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

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1 LBSGU MASUR AND L2 SPAC. ANNI WANG Abstract. This paper begins with an introduction to measure spaces and the Lebesgue theory of measure and integration. Several important theorems regarding the Lebesgue integral are then developed. Finally, we prove the completeness of the L 2 (µ) space and show that it is a metric space, and a Hilbert space. Contents 1. Measure Spaces 1 2. Lebesgue Integration 2 3. L 2 Space 4 Acknowledgments 9 References 9 1. Measure Spaces Definition 1.1. Suppose is a set. Then is said to be a measure space if there exists a σ-ring M (that is, M is a nonempty family of subsets of closed under countable unions and under complements)of subsets of and a non-negative countably additive set function µ (called a measure) defined on M. If M, then is said to be a measurable space. For example, let = R p, M the collection of Lebesgue-measurable subsets of R p, and µ the Lebesgue measure. Another measure space can be found by taking to be the set of all positive integers, M the collection of all subsets of, and µ() the number of elements of. We will be interested only in a special case of the measure, the Lebesgue measure. The Lebesgue measure allows us to extend the notions of length and volume to more complicated sets. Definition 1.2. Let R p be a p-dimensional uclidean space. We denote an interval of R p by the set of points x = (x 1,..., x p ) such that (1.3) a i x i b i (i = 1,..., p) Definition 1.4. Let I be an interval in R p and define p (1.5) m(i) = (b i a i ) i=1 Date:

2 2 ANNI WANG If A is a union of finite intervals A = I 1... I n and these intervals are pairwise disjoint, we set (1.6) m(a) = m(i 1 ) m(i n ) Definition 1.7. Let f be a function defined on a measurable space with values in the extended real number system. The function f is said to be measurable if the set (1.8) {x f(x) > a} is measurable for all real a. Measurability is also preserved with respect to addition, multiplication, and limit processes of measurable functions. 2. Lebesgue Integration Definition 2.1. A real-valued function s defined on is called a simple function if the range of s is finite. Let and define { 1 x (2.2) K (x) = 0 x / K is called the characteristic function or indicator function of. Any simple function can be written as a finite linear combination of characteristic functions. Suppose s is a simple function which takes on values c 1,..., c n and let (2.3) i = {x s(x) = c i } i = 1,..., n Then (2.4) s = n c i K i i=1 Theorem 2.5. Let f be a real function on. For every x, there exists a sequence {s n } such that s n (x) f(x) as n. If f is measurable {s n } may be chosen as a sequence of measurable functions. If f is nonnegative, {s n } may be chosen to be a monotonically increasing sequence. This theorem shows that any measurable function can be approximated by simple functions, and therefore also by linear combinations of simple functions. We will use this to define the Lebesgue integral. Definition 2.6. Let (2.7) s(x) = n c i K i (x) (x, c i > 0) i=1 as in (2.1). Suppose s is measurable, and suppose M. Define n (2.8) I (s) = c i µ( i ) i=1

3 LBSGU MASUR AND L2 SPAC. 3 If f is measurable and nonnegative, we define (2.9) f dµ = sup I (s) where the sup is taken over all measurable simple functions s such that 0 s f The left side of (2.9) is called the Lebesgue integral of f (with respect to the measure µ over the set ). To extend the integral to functions that are not nonnegative is an easy addition. Definition Let f be defined on a measure space with values in the extended real numbers. We may write (2.11) f = f + f where (2.12) f + (x) = { f(x) if f(x) 0 0 otherwise (2.13) f (x) = { f(x) if f(x) 0 0 otherwise Then f + and f are nonnegative measurable functions. We define (2.14) f = f + + f Integration can also be extended to complex functions. Definition Let f be a complex-valued function defined on a measure space and f = u + iv, with u and v real. We say that f is measurable if and only if u and v are measurable. Suppose µ is a measure on, is a measurable subset of, and f is a complex function on. We say that f L(µ) on (and f is complex square-integrable) if f is measurable and (2.16) f dµ < + We define the integral of f (with respect to µ and over ) as (2.17) f dµ = u dµ + i v dµ Intuitively, the Lebesgue integral measures the area under a function by making partitions of the range of the function, whereas the Riemann integral partitions the domain. The Lebesgue integral has many desirable properties compared to the Riemann integral. The one we will be most concerned with is the fact that any Lebesgue measurable function is Lebesgue integrable (while a function is Riemann integrable on an interval if and only if it is continuous almost everywhere). This allows us to integrate additional functions on much more diverse sets.

4 4 ANNI WANG 3. L 2 Space The L 2 space is a special case of an L p space, which is also known as the Lebesgue space. Definition 3.1. Let be a measure space. Given a complex function f, we say f L 2 on if f is (Lebesgue) measurable and if (3.2) f 2 dµ < + Then the function f is also said to be square-integrable. In other words, L 2 is the set of square-integrable functions. For f L 2 (µ) define ( ) 1/2 (3.3) f = f 2 dµ We call f the L 2 (µ) norm of f. To give a notion of distance in L 2 (µ), we define the distance between between two functions f and g in L 2 (µ) as (3.4) d(f, g) = f g We define the L p space and the L p norm similarly (merely switching the 2 above with p ). Definition 3.5. Let be a measure space. The measurable function f is said to be in L p if it is p-integrable; that is, if (3.6) f p dµ < + The L p norm of f is defined by ( (3.7) f p = ) 1/p f p dµ We also identify functions which differ only on a set of measure zero. This allows L 2 (µ) to satisfy the properties of a metric space, namely f f = 0 f g > 0 if f g f g = g f f g f h + g h for any h L 2 (µ) Theorem 3.8. (Schwarz inequality) Suppose f L 2 (µ) and g L 2 (µ). fg L 2 (µ), and (3.9) fg dµ f g Proof. We have for all real λ, (3.10) 0 ( f + λ g ) 2 dµ = f 2 + 2λ For g 0, let λ = f g to obtain the desired inequality. fg dµ + λ 2 g 2 Then

5 LBSGU MASUR AND L2 SPAC. 5 Theorem (Lebesgue s monotone convergence theorem) Suppose M. Let {f n } be a sequence of measurable functions such that (3.12) 0 f 1 (x) f 2 (x)... (x ) Let f be such that (3.13) f n (x) f(x) (x ) as n. Then (3.14) Proof. We have (3.15) f n dµ as n. Since f n f, we have (3.16) α f n dµ α f dµ f dµ Choose c such that 0 < c < 1 and s a simple measurable function such that 0 s f. Let (3.17) n = {x f n (x) cs(x)} (n = 1, 2, 3,...) Then By (), (3.18) = For every n, (3.19) Letting n, we obtain (3.20) α c Letting c 1, we see that (3.21) α which by (3.13) implies (3.22) α c n=1 n f n dµ f n dµ c n s dµ n s dµ s dµ f dµ The theorem then follows from (3.15), (3.16), and (3.22). Theorem Suppose M. If {f n } is a sequence of nonnegative measurable functions and (3.24) f(x) = f n (x) x n=1

6 6 ANNI WANG then (3.25) f dµ = f n dµ Proof. Apply Lebesgue s monotone convergence theorem to the partial sums of (3.15). Theorem (Fatou s theorem) Suppose M. If f n is a sequence of nonnegative measurable functions and n=1 (3.27) f(x) = lim inf f n(x) (x ) then (3.28) Proof. Let g n be such that fdµ lim inf f n dµ (3.29) g n (x) = inff i (x) (i n, n = 1, 2, 3,...) Then g n is measurable on and (3.30) 0 g 1 (x) g 2 (x)... (3.31) g n (x) f n (x) (3.32) g n (x) f(x) (n ) By Lebesgue s monotone convergence theorem, it follows that (3.33) g n dµ f dµ which, with (3.31), implies the desired inequality. Theorem If f, g L 2 (µ), then f + g L 2 (µ), and (3.35) f + g f + g Theorem (Lebesgue s dominated convergence theorem) Suppose M. Let {f n } be a sequence of measurable functions such that (3.37) f n (x) f(x) (x ) as n. If there exists a function g in Lµ on such that (3.38) f n (x) g(x) (x, n = 1, 2, 3,...) then (3.39) lim f n dµ = f dµ. Proof. We have that f n L(µ) and f L(µ) on. Since f n + g 0, Fatou s theorem implies that (3.40) (f + g) dµ lim inf (f n + g)dµ or (3.41) f dµ lim inf f n dµ

7 LBSGU MASUR AND L2 SPAC. 7 Since g f n 0, we also have that (3.42) (g f) dµ lim inf Therefore (3.43) which is the same as saying (3.44) (g f n )dµ [ ] f dµ lim inf f n dµ f dµ lim sup f n dµ The conclusion of the theorem then follows from (3.40) and (3.44). Theorem (The set of continuous functions is dense in L 2 on [a, b].) For any f L 2 on [a, b] and any ɛ > 0, there exists continuous function g on [a, b] such that { 1/2 b (3.46) f g = (f g) dx} 2 < ɛ a Proof. Let A be a closed subset of [a, b] and K A be its characteristic function. Let (3.47) t(x) = inf x y (y A)and (3.48) g n (x) = 1/1 + nt(x) (n = 1, 2, 3,...) Then g n is continuous on [a, b], g n (x) = 1, and g n (x) 0 on B, defined by B = [a, b] A. By Lebesgue s dominated convergence theorem, it follows that ( 1/2 (3.49) g n K A = g n (x) dx) 2 0 Therefore characteristic functions of closed sets can be approximated by continuous functions in L 2, which implies the same for simple measurable functions. If f is nonnegative and f L 2, by Theorem 2.5 we may let {s n (x)} be a monotonically increasing sequence of simple nonnegative measurable functions such that s n (x) f(x). Since f s n 2 f 2, Lebesgue s dominated convergence theorem shows that f s n 0. The general case follows from (2.10). Definition Let f, f n L 2 (µ)(n = 1, 2, 3,...). We say that {f n } converges to f in L 2 (µ) if f n f 0. We say that {f n } is a Cauchy sequence in L 2 (µ) if for every ɛ > 0 there exists an integer N such that n N, m N implies f n f m < ɛ We will now give a proof that every Cauchy sequence in L 2 (µ) converges - that is, that L 2 (µ) is complete. Theorem (L 2 (µ) is complete.) If {f n } is a Cauchy sequence in L 2 (µ), then there exists a function f L 2 (µ) such that {f n } converges to f in L 2 (µ). Proof. By definition, there exists a sequence {n k }(k N) such that (3.52) f ( n k ) f ( n k+1 ) < 1 (k N) 2k

8 8 ANNI WANG Choose a function g L 2 (µ). By the Schwarz inequality, (3.53) g(f ( n k ) f ( n k+1 ) dµ f ( n k ) f ( n k+1 g g 2 k Adding the inequalities (3.54) for each k, we obtain (3.55) k=1 g(f ( n k ) f ( n k+1 ) dµ g 2 k g(f ( n k ) f ( n k+1 ) dµ g By Theorem 3.15 we may interchange the summation and integration above. It follows that (3.56) g(x) f ( n k )(x) f ( n k+1 (x) < + k=1 almost everywhere on (since its integral is also finite). Then (3.57) f ( n k )(x) f ( n k+1 (x) < + k=1 almost everywhere on. If the series above diverged on a set of positive measure, then we could take g(x) to be nonzero on a subset of of positive measure, contradicting (3.56). Since the kth partial sum of the telescoping series (3.58) f nk+1 (x) f nk (x) < + k=1 which converges almost everywhere on, is (3.59) f nk+1 (x) f n1 (x) we see that (3.60) f(x) = lim k f n k (x) defines f(x) for almost all x. Now let ɛ > 0 be given and choose an integer N such that n N, m N implies f n f m ɛ. By Fatou s theorem, if n k > N then (3.61) f f nk lim inf f n i f nk (x) ɛ k Thus f f nk L (µ). Since f = (f f nk ) + f nk, we see that f L (µ) also. Since ɛ was arbitrary, (3.62) lim k f f n k = 0. Finally, the inequality (3.63) f f n f f nk + f nk f n shows that f n converges to f in L (µ), since by taking n and n k large enough each of the terms on the right can be made arbitrarily small.

9 LBSGU MASUR AND L2 SPAC. 9 In addition, by defining the inner product for L 2 of two functions f and g on a measure space with (3.64) f, g = fgdµ L 2 (µ) becomes a Hilbert space. A Hilbert space is a complete vector space with an inner product, and is also a complete metric space. In particular, L 2 is an infinite-dimensional vector space. The space L 2 is unique among L p spaces as a Hilbert space. Hilbert spaces have many useful properties. In particular, its similarity to uclidean space makes possible the use of geometric notions such as distance and orthogonality. The Pythagorean identity also holds true in L 2. In addition, Hilbert spaces, and in particular the L 2 Hilbert space, are crucial to and arise naturally in areas as diverse as quantum mechanics, Fourier series, and stochastic calculus. Acknowledgments. I would like to thank my mentor, Daping Weng, for helping me understand and gain intuition for many otherwise confusing proofs and concepts; this paper would have been much more difficult to complete without him. His impressive mathematical knowledge and clarity were inspiring. I also thank the University of Chicago Mathematics department for making this RU program possible. References [1] Rudin, Walter. Principles of Mathematical Analysis. McHraw-Hilll, Inc [2] Taylor, Michael. Measure Theory and Integration. American Mathematical Society [3] Tung, Diana. Lebesgue Integration and Measure:Its Properties and Implications. cims.nyu.edu/vigrenew/ug_research/dianatung05.pdf [4] Bieren, Herman J. Introduction to Hilbert Spaces. HILBRT.PDF