LEBESGUE MEASURE AND L2 SPACE. ANNIE 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 L2(µ) space and show that it is a metric space, and a Hilbert space. Contents 1. Measure Spaces 1 2. Lebesgue Integration 2 3. L2 Space 4 Acknowledgments 9 References 9 1. Measure Spaces Definition 1.1. Suppose X is a set. Then X is said to be a measure space if there exists a σ-ring M (that is, M is a nonempty family of subsets of X closed under countable unions and under complements)of subsets of X and a non-negative countably additive set function µ (called a measure) defined on M . If X 2 M, then X is said to be a measurable space. For example, let X = Rp, M the collection of Lebesgue-measurable subsets of Rp, and µ the Lebesgue measure. Another measure space can be found by taking X to be the set of all positive integers, M the collection of all subsets of X, and µ(E) the number of elements of E. 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 Rp be a p-dimensional Euclidean space . We denote an interval p of R by the set of points x = (x1; :::; xp) such that (1.3) ai ≤ xi ≤ bi (i = 1; : : : ; p) Definition 1.4. Let I be an interval in Rp and define p Y (1.5) m(I) = (bi − ai) i=1 Date: 2011. 1 2 ANNIE WANG If A is a union of finite intervals A = I1 [ ::: [ In and these intervals are pairwise disjoint, we set (1.6) m(A) = m(I1) + ::: + m(In) Definition 1.7. Let f be a function defined on a measurable space X with values in the extended real number system. The function f is said to be measurable if the set (1.8) fxjf(x) > ag 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 X is called a simple function if the range of s is finite. Let E 2 X and define 1 x 2 E (2.2) K (x) = E 0 x2 = E KE is called the characteristic function or indicator function of E. Any simple function can be written as a finite linear combination of characteristic functions. Suppose s is a simple function which takes on values c1; : : : ; cn and let (2.3) Ei = fxjs(x) = cig i = 1; : : : ; n Then n X (2.4) s = ciKEi i=1 Theorem 2.5. Let f be a real function on X. For every x 2 X, there exists a sequence fsng such that sn(x) ! f(x) as n ! 1. If f is measurable fsng may be chosen as a sequence of measurable functions. If f is nonnegative, fsng 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 n X (2.7) s(x) = ciKEi (x)(x 2 X; ci > 0) i=1 as in (2.1). Suppose s is measurable, and suppose E 2 M. Define n X (2.8) IE(s) = ciµ(E \ Ei) i=1 LEBESGUE MEASURE AND L2 SPACE. 3 If f is measurable and nonnegative, we define Z (2.9) f dµ = sup IE(s) E 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 E). To extend the integral to functions that are not nonnegative is an easy addition. Definition 2.10. Let f be defined on a measure space X with values in the ex- tended real numbers. We may write (2.11) f = f + − f − where f(x) if f(x) > 0 (2.12) f +(x) = 0 otherwise −f(x) if f(x) < 0 (2.13) f −(x) = 0 otherwise Then f + and f − are nonnegative measurable functions. We define Z Z Z (2.14) f = f + + f − E E E Integration can also be extended to complex functions. Definition 2.15. Let f be a complex-valued function defined on a measure space X 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 X, E is a measurable subset of X, and f is a complex function on X. We say that f 2 L(µ) on E (and f is complex square-integrable) if f is measurable and Z (2.16) jfj dµ < +1 E We define the integral of f (with respect to µ and over E) as Z Z Z (2.17) f dµ = u dµ + i v dµ E E E 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 ANNIE WANG 3. L2 Space The L2 space is a special case of an Lp space, which is also known as the Lebesgue space. Definition 3.1. Let X be a measure space. Given a complex function f, we say f 2 L2 on X if f is (Lebesgue) measurable and if Z (3.2) jfj2 dµ < +1 X Then the function f is also said to be square-integrable. In other words, L2 is the set of square-integrable functions. For f 2 L2(µ) define Z 1=2 (3.3) kfk = jfj2 dµ X We call kfk the L2(µ) norm of f. To give a notion of distance in L2(µ), we define the distance between between two functions f and g in L2(µ) as (3.4) d(f; g) = kf − gk We define the Lp space and the Lp norm similarly (merely switching the "2" above with "p"). Definition 3.5. Let X be a measure space. The measurable function f is said to be in Lp if it is p-integrable; that is, if Z (3.6) jfjp dµ < +1 X The Lp norm of f is defined by Z 1=p p (3.7) kfkp = jfj dµ X We also identify functions which differ only on a set of measure zero. This allows L2(µ) to satisfy the properties of a metric space, namely • kf − fk = 0 • kf − gk > 0 if f 6= g • kf − gk = kg − fk • kf − gk ≤ kf − hk + kg − hk for any h 2 L2(µ) Theorem 3.8. (Schwarz inequality) Suppose f 2 L2(µ) and g 2 L2(µ). Then fg 2 L2(µ), and Z (3.9) jfgj dµ ≤ kfkkgk X Proof. We have for all real λ, Z Z (3.10) 0 ≤ (jfj + λjgj)2 dµ = kfk2 + 2λ jfgj dµ + λ2kgk2 X X kfk For kgk 6= 0, let λ = − to obtain the desired inequality. kgk LEBESGUE MEASURE AND L2 SPACE. 5 Theorem 3.11. (Lebesgue's monotone convergence theorem) Suppose E 2 M. Let ffng be a sequence of measurable functions such that (3.12) 0 ≤ f1(x) ≤ f2(x) ≤ ::: (x 2 E) Let f be such that (3.13) fn(x) ! f(x)(x 2 E) as n ! 1. Then Z Z (3.14) fn dµ ! f dµ E E Proof. We have Z (3.15) fn dµ ! α E R R as n ! 1. Since fn ≤ f, we have Z (3.16) α ≤ f dµ E Choose c such that 0 < c < 1 and s a simple measurable function such that 0 ≤ s ≤ f. Let (3.17) En = fxjfn(x) ≥ cs(x)g (n = 1; 2; 3; :::) Then E1 ⊂ E2 ⊂ E3 ⊂ :::. By (), 1 [ (3.18) E = En n=1 For every n, Z Z Z (3.19) fn dµ ≥ fn dµ ≥ c s dµ E En En Letting n ! 1, we obtain Z (3.20) α ≥ c s dµ E Letting c ! 1, we see that Z (3.21) α ≥ s dµ E which by (3.13) implies Z (3.22) α ≥ c f dµ E The theorem then follows from (3.15), (3.16), and (3.22). Theorem 3.23. Suppose E 2 M. If ffng is a sequence of nonnegative measurable functions and 1 X (3.24) f(x) = fn(x) 8x 2 E n=1 6 ANNIE WANG then 1 Z X (3.25) f dµ = fn dµ E n=1 Proof. Apply Lebesgue's monotone convergence theorem to the partial sums of (3.15). Theorem 3.26. (Fatou's theorem) Suppose E 2 M. If fn is a sequence of non- negative measurable functions and (3.27) f(x) = lim inf fn(x)(x 2 E) n!1 then Z Z (3.28) fdµ ≤ lim inf fndµ E n!1 E Proof.