belongs to fJI. The members of fJI are called the Borel sets of X. In particular, closed sets are Borel sets (being, by definition, the complements of open sets), and so are all countable unions of closed sets and all countable intersections of open sets. These last two are called F,r's and G,/s, respectively, and playa considerable role. The notation is due to Hausdorff. The letters F and G were used for closed and open sets, respectively, and a refers to union (Summe), {j to int~rsection (Durchschnitt). For example, every half-open interval [a, b) is a G6 and an Fa in RI. Since fJI is a a-algebra, we may now regard X as a measurable space, with the Borel sets playing the role of the measurable sets; more concisely, we consider the measurable space (X, BB). Iff: X -+ Y is a continuous mapping of X, where Y is any topological space, then it is evident from the definitions thatf-I(V) E fJI for every open set V in Y. In other words, every continuous mapping of X is Borel mea~urable. Borel measurable mappings are often called Borel mappings, or Borel functions. ABSTRACT INTEGRATION 13INTEGRATION 13 1.12 Theorem Suppose rol is a u-algebra in X, and Y is a topological space. Letfmap X into Y. (a) If n is the collection of all sets E c Y such that f -1(E) E rol, then n is a u-algebra in Y. (b) Iffis measurable and E is a Borel set in Y, thenf-1(E) E rol. (c) If Y = [-00,00] andf1((IX, 00]) E rolfor every real IX, thenfis measurable. (d) If f is measurable, if Z is a topological space, if g: Y --+ Z is a Borel mapping, and if h = 9 0 f, then h: X --+ Z is measurable. Part (c) is a frequently used criterion for the measurability of real-valued functions. (See also Exercise 3.) Note that (d) generalizes Theorem 1.7(b). PROOF (a) follows from the relations f-1(Y) = X, f-1(y A) = X f1(A), and f-1(A 1 u A2 U ... ) =f1(A 1) uf1(A 2) u .... To prove (b), let n be as in (a); the measurability of f implies that n contains all open sets in Y, and since n is au-algebra, n contains all Borel sets in Y. To prove (c), let n be the collection of all E c [ 00, 00] such that f -1(E) E rol. Choose a real number IX, and choose IXn < IX so that IXn --+ IX as n --+ 00. Since (IXn' 00] E n for each n, since 00 00 [-00, IX) = U [-00, IXn] = U (IXn' 00]<, n= 1 n= 1 and since (a) shows that n is a u-algebra, we see that [ 00, IX) E n. The same is then true of (IX, {J) = [ 00, {J) () (IX, 00]. Since every open set in [00, 00] is a countable union of segments of the above types, n contains every open set. Thusfis measurable. To prove (d), let V c Z be open. Then g-1(V) is a Borel set of Y, and since (b) shows that h1(V) E rol. IIII 1.13 Definition Let {an} be a sequence in [ 00, 00], and put (k = 1, 2, 3, ... ) (1) 14 REAL AND COMPLEX ANALYSIS and P = inf {b 1, bz , b3 , ••• }. (2) We call p the upper limit of {an}, and write P = lim sup an' (3) n-+ co The following properties are easily verified: First, b1 ~ bz ~ b3 ~ • ", so that bk pas k00; secondly, there is a subsequence {anJ of {an} such that an;p as i00, and p is the largest number with this property. The lower limit is defined analogously: simply interchange sup and inf in (1) and (2). Note that lim inf an = -lim sup (-aJ. (4) n-+ co n-+ co If {an} converges, then evidently lim sup an = lim inf an = lim an' (5) n-+co n-+ 00 n-+ 00 Suppose Un} is a sequence of extended-real functions on a set X. Then sup /" and lim sup /" are the functions defined on X by n 11-+ 00 ( s~p /" )<X) = s~p (fn(x)), (6) ( lim sup /,,)(X) = lim sup (fn(x)). 11-+00 n-+oo (7) If f(x) = lim /"(x), (8) n-+ co the limit being assumed to exist at every x E X, then we call f the pointwise limit of the sequence {/,,}. 1.14 Theorem Iffn: X[-00, 00] is measurable,for n = 1,2, 3, ... , and g = supfn' n;,1 then g and h are measurable. h = lim sup/", n-+ co PROOF g-I((IX, 00]) = U:"=1 f;I((IX, 00]). Hence Theorem lJ2(c) implies that g is measurable. The same result holds of course with inf in place of sup, and since h = inf {sup h}, 1;;,1 1;,1; it follows that h is measurable. IIII ABSTRACT INTEGRATION 15INTEGRATION 15 Corollaries (a) The limit of every pointwise convergent sequence of complex measurable functions is measurable. (b) Iff and g are measurable (with range in [00, 00]), then so are max {J, g} and min {J, g}. In particular, this is true of the functions f+ = max {J, O} and f= -min {J, O}. 1.15 The above functionsf+ andfare called the positive and negative parts off. We have I f I = f + + f and f = f + f -, a standard representation of f as a difference of two nonnegative functions, with a certain minimality property: Proposition Iff = g h, g ~ 0, and h ~ 0, then f + '5. g and f '5. h. PROOF f'5. g and 0 '5. g clearly implies max {J, O} '5. g. IIII