Constantin Carathéodory ( )

Transcription

1 Constantin Carathéodory ( ) Constantin Carathéodory was born on September 13, 1873, in Berlin, Germany, to Greek parents. His father, Stephanos, was an Ottoman Greek who had studied law in Berlin; his mother, Despina, came from a Greek family of businessmen. At the time of Carathéodory s birth, his family was in Berlin because his father had been appointed there as First Secretary to the Ottoman Legation. Carathéodory was a student at the Belgian Military Academy from and then worked as an assistant engineer with the British Asyūt Dam project in gypt. Returning to Germany, Carathéodory entered the University of Berlin in 1900 to begin his study of mathematics. In 1902, he transferred to the University of Göttingen, where he received his doctorate in 1904 under Hermann Minkowski. The works of Carathéodory encompassed many disciplines. Among them were a simplified proof of a central theorem in conformal representation, extensions of findings by Picard on function theory, and the amplification of measure theory begun by Émile Borel and Henri Lebesgue. Regarding the latter discipline, two particularly important results in measure theory are the Carathéodory criterion for measurability and the Carathéodory extension theorem. Carathéodory wrote many outstanding books. Two of particular relevance here are Vorlesungen Über Reelle Funktionen (Lectures on Real Functions) and Mass und Integral und Ihre Algebraisierung (Measure and Integral and their Algebraisation). Carathéodory taught at the Universities of Hannover (1909), Breslau ( ), Göttingen ( ), Berlin ( ), all in Germany. In 1920, he went to Anatolia, Greece, to establish the University of Smyrna. After the Turks burned Smyrna in 1922, he went to the University of Athens. In 1924, he returned to the University of Munich where he remained, except for one year ( ) at the University of Wisconsin, until his death on February 2,

2 5 lements of Measure Theory In Chapter 3, we expanded the collection of continuous functions to the collection of Borel measurable functions, the smallest algebra that contains the continuous functions and is closed under pointwise limits. Subsequently, in Chapter 4, we extended the Riemann integral so that it applies to all Borel measurable functions and, in doing so, we encountered Lebesgue measure, the collection of Lebesgue measurable functions, and the Lebesgue integral. We will discover, in this chapter, that the concepts and methods of Chapters 3 and 4 lend themselves to considerable generalization with relatively little effort and huge rewards. This generalized theory has extensive applications throughout mathematics and, as well, to a large variety of fields outside of mathematics. 5.1 MASUR SPACS A Course in Real Analysis. Second edition. Copyright 1999, 2013, lsevier Inc. All rights reserved. When we examine the definition of the Lebesgue integral carefully, we find that it depends ultimately on the concept of measure. More precisely, the mathematical framework requires a set, a σ-algebra of subsets, and a set function that assigns to each set in the σ-algebra a nonnegative number (its measure). In Chapter 3, this consisted, respectively, of R, M, and λ. But we can abstract the mathematical framework to provide a broader setting for the integral. We begin by considering the general concept of measure. In developing Lebesgue measure, we imposed three conditions; namely, Conditions (M1) (M3) on page 91. The first two conditions are specific to the generalization of length; but the third is not. In fact, Condition (M3), the countable-additivity condition, is the primary property of an abstract measure. 145

3 146 Chapter 5 lements of Measure Theory DFINITION 5.1 Measure, Measurable Space, Measure Space Let be a set and A a σ-algebra of subsets of. A measure μ on A is an extended real-valued function satisfying the following conditions: a) μ(a) 0 for all A A. b) μ( ) =0. c) If A 1, A 2,... are in A, with A i A j = for i j, then ( ) μ A n = μ(a n ). n n The pair (, A) is called a measurable space and the triple (, A,μ) is called a measure space. Note: We will often refer to members of A as A-measurable sets. We should point out the following fact: If μ satisfies (a) and (c) of Definition 5.1, then it is a measure (i.e., also satisfies (b)) if and only if there is an A Asuch that μ(a) <. We leave the proof of this fact to the reader. XAMPL 5.1 Illustrates Definition 5.1 a) (R, M,λ) is a measure space, the one that we studied in Chapter 3. b) (R, B,λ B ) is a measure space. c) Let (, A,μ) be a measure space. For D A, define A D = { D A : A A} and μ D = μ AD. Then A D is a σ-algebra of subsets of D, μ D is a measure on A D, and, hence, (D, A D,μ D ) is a measure space. d) Referring to part (c), let = R, A = M, μ = λ, and D =[0, 1]. Then ([0, 1], M [0,1],λ [0,1] ) is a measure space. λ [0,1] is called Lebesgue measure on [0, 1]. More generally, if D is a Lebesgue measurable set, then (D, M D,λ D ) is a measure space and λ D is called Lebesgue measure on D. e) Refer to part (c). By Theorem 3.7 on page 88, if D B, then B D = B(D). f) Let be a nonempty set and A = P(). Define μ on A by { N(), if is finite; μ() =, if is infinite. where N() denotes the number of elements of. Then μ is a measure on A and is called counting measure. g) Let = N, A = P(N ), and μ be counting measure on A, as defined in part (f). Then, for instance, μ(n )= and μ({1, 3}) = 2. We will see later that (N, P(N ),μ) is the appropriate measure space for the analysis of infinite series. h) Suppose that (, A,μ) is a measure space. If μ() = 1, then (, A,μ)is called a probability space and μ a probability measure. Furthermore, μ is usually replaced by a P (for probability). Two simple examples are as follows: (i) ([0, 1], M [0,1],λ [0,1] ) is a probability space since λ([0, 1])=1. Itisan appropriate measure space for analyzing the experiment of selecting a number at random from the unit interval.

4 5.1 Measure Spaces 147 (ii) Consider the experiment of tossing a coin twice. The set of possible outcomes for that experiment is = {HH, HT, TH, TT} where, for instance, HT denotes the outcome of a head on the first toss and a tail on the second toss. Set A = P() and, for A, define P () =N()/4 where, as before, N() denotes the number of elements of. Then (, A,P) is a probability space the appropriate measure space to use when the coin is balanced (i.e., equally likely to come up heads or tails). To illustrate: The probability of getting at least one head in two tosses of a balanced coin is P ({HH, HT, TH}) =3/4. i) Let be a nonempty set, {x n } n a sequence of distinct elements of, and {a n } n a sequence of nonnegative numbers. For, define μ() = a n, (5.1) x n where the notation x n means the sum over all indices n such that x n. Then μ is a measure on P() and, consequently, (, P(),μ) is a measure space. Here are two special cases: (i) If is countable, {x n } n is an enumeration of, and a n = 1 for all n, then the measure μ defined in (5.1) is counting measure. (ii) If the sequence {x n } n consists of only one element, say x 0, and if a 0 =1, then the measure μ defined in (5.1) takes the form μ() = { 1, if x0 ; 0, if x 0. This measure is denoted by δ x0 and is called the unit point mass or Dirac measure concentrated at x 0. Note that δ x0 is a probability measure. j) Let (, A) be a measurable space such that {x} A for each x. A measure μ on A is called discrete if there is a countable set K such that μ(k c ) = 0. It is not too difficult to show that if μ is a discrete measure, then we can write μ = x K μ({x})δ x. See xercises 5.6 and 5.19 for more on discrete measures. The following theorem provides some important properties of measures. We leave the proof as an exercise for the reader. THORM 5.1 Suppose that (, A,μ) is a measure space and that A and B are A-measurable sets. Then the following properties hold: a) If μ(a) < and A B, then μ(b \ A) =μ(b) μ(a). b) A B μ(a) μ(b). (monotonicity) c) If { n } Awith 1 2 and μ( 1 ) <, then ( μ n ) = lim μ( n).

5 148 Chapter 5 lements of Measure Theory d) If { n } Awith 1 2, then ( μ n ) = lim μ( n). e) If { n } n A, then ( ) μ n μ( n ). n n This property is called countable subadditivity. Almost verywhere and Complete Measure Spaces Recall from Section 4.4 that a property holds Lebesgue almost everywhere (λ-ae) if it holds except on a set of Lebesgue measure zero. That concept can be generalized to apply to any measure space. DFINITION 5.2 Almost verywhere A property is said to hold μ almost everywhere, or μ-ae for short, if it holds except on a set of μ-measure zero, that is, except on a set N with μ(n) =0. Note: Several terms are used synonymously for almost everywhere. Here are a few: almost always, for almost all x, and, in probability theory, almost surely, with probability one, and almost certainly. Proposition 3.4 on page 106 implies that subsets of Lebesgue measurable sets of Lebesgue measure zero are also Lebesgue measurable sets. On the other hand, xercise 4.48 on page 142 indicates that there exist subsets of Borel sets of Lebesgue measure zero that are not Borel sets. Those two facts have relevance to almost-everywhere (ae) properties of measurable functions. For instance, by Proposition 4.7 on page 140, if f is Lebesgue measurable and g = fλ-ae, then g is Lebesgue measurable. However, as xercise 4.49 on page 142 shows, that result is not true for Borel measurable functions. We now see that it is important to know whether subsets of sets of measure zero are measurable sets. Hence, we make the following definition. DFINITION 5.3 Complete Measure Space A measure space (, A,μ) is said to be complete if all subsets of A-measurable sets of μ-measure zero are also A-measurable; in other words, if A A and μ(a) = 0, then B Afor all B A. Thus, (R, M,λ) is a complete measure space, whereas (R, B,λ B ) is not a complete measure space.

6 5.1 Measure Spaces 149 The following theorem shows that any measure space can be extended to a complete measure space. We leave the proof of the theorem as an exercise for the reader. THORM 5.2 Let (, A,μ) be a measure space. Denote by A the collection of all sets of the form B A where B Aand A C for some C Awith μ(c) =0. For such sets, define μ(b A) =μ(b). Then A is a σ-algebra, μ is a measure on A, and (, A, μ) is a complete measure space. Furthermore, A A and μ A = μ. (, A, μ) is called the completion of (, A,μ). It can be shown that the measure space (R, M,λ) is the completion of the measure space (R, B,λ B ). See xercise xercises for Section 5.1 Note: A denotes an exercise that will be subsequently referenced. 5.1 Suppose that (, A,μ) is a measure space and that D is an A-measurable set. Define A D = { D A : A A}and μ D = μ AD. Show that (D, A D,μ D) is a measure space. 5.2 Let be a nonempty set and A = P(). Define μ on A by { N(), if is finite; μ() =, if is infinite. where N() denotes the number of elements of. Prove that μ is a measure on A. 5.3 Consider the experiment of selecting a number at random from the interval [ 1, 1]. a) Construct an appropriate probability space for this experiment. b) Determine the probability that the number selected exceeds 0.5. c) Determine the probability that the number selected is rational. 5.4 Let (, A) be a measurable space, μ and ν measures on A, and α>0. Define set functions μ + ν and αμ on A by (μ + ν)(a) =μ(a)+ν(a), (αμ)(a) =αμ(a). a) Show that μ + ν is a measure on A. b) Show that αμ is a measure on A. 5.5 Let (, A) be a measurable space, {μ n} a sequence of measures on A, and {αn} a sequence of nonnegative real numbers. Define αnμn on A by ( α nμ n )(A) = α nμ n(a). Prove that αnμn is a measure on A. 5.6 Refer to xample 5.1(j). Let (, A) be a measurable space and suppose that {x} A for each x. Show that a measure μ on A is discrete if and only if there is a countable subset K of such that μ = μ({x})δx. x K 5.7 Suppose that a balanced coin is tossed three times. a) Construct a probability space for this experiment in which each possible outcome is equally likely.

7 150 Chapter 5 lements of Measure Theory b) Determine the probability of obtaining exactly two heads. c) xpress the probability measure P as a finite linear combination of Dirac measures. 5.8 Let be a nonempty set, {x n} n a sequence of distinct elements of, and {a n} n a sequence of nonnegative real numbers. For, define μ() = a n. x n a) Show that μ is a measure on P(). b) Interpret the a ns in terms of the measure μ. c) xpress μ as a linear combination of Dirac measures. 5.9 Suppose that two balanced dice are thrown. a) Construct a probability space for this experiment in which each possible outcome is equally likely. b) Use part (a) to determine the probability that the sum of the dice is seven or 11. c) Construct a probability space for this experiment in which the outcomes consist of the possible sums of the two dice. d) Use part (c) to determine the probability that the sum of the dice is seven or Prove Theorem Let (, A) be a measurable space. A measure μ on A is called a finite measure if μ() <. A measure space (, A,μ) is called a finite measure space if μ is a finite measure. For a finite measure space, prove the following: a) If A and B are A-measurable sets, then μ(a B) =μ(a)+μ(b) μ(a B). b) Generalize part (a) to an arbitrary finite number of A-measurable sets Let { n} be a sequence of A-measurable sets. Prove that ( ( ) ) μ k lim inf μ(n). k=n 5.13 Let { n} be a sequence of A-measurable sets with μ ( ) n <. Prove that ( ( ) ) μ k lim sup μ( n). k=n 5.14 Let (, A,μ) be a measure space and { n} a sequence of A-measurable sets. Define = { x : x n for infinitely many n }. a) Prove that = ( k=n k). b) Prove that μ(n) < μ() = Prove Theorem Prove that (R, M,λ) is the completion of (R, B,λ B ). Use the following steps: a) Verify that B Mby employing xercise 3.32 on page 107. b) Show that B Mby applying xercise 3.44 on page 108. c) Prove that λ = λ B. Hint: Use the fact established in parts (a) and (b) that M = B Let (, A,μ) be a measure space. Suppose that (, F,ν) is a complete measure space with F Aand ν A = μ. Prove that F Aand that ν A = μ. Conclude that (, A, μ) is the smallest complete measure space that contains (, A,μ).

8 5.2 Measurable Functions Let f be a nonnegative M-measurable function. Define μ f on M by μ f () = fdλ. Prove that μ f is a measure on M Let (, A,μ) be a measure space such that {x} Afor each x. An element x is said to be an atom of μ if μ({x}) > 0. Assume now that μ is a finite measure, that is, μ() <. Prove the following facts. a) μ has only countably many atoms. b) μ can be expressed uniquely as the sum of two measures, μ c and μ d, where μ c has no atoms and μ d is discrete. Moreover, we have that μ d = μ({x})δx, where x K K is the set of atoms of μ. 5.2 MASURABL FUNCTIONS The next step in developing the abstract Lebesgue integral is to introduce the concept of measurability for functions defined on an abstract space. In addition to real-valued functions, we will also consider complex-valued and extended realvalued functions. We begin with real-valued functions. Real-Valued Measurable Functions Let (, A) be a measurable space and f: R. We want to specify when f is measurable. In previous chapters, we discussed two kinds of measurable functions: Borel measurable functions and Lebesgue measurable functions. Recall that a real-valued function f is Borel measurable if and only if f 1 (O) Bfor each open set O Rand it is Lebesgue measurable if and only if f 1 (O) Mfor each open set O R. Hence, it is quite natural to make the following definition. DFINITION 5.4 Real-Valued Measurable Function Let (, A) be a measurable space. A real-valued function f on is said to be an A-measurable function if the inverse image of each open subset of R under f is an A-measurable set, that is, if f 1 (O) Afor all open sets O R. XAMPL 5.2 Illustrates Definition 5.4 a) Let = R. Then, as we know from Chapters 3 and 4, the Borel measurable functions are the B-measurable functions, and the Lebesgue measurable functions are the M-measurable functions. b) Let (, A) be a measurable space, D A, and A D = { D A : A A}. Then a function f: D Ris A D -measurable if and only if for each open subset O of R, f 1 (O) is of the form D A for some A A. c) very real-valued function on a nonempty set is P()-measurable. An important special case: If = N, then A is usually taken to be P(N ); hence, all functions f: N Rare A-measurable. But functions on N are infinite sequences. Consequently, in this case, the A-measurable functions are precisely the infinite sequences.

9 152 Chapter 5 lements of Measure Theory The following proposition provides some useful equivalent conditions for a function to be A-measurable. To prove the proposition, we proceed in a similar manner as we did in the proof of Lemma 3.5 on page 85. PROPOSITION 5.1 Let (, A) be a measurable space and f a real-valued function on. Then the following statements are equivalent: a) f is A-measurable. b) For each a R, f 1( (,a) ) A. c) For each a R, f 1( (a, ) ) A. d) For each a R, f 1( (,a] ) A. e) For each a R, f 1( [a, ) ) A. Theorem 5.3, which we prove next, gives several important properties of realvalued A-measurable functions. Theorem 4.1 on page 113 is a special case. PROOF THORM 5.3 Let (, A) be a measurable space. The collection of all real-valued A-measurable functions forms an algebra. In other words, if f and g are A-measurable and α R, then a) f + g is A-measurable. b) αf is A-measurable. c) f g is A-measurable. a) By Proposition 5.1, to prove that f + g is A-measurable, it suffices to show that { x : f(x)+g(x) >a} Afor each a R.Now, { x : f(x)+g(x) >a} = { x : f(x) >a g(x) } = { x : f(x) >r>a g(x) } r Q = ( f 1( (r, ) ) g 1( (a r, ) )). r Q This last union is an A-measurable set since f and g are A-measurable functions, A is a σ-algebra, and Q is countable. Consequently, f + g is an A-measurable function. b) If α = 0, then αf 0, which is A-measurable (why?). So, assume α 0 and let O be any open set in R. Then α 1 O = { α 1 y : y O } is open. Therefore, because f is A-measurable, (αf) 1 (O) =f 1 (α 1 O) A. This fact proves that αf is A-measurable. c) First we show that if f is A-measurable, then so is f 2. If a<0, then (f 2 ) 1 ((a, ))= A.Ifa 0, then we have (f 2 ) 1( (a, ) ) = { x : f(x) 2 >a} = { x : f(x) > a } {x : f(x) < a } = f 1( ( a, ) ) f 1( (, a) ). This last union is an A-measurable set because f is A-measurable. Hence, f 2 is an A-measurable function whenever f is.

10 5.2 Measurable Functions 153 Now, for any two functions f and g, we can write f g = 1 4( (f + g) 2 (f g) 2). Applying parts (a) and (b) of this theorem and the fact that the square of an A-measurable function is A-measurable, we conclude that f g is an A-measurable function. We should emphasize that the measurability (or nonmeasurability) of a function depends only on the σ-algebra, A, of subsets of ; that is, it has nothing to do with a measure. Nonetheless, if (, A,P) is a probability space, then the A-measurable functions are called random variables. Thus, an A-measurable function is a random variable only when considered in the context of a probability space. By the way, in probability theory, random variables are usually denoted by uppercase italicized nglish-alphabet letters that are near the end of the alphabet (e.g., X, Y, and Z) instead of the more usual f, g, and h. XAMPL 5.3 Illustrates Random Variables Let (, A, P) be the probability space from subpart (ii) of xample 5.1(h) on page 147. Define X(HH) = 2, X(HT) = X(TH) = 1, andx(tt) = 0. Then X: Ris a random variable. It indicates the number of heads obtained when a balanced coin is tossed twice. Our next result is a generalization of Proposition 4.7 on page 140 to an arbitrary complete measure space. Its proof is essentially identical to that of Proposition 4.7. PROPOSITION 5.2 Suppose that (, A,μ) is a complete measure space. If f is A-measurable and g = fμ-ae, then g is A-measurable. Complex-Valued Measurable Functions In applying real analysis, we often encounter complex-valued functions. This occurs, for instance, in Fourier analysis. We will denote the set of all complex numbers by C. Here now is the definition of measurability for complex-valued functions. DFINITION 5.5 Complex-Valued Measurable Function Let (, A) be a measurable space. A complex-valued function f on is said to be an A-measurable function if the inverse image of each open subset of C under f is an A-measurable set, that is, if f 1 (O) Afor all open sets O C. The following theorem provides a useful characterization of measurability for complex-valued functions. We leave the proof of the theorem as an exercise for the reader.

11 154 Chapter 5 lements of Measure Theory THORM 5.4 A complex-valued function f on is A-measurable if and only if both its real part Rf and its imaginary part If are (real-valued) A-measurable functions. XAMPL 5.4 Illustrates Complex-Valued Measurable Functions a) If f is a real-valued A-measurable function on, then it is also a complexvalued A-measurable function. b) Let = R and A = B. Define f: R Cbyf(x) =e ix. The real and imaginary parts of f(x) are cos x and sin x, respectively. Since those two functions are continuous, they are B-measurable. Consequently, by Theorem 5.4, f is a complex-valued B-measurable function. c) If g and h are real-valued A-measurable functions, then, by Theorem 5.4, the complex-valued function f = g + ih is also A-measurable. d) Let {a n } be a sequence of complex numbers and define f: N Cby f(n) =a n. Then f is a complex-valued P(N )-measurable function. Theorem 5.3 holds also for complex-valued A-measurable functions. That is, the collection of complex-valued A-measurable functions forms a (complex) algebra. See xercise xtended Real-Valued Measurable Functions In addition to real- and complex-valued functions, we frequently must deal with extended real-valued functions, in other words, functions that take values in R = R {, }. This is especially so when considering suprema, infima, and limits. For instance, define f n (x) = n 2π e (nx)2 /2 for x R and n N. Then, as n, f n (x) 0ifx 0 and f n (0). Consequently, the sequence {f n } of real-valued functions converges pointwise to the extended real-valued function f, where { 0, if x 0; f(x) =, if x =0. Thus, we next consider measurability for extended real-valued functions. Recall that, by definition, a real-valued function f is A-measurable if f 1 (O) A for all open sets O R. Also, by definition, a complex-valued function f is A-measurable if f 1 (O) A for all open sets O C. Hence, once we identify the open sets of R, we have a natural way to define extended real-valued A-measurable functions. DFINITION 5.6 Open Subsets of the xtended Real Numbers A subset of R is said to be open if it can be expressed as a union of intervals of the form (a, b), [,b), and (a, ], where a, b R.

12 5.2 Measurable Functions 155 DFINITION 5.7 xtended Real-Valued Measurable Function Let (, A) be a measurable space. An extended real-valued function f on is said to be an A-measurable function if the inverse image of each open subset of R under f is an A-measurable set, that is, if f 1 (O) Afor all open sets O R. The next proposition provides the analogue of Proposition 5.1 for extended real-valued functions. Its proof is left as an exercise. PROPOSITION 5.3 Let (, A) be a measurable space and f an extended real-valued function on. Then the following statements are equivalent: a) f is A-measurable. b) For each a R, f 1( [,a) ) A. c) For each a R, f 1( (a, ] ) A. d) For each a R, f 1( [,a] ) A. e) For each a R, f 1( [a, ] ) A. Theorem 5.3 shows that the collection of real-valued A-measurable functions forms an algebra. In the case of extended real-valued functions, if we adopt the convention that is some fixed extended real number, then the collection of extended real-valued A-measurable functions is closed under addition, scalar multiplication, and multiplication. See xercises 5.39 and The following theorem establishes that the collection of extended real-valued A-measurable functions is closed under maxima, minima, suprema, infima, and pointwise limits. Note that Theorem 4.2 on page 113 is an immediate consequence. THORM 5.5 Suppose that f and g are extended real-valued A-measurable functions and that {f n } is a sequence of extended real-valued A-measurable functions. Then a) f g and f g are A-measurable. b) sup n f n and inf n f n are A-measurable. c) lim sup f n and lim inf f n are A-measurable. d) If {f n } converges pointwise, then lim f n is A-measurable. PROOF a) Let h = f g and a R. Then h 1( (a, ] ) = f 1( (a, ] ) g 1( (a, ] ). This union is in A because f and g are A-measurable functions. Thus, f g is A-measurable. Similarly, f g is A-measurable. b) Let h = sup n f n and a R. Then h 1( (a, ] ) = f ( ) n 1 (a, ]. This union is in A because each f n is an A-measurable function. Hence, we see that sup n f n is A-measurable. Similarly, inf n f n is A-measurable.

13 156 Chapter 5 lements of Measure Theory c) Noting that lim sup f n = inf n sup k n f k, it follows from part (b) that lim sup f n is A-measurable. Using an entirely similar argument, we find that lim inf f n is A-measurable. d) If {f n } converges pointwise, then we have lim f n = lim sup f n. So, lim f n is A-measurable by part (c). A common application of Theorem 5.5 occurs when {f n } is a sequence of real-valued A-measurable functions, but where at least one of the functions inf n f n, sup n f n, lim inf f n, lim sup f n, and lim f n is an extended real-valued A-measurable function. XAMPL 5.5 Illustrates Theorem 5.5 a) Let (, A,μ)=(R, M,λ). For x Rand n N, define f n (x) = n 2π e (nx)2 /2. Then {f n } is a sequence of real-valued M-measurable functions. Moreover, f n f pointwise, where { 0, if x 0; f(x) =, if x =0. By Theorem 5.5(d), f is an extended real-valued A-measurable function, a fact that we can easily verify directly. b) Let f be an extended real-valued A-measurable function. By Theorem 5.5(a), f is A-measurable since f = f f. Theorem 5.5(d) shows that if a sequence {f n } of A-measurable functions converges pointwise to a function f, then f is an A-measurable function. What if the convergence is only almost everywhere? In general, we cannot conclude that f is A-measurable; however, for complete measure spaces we can. PROOF PROPOSITION 5.4 Let (, A,μ) be a complete measure space. Suppose that {f n } is a sequence of complex-valued or extended real-valued A-measurable functions and that f n fμ-ae. Then f is an A-measurable function. The proof is essentially identical to that of Proposition 4.8 on page 140 and is left to the reader. xercises for Section Prove Proposition 5.1 on page Let (, A) be a measurable space and f a real-valued function on. Prove that f is A-measurable if and only if f 1 (B) Afor each B B Suppose that (, A) is a measurable space and that f: R is an A-measurable function. Further suppose that g: R Ris a Borel measurable function. Prove that g f is A-measurable.

14 5.2 Measurable Functions Let D B. Show that Ĉ(D), the collection of Borel measurable functions on D, is precisely the collection of B D-measurable functions Let (, A) be a measurable space, D A, and A D = { D A : A A}. a) If f: Ris A-measurable, show that f D is A D-measurable. b) Suppose that g: D Ris A D-measurable. Define f: Rby { g(x), x D; f(x) = 0, x / D. Prove that f is A-measurable. (This result shows that every A D-measurable function can be extended to an A-measurable function.) 5.25 Prove Proposition 5.2 on page Provide an example to show that the hypothesis of completeness cannot be omitted from Proposition If O is an open subset of R and α is a nonzero real number, show that α 1 O is an open subset of R Prove Theorem 5.4 on page 154. Hint: Use the fact that each open set in Cisa countable union of open rectangles. [An open rectangle in C is a set of the form { u + iv C:a<u<b, c<v<d}.] 5.29 Show that every real-valued A-measurable function is a complex-valued A-measurable function The collection B 2 of Borel sets of C is defined to be the smallest σ-algebra of subsets of C that contains all the open subsets of C. Show that f: CisA-measurable if and only if f 1 (B) Afor all B B Let (, A,P) be a probability space, X a random variable on, and t a fixed real number. Define g: Cbyg = e itx ; that is, for each x, g(x) =e itx(x). Prove that g is A-measurable. Is g a random variable? xplain your answer Prove that the collection of complex-valued A-measurable functions forms a complex algebra. That is, if f and g are complex-valued A-measurable functions and α C, show that f + g, αf, and f g are complex-valued A-measurable functions Show that each open subset of R is also an open subset of R Prove Proposition 5.3 on page 155. Hint: Show that each open set in R can be written as a countable union of intervals of the form (a, b), [,b), and (a, ], where a, b R Show that f: R is A-measurable if and only if (i) f 1 ({ }) and f 1 ({ }) are in A and (ii) f 1 (B) Afor all B B Show that every real-valued A-measurable function is also an extended real-valued A-measurable function Show that a set O Ris open in R if and only if there is an open subset U of R such that O = R U Suppose that f and g are extended real-valued A-measurable functions. Prove that the following three sets are A-measurable: a) { x : f(x) >g(x) }. b) { x : f(x) g(x) }. c) { x : f(x) =g(x) } Let f and g be extended real-valued A-measurable functions and let β R. Set = { x : f(x) =, g(x) = } {x : f(x) =, g(x) = }. For x, define (f + g)(x) =β; otherwise, define (f + g)(x) =f(x)+g(x), as usual. Prove that f + g is A-measurable.

15 158 Chapter 5 lements of Measure Theory 5.40 With the convention established in the preceding exercise, prove that the collection of extended real-valued A-measurable functions is closed under scalar multiplication and multiplication Suppose that {f n} is a sequence of extended real-valued A-measurable functions. Verify that { x : lim f n(x) exists } is an A-measurable set Suppose {f n} is a sequence of complex-valued A-measurable functions that converges pointwise to a complex-valued function f. Prove that f is A-measurable Construct a sequence {f n} of A-measurable functions that converges almost everywhere to a function f that is not A-measurable. Hint: Take (, A,μ)=(R, B,λ B ) and do something with a non-borel measurable subset of the Cantor set Prove Proposition 5.4 on page Suppose that {f n} is a sequence of complex-valued A-measurable functions. Define { lim f(x) = if lim fn(x) exists; 0, otherwise. Prove that f is A-measurable Suppose that {f n} is a sequence of complex-valued A-measurable functions and that f n gμ-ae. Prove that there exists an A-measurable function f such that f n fμ-ae. Note: g need not be A-measurable unless, of course, (, A,μ) is complete Suppose that is an open subset of C and that g is a real-valued continuous function on. Further suppose that f is a complex-valued A-measurable function on with the range of f being a subset of. Prove that g f is a real-valued A-measurable function on. Repeat the proof if is a closed subset of C Suppose that f: CisA-measurable. Verify that f can be written in the polar form, f = Re iθ, where R: [0, ) and Θ: Rare A-measurable functions. 5.3 TH ABSTRACT LBSGU INTGRAL FOR NONNGATIV FUNCTIONS Now that we have discussed measure spaces and measurable functions, we can proceed to develop the abstract Lebesgue integral, that is, the Lebesgue integral on an arbitrary measure space (, A,μ). As we will see, the development of the abstract Lebesgue integral is almost identical to that of the Lebesgue integral on the real line, that is, on (R, M,λ), given in Chapter 4. Consequently, many of the proofs will be left to the reader. Following the procedure used in Chapter 4, we will first define the abstract Lebesgue integral of a simple function, then of a nonnegative A-measurable function, and then of a real-valued A-measurable function. In addition, we will also define the abstract Lebesgue integral of extended real-valued and complex-valued A-measurable functions. Nonnegative functions will be considered in this section and general functions in the next. The Lebesgue Integral of a Nonnegative Simple Function Let (, A,μ) be a measure space. An A-measurable function on is called a simple function if it takes on only finitely many values. More precisely, we have the following definition.

16 5.3 The Abstract Lebesgue Integral for Nonnegative Functions 159 DFINITION 5.8 Simple Function and Canonical Representation An A-measurable function s is said to be a simple function if its range is a finite set. Let a 1, a 2,..., a n denote the distinct nonzero values of s and set A k = { x : s(x) =a k },1 k n. Then n s = a k χ Ak. k=1 This is called the canonical representation of s. We leave it as an exercise for the reader to show that the sets A 1, A 2,..., A n, appearing in the canonical representation of an A-measurable simple function, are A-measurable and pairwise disjoint. XAMPL 5.6 Illustrates Definition 5.8 a) The Lebesgue measurable simple functions introduced in Chapter 4 are M- measurable simple functions in the sense of Definition 5.8. b) If is a finite set, then every A-measurable function is simple. We now give the definition of the abstract Lebesgue integral of a nonnegative A-measurable simple function. It is a straightforward generalization of the definition presented in Chapter 4 for the Lebesgue integral of a nonnegative Lebesgue measurable simple function. DFINITION 5.9 Integral of a Nonnegative Simple Function Let (, A,μ) be a measure space and s a nonnegative A-measurable simple function on with canonical representation s = n k=1 a kχ Ak. Then the (abstract) Lebesgue integral of s over with respect to μ is defined by n s(x) dμ(x) = a k μ(a k ). k=1 If A, then the (abstract) Lebesgue integral of s over with respect to μ is defined by s(x) dμ(x) = χ (x)s(x) dμ(x). Note: The notations sdμ and s(x) μ(dx) are commonly used in place of s(x) dμ(x). The next proposition shows how we can obtain the abstract Lebesgue integral of a nonnegative simple function from a possibly noncanonical representation. The proof is identical to that of Proposition 4.2 on page 114.

17 160 Chapter 5 lements of Measure Theory PROPOSITION 5.5 Let s be a nonnegative A-measurable simple function that can be expressed in the form s = m k=1 b kχ Bk, where this representation is not necessarily canonical but B k Afor 1 k m and B i B j = for i j. Then m s(x) dμ(x) = b k μ(b k ). More generally, we have for each A. s(x) dμ(x) = k=1 m b k μ(b k ) The following fact is proved in precisely the same way as Lemma 4.1 on page 116. k=1 PROPOSITION 5.6 Suppose that s and t are nonnegative A-measurable simple functions and that α, β 0. Then αs + βt is a nonnegative A-measurable simple function and (αs + βt) dμ = α sdμ+ β tdμ for each A. The Lebesgue Integral of a Nonnegative A-measurable Function The next thing on the agenda is the definition of the abstract Lebesgue integral for a nonnegative extended real-valued A-measurable function. Proposition 5.7, whose proof is left to the reader as an exercise, provides the motivation for that definition. PROPOSITION 5.7 a) Suppose that f is a nonnegative extended real-valued A-measurable function on. Then there is a nondecreasing sequence of nonnegative A-measurable simple functions that converges pointwise to f. In other words, there is a sequence {s n } of nonnegative A-measurable simple functions such that, for all x, s 1 (x) s 2 (x) and lim s n (x) =f(x). b) If {s n } is a sequence of nonnegative A-measurable simple functions that converges pointwise on to a function f, then f is a nonnegative extended real-valued A-measurable function. Proposition 5.7 shows that the functions that can be approximated by nonnegative A-measurable simple functions are precisely the nonnegative extended real-valued A-measurable functions. Thus, we make the following definition.

18 5.3 The Abstract Lebesgue Integral for Nonnegative Functions 161 DFINITION 5.10 Lebesgue Integral of a Nonnegative Function Let f be a nonnegative extended real-valued A-measurable function on. Then the (abstract) Lebesgue integral of f over with respect to μ is defined by f(x) dμ(x) = sup s(x) dμ(x), s where the supremum is taken over all nonnegative A-measurable simple functions that are dominated by f. If A, then the (abstract) Lebesgue integral of f over with respect to μ is defined by f(x) dμ(x) = χ (x)f(x) dμ(x). Note: The abstract Lebesgue integral of a nonnegative M-measurable function with respect to λ is identical to its Lebesgue integral, as defined in Chapter 4. Some of the more important properties of the abstract Lebesgue integral for nonnegative extended real-valued A-measurable functions are provided in Proposition 5.8. The proof is left as an exercise for the reader. PROPOSITION 5.8 Let f and g be nonnegative extended real-valued A-measurable functions on, α 0, and A. Then a) f gμ-ae fdμ gdμ. b) B and B A B fdμ fdμ. c) f(x) =0for all x fdμ=0. d) μ() =0 fdμ=0. e) αf dμ = α fdμ. Convergence Properties of the Abstract Lebesgue Integral for Nonnegative A-measurable Functions We now present two major convergence theorems for the abstract Lebesgue integral of nonnegative extended real-valued A-measurable functions the monotone convergence theorem (MCT) and Fatou s lemma. The proofs are similar to those given in Section 4.2 (page 121 onward). The MCT is stated first. Observe that it applies to extended real-valued A-measurable functions as well as to real-valued A-measurable functions. THORM 5.6 Monotone Convergence Theorem (MCT) Suppose that {f n } is a monotone nondecreasing sequence of nonnegative extended real-valued A-measurable functions. Then lim f n dμ = lim f n dμ for each A.

19 162 Chapter 5 lements of Measure Theory COROLLARY 5.1 Let f, g, f 1, f 2,... be nonnegative extended real-valued A-measurable functions and let A. Then a) (f + g) dμ = fdμ+ g dμ. b) f n dμ = f n dμ. c) If { n } n Aare pairwise disjoint, then n fdμ= n n n fdμ. Proposition 5.8(e) and Corollary 5.1(a) together imply that if f and g are nonnegative extended real-valued A-measurable functions and α, β 0, then (αf + βg) dμ = α fdμ+ β g dμ. (5.2) quation (5.2), Proposition 5.7, and the MCT are frequently used together for bootstrapping arguments. That is, suppose we want to prove that a certain Lebesgue-integral property holds for all nonnegative A-measurable functions. To bootstrap, we employ three steps: First we show that the property holds for characteristic functions of A-measurable sets; next we apply (5.2) to conclude that the property holds for nonnegative simple functions; and then we use Proposition 5.7(a) and the MCT to deduce that the property holds for all nonnegative A-measurable functions. xercises 5.60 and 5.61 provide illustrations of bootstrapping. Next we state Fatou s lemma. This version of Fatou s lemma not only generalizes to arbitrary measure spaces the version presented in Theorem 4.6 on page 126 but its hypotheses are less restrictive. Specifically, it does not impose any convergence conditions on {f n }. THORM 5.7 Fatou s Lemma Let {f n } be a sequence of nonnegative extended real-valued A-measurable functions. Then lim inf f n dμ lim inf f n dμ for each A. XAMPL 5.7 Illustrates the Abstract Lebesgue Integral a) Let (, A,μ) be a measure space and f a nonnegative extended real-valued A-measurable function on. Suppose that x 0 and that {x 0 } A. We claim that fdμ= f(x 0 )μ({x 0 }). (5.3) {x 0} To see this, note that χ {x0}f is the simple function f(x 0 )χ {x0} and, hence, by Definition 5.9 on page 159, fdμ= χ {x0}f dμ= f(x 0 )χ {x0} dμ = f(x 0 )μ({x 0 }). {x 0}

20 5.3 The Abstract Lebesgue Integral for Nonnegative Functions 163 More generally, let C = {x n } n be a countable subset of with {x n } A for each n. Then, by Corollary 5.1(c) and (5.3), fdμ= fdμ= fdμ= f(x n )μ({x n }). (5.4) C n {xn} n {x n} n b) Let μ be counting measure on P(N ). Then, as we learned in xample 5.2(c), a nonnegative real-valued P(N )-measurable function f on N is a nonnegative infinite sequence {a n }, where we have let a n = f(n). Thus, by (5.4), fdμ= f(n)μ({n}) = a n. N Hence, we can apply abstract measure theory to study infinite series. c) Let (, A,P) be a probability space and X a nonnegative random variable. Then the abstract Lebesgue integral of X over with respect to P is called the mean (expectation, expected value) of X. The mean of X is denoted by (X). Thus, (X) = XdP. For instance, consider the random experiment of tossing a balanced coin twice. An appropriate probability space for that experiment is (, A, P), where = {HH, HT, TH, TT}, A = P() and, for A, P () =N()/4. Let X denote the number of heads obtained. Then, by (5.4), the mean of X equals (X) = XdP = X(HH)P ({HH})+X(HT)P ({HT}) + X(TH)P ({TH})+X(TT)P ({TT}) = =1, which is intuitively what it should be. d) Let be a set, {x n } n a sequence of distinct elements of, and {b n } n a sequence of nonnegative real numbers. For, define μ() = b n. x n Then μ is a measure on P(). Let f be a nonnegative function on and set C = {x n } n. Then, by Corollary 5.1(c), Proposition 5.8(d) on page 161, and (5.4), fdμ= fdμ+ fdμ= fdμ+0 C C c C = f(x n )μ({x n })= (5.5) f(x n )b n. n n We will employ (5.5) frequently.

21 164 Chapter 5 lements of Measure Theory xercises for Section 5.3 e) Let be a set, A = P(), and μ counting measure on A. Iff is a nonnegative function on, then fdμ= f(x), x where x f(x) = sup { x F f(x) :F finite, F }. The verification of this fact is left to the reader stablish that the sets appearing in the canonical representation of an A-measurable simple function are A-measurable and pairwise disjoint Prove Proposition 5.7 on page 160. Hint: Refer to Proposition 4.3 on page Suppose that f is a nonnegative extended real-valued A-measurable function on, c>0, and A c = { x : f(x) c }. Prove that μ(a c) 1 c f dμ Let f be a nonnegative extended real-valued A-measurable function on and A. Prove that fdμ= 0 if and only if f =0μ-ae on Suppose that f is a nonnegative extended real-valued A-measurable function on and that fdμ<. Show that f is finite μ-ae Prove Proposition 5.8 on page 161. Hint: Refer to Proposition 4.4 on page Prove the MCT, Theorem 5.6 on page Show that for a fixed A, the conclusion of the MCT remains valid if the hypotheses are satisfied only on Prove Corollary 5.1 on page Suppose that f is a nonnegative extended real-valued A-measurable function on. Also, suppose that { n} Awith 1 2. Prove that fdμ= lim f dμ. n n 5.59 Prove Fatou s lemma, Theorem 5.7 on page Suppose that (, A,μ) is a measure space, D A, and f is a nonnegative extended realvalued A-measurable function on. Let (D, A D,μ D) be as defined in xample 5.1(c) on page 146. Show that fdμ= D f D dμ D. D Hint: Use bootstrapping Let (, A,μ) be a measure space and g a nonnegative A-measurable function on. For A, define ν() = g dμ. a) Show that ν is a measure on A.

22 5.4 The General Abstract Lebesgue Integral 165 b) Show that fdν= fgdμ for each nonnegative A-measurable function f. Hint: Bootstrap Let {a mn} m, be a double sequence of nonnegative numbers. Prove that a mn = a mn. m=1 m=1 Hint: Refer to xample 5.7(b) on page Let f: [0, 1] be an A-measurable function. a) Prove that lim f n 1 dμ = μ (f 1( (0, 1] )). b) If μ() <, prove that lim f n dμ = μ ( f 1 ({1}) ). 5.4 TH GNRAL ABSTRACT LBSGU INTGRAL In the previous section, we discussed the abstract Lebesgue integral for nonnegative extended real-valued A-measurable functions. We will now expand the definition of the abstract Lebesgue integral so that it applies to A-measurable functions that are not necessarily nonnegative. We begin with extended realvalued functions. Lebesgue Integral of an xtended Real-Valued Function Let (, A,μ) be a measure space. To define the abstract Lebesgue integral of an extended real-valued A-measurable function f on, we follow the procedure used in Section 4.3 for defining the Lebesgue integral of a real-valued Lebesgue measurable function on R. DFINITION 5.11 Integral of an xtended Real-Valued Function Let f be an extended real-valued A-measurable function on and A. Then the (abstract) Lebesgue integral of f over with respect to μ is defined by fdμ= f + dμ f dμ (5.6) provided that the right-hand side makes sense; that is, at least one of the integrals on the right-hand side of (5.6) is finite. Here f + = f 0 and f = (f 0) denote the positive and negative parts of f, respectively. In addition, we say that f is Lebesgue integrable over if both integrals on the right-hand side of (5.6) are finite or, equivalently, if f dμ = f + dμ + f dμ <. (5.7) If f is Lebesgue integrable over, then we say that f is Lebesgue integrable. We should mention that if f is Lebesgue integrable (over ), then it is Lebesgue integrable over every A. Here are some examples.

23 166 Chapter 5 lements of Measure Theory XAMPL 5.8 Illustrates Definition 5.11 a) Let (, A,μ)=(R, M,λ) and f(x) =x. Then f + (x) = { x, x 0; 0, x < 0. and f (x) = { 0, x 0; x, x < 0. (i) If = R, then R f + dλ = R f dλ =. Hence, the integral R fdλ is not defined. (ii) If =[ 1, 2], then f + dλ = 2 and f dλ =1/2 and, consequently, fdλ=2 1/2 =3/2. And, because f dλ =2+1/2 =5/2 <, we see that f is Lebesgue integrable over [ 1, 2]. (iii) If =(, 1), then f + dλ =1/2 and f dλ = and, consequently, fdλ=1/2 =. But, as f dλ =1/2+ =, we see that f is not Lebesgue integrable over (, 1). b) Let (, A,μ)=(N, P(N ),μ), where μ is counting measure on P(N ). Then real-valued A-measurable functions are simply infinite sequences of real numbers. Referring to xample 5.7(b) on page 163, we see that a sequence of real numbers {a n } is Lebesgue integrable (over N ) if and only if a n <, (5.8) that is, if and only if the series is absolutely convergent. For instance, the sequence {( 1) n /n p } is Lebesgue integrable if and only if p>1. Note that, although ( 1)n /n converges, {( 1) n /n} is not Lebesgue integrable as the series is not absolutely convergent. Lebesgue Integral of a Complex-Valued Function Next we define the abstract Lebesgue integral for complex-valued A-measurable functions. First we present some preliminaries. DFINITION 5.12 Modulus of a Complex-Valued Function Let f be a complex-valued function on. Then the modulus of f, denoted by f, is defined to be the real-valued function f = (Rf) 2 +(If) 2. In other words, f (x) = f(x), where f(x) denotes the modulus of the complex number f(x). The following two propositions will be required. We leave the proofs as exercises for the reader.

24 5.4 The General Abstract Lebesgue Integral 167 PROPOSITION 5.9 Let f be a complex-valued function on. Then a) f Rf + If. b) Rf f and If f. c) f is A-measurable if f is. PROPOSITION 5.10 Let f be a complex-valued A-measurable function on and A. Then f is Lebesgue integrable over if and only if both Rf and If are. In view of Proposition 5.10 and the fact that f = Rf + iif, it is reasonable to make the following definition. DFINITION 5.13 Integral of a Complex-Valued Function Let f be a complex-valued A-measurable function on and A. We say that f is Lebesgue integrable over with respect to μ if f is Lebesgue integrable over with respect to μ; that is, f dμ <. In that case, the (abstract) Lebesgue integral of f over with respect to μ is defined by fdμ= (Rf) dμ + i (If) dμ. If f is Lebesgue integrable over, then we say that f is Lebesgue integrable. For a measure space (, A, μ), the collection of all complex-valued Lebesgue integrable functions is denoted by L 1 (, A,μ). When no confusion will arise, we write L 1 (μ) for L 1 (, A,μ). XAMPL 5.9 Illustrates Definition 5.13 a) Let (, A,μ)=(R, M,λ) and consider the complex-valued function f defined by f(x) =e ix /(1 + x 2 ). Then it is easy to see that f is measurable. Moreover, Rf(x) = cos x/(1 + x 2 ), If(x) = sin x/(1 + x 2 ), and f(x) =1/(1 + x 2 ). By xercise 4.20 on page 127 and Theorem 4.9 on page 135, f(x) dλ(x) = (1 + x 2 ) 1 dλ(x) R R n = lim (1 + x 2 ) 1 dx dλ(x) = lim [ n,n] n (1 + x 2 ) ( ) = lim arctan(n) arctan( n) = π<. Therefore, f L 1 (λ).

25 168 Chapter 5 lements of Measure Theory b) Let (, A,μ)=(N, P(N ),μ), where μ is counting measure on P(N ). Then complex-valued A-measurable functions are simply infinite sequences of complex numbers. Referring to xample 5.7(b) on page 163, we see that a sequence of complex numbers {a n } is in L1 (μ) if and only if the series a n converges absolutely. We point out here that the notations l 1 or l 1 (N ) are generally used in place of L 1 (N, P(N ),μ). c) Let (, A) be a measurable space. A measure μ on A is said to be a finite measure if μ() <. If μ is a finite measure, then (, A,μ) is called a finite measure space. For a finite measure space, each bounded complexvalued A-measurable function f is in L 1 (μ). Indeed, if f M, then by Proposition 5.8(a) on page 161, f dμ Mdμ= Mμ() <. Note that boundedness is a sufficient but not necessary condition for integrability. For instance, let (, A,μ)= ( (0, 1), M (0,1),λ (0,1) ) and f(x) =x 1/2. Then f is not bounded on (0, 1) but is in L 1( λ (0,1) ). d) If (, A, P) is a probability space, then the integrable functions, that is, members of L 1 (P ), are called random variables with finite mean or finite expectation. The following theorem, whose proof is left as an exercise, provides some important properties of Lebesgue integrable functions. THORM 5.8 Suppose that f and g are in L 1 (, A,μ) and that α C. Then a) f + g L 1 (μ) and (f + g) dμ = fdμ+ g dμ. b) αf L 1 (μ) and αf dμ = α f dμ. c) If f and g are real-valued and f g on, then fdμ gdμ. d) fdμ f dμ. e) μ() =0 fdμ=0. f) If A and B are disjoint A-measurable sets, then fdμ= fdμ+ f dμ. A B Remark: Parts (a) and (b) of Theorem 5.8 together imply that if α, β C and f, g L 1 (μ), then (αf + βg) dμ = α fdμ+ β g dμ. This result is called the linearity property of the abstract Lebesgue integral. A B