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

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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 ∈ 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) {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 X is called a simple function if the range of s is finite. Let E ∈ X and define 1 x ∈ E (2.2) K (x) = E 0 x∈ / 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 = {x|s(x) = ci} 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 ∈ X, there exists a sequence {sn} such that sn(x) → f(x) as n → ∞. If f is measurable {sn} may be chosen as a sequence of measurable functions. If f is nonnegative, {sn} 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 ∈ X, ci > 0) i=1 as in (2.1). Suppose s is measurable, and suppose E ∈ 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 extended 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 ∈ L(µ) on E (and f is complex square-integrable) if f is measurable and Z (2.16) |f| dµ < +∞ 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 ∈ L2 on X if f is (Lebesgue) measurable and if Z (3.2) |f|2 dµ < +∞ 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 ∈ L2(µ) define Z 1/2 (3.3) kfk = |f|2 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) |f|p dµ < +∞ X The Lp norm of f is defined by Z 1/p p (3.7) kfkp = |f| 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 ∈ L2(µ) Theorem 3.8. (Schwarz inequality) Suppose f ∈ L2(µ) and g ∈ L2(µ). Then fg ∈ L2(µ), and Z (3.9) |fg| dµ ≤ kfkkgk X Proof. We have for all real λ, Z Z (3.10) 0 ≤ (|f| + λ|g|)2 dµ = kfk2 + 2λ |fg| 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 ∈ M. Let {fn} be a sequence of measurable functions such that

(3.12) 0 ≤ f1(x) ≤ f2(x) ≤ ... (x ∈ E) Let f be such that

(3.13) fn(x) → f(x)(x ∈ E) as n → ∞. 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 → ∞. 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 = {x|fn(x) ≥ cs(x)} (n = 1, 2, 3, ...)

Then E1 ⊂ E2 ⊂ E3 ⊂ .... By (), ∞ [ (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 → ∞, 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 ∈ M. If {fn} is a sequence of nonnegative measurable functions and ∞ X (3.24) f(x) = fn(x) ∀x ∈ E n=1 6 ANNIE WANG then ∞ 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 ∈ M. If fn is a sequence of nonnegative measurable functions and

(3.27) f(x) = lim inf fn(x)(x ∈ E) n→∞ then Z Z (3.28) fdµ ≤ lim inf fndµ E n→∞ E

Proof. Let gn be such that

(3.29) gn(x) = inffi(x)(i ≥ n, n = 1, 2, 3, ...)

Then gn is measurable on E and

(3.30) 0 ≤ g1(x) ≤ g2(x) ≤ ...

(3.31) gn(x) ≤ fn(x)

(3.32) gn(x) → f(x)(n → ∞) By Lebesgue’s monotone convergence theorem, it follows that Z Z (3.33) gn dµ → f dµ E E which, with (3.31), implies the desired inequality. Theorem 3.34. If f, g ∈ L2(µ), then f + g ∈ L2(µ), and (3.35) kf + gk ≤ kfk + kgk Theorem 3.36. (Lebesgue’s dominated convergence theorem) Suppose E ∈ M. Let {fn} be a sequence of measurable functions such that

(3.37) fn(x) → f(x)(x ∈ E) as n → ∞. If there exists a function g in Lµ on E such that

(3.38) |fn(x)| ≤ g(x)(x ∈ E, n = 1, 2, 3, ...) then Z Z (3.39) lim fn dµ = f dµ. n→∞ E E

Proof. We have that fn ∈ L(µ) and f ∈ L(µ) on E. Since fn + g ≥ 0, Fatou’s theorem implies that Z Z (3.40) (f + g) dµ ≤ lim inf (fn + g)dµ E n→∞ E or Z Z (3.41) f dµ ≤ lim inf fndµ E n→∞ E LEBESGUE MEASURE AND L2 SPACE. 7

Since g − fn ≥ 0, we also have that Z Z (3.42) (g − f) dµ ≤ lim inf (g − fn)dµ E n→∞ E Therefore Z Z (3.43) − f dµ ≤ lim inf − fn dµ E n→∞ E which is the same as saying Z Z (3.44) f dµ ≥ lim sup fn dµ E n→∞ E The conclusion of the theorem then follows from (3.40) and (3.44). Theorem 3.45. (The set of continuous functions is dense in L2 on [a, b].) For any f ∈ L2 on [a, b] and any > 0, there exists continuous function g on [a, b] such that 1/2 (Z b ) (3.46) kf − gk = (f − g)2 dx < a

Proof. Let A be a closed subset of [a, b] and KA be its characteristic function. Let (3.47) t(x) = inf|x − y| (y ∈ A)and

(3.48) gn(x) = 1/1 + nt(x)(n = 1, 2, 3, ...)

Then gn is continuous on [a, b], gn(x) = 1, and gn(x) → 0 on B, deﬁned by B = [a, b] − A. By Lebesgue’s dominated convergence theorem, it follows that Z 1/2 2 (3.49) kgn − KAk = gn(x) dx → 0 X Therefore characteristic functions of closed sets can be approximated by continuous functions in L2, which implies the same for simple measurable functions. 2 If f is nonnegative and f ∈ L , by Theorem 2.5 we may let {sn(x)} be a monotonically increasing sequence of simple nonnegative measurable functions such that 2 2 sn(x) → f(x). Since |f − sn| ≤ f , Lebesgue’s dominated convergence theorem shows that kf − snk → 0. The general case follows from (2.10). 2 Deﬁnition 3.50. Let f, fn ∈ L (µ)(n = 1, 2, 3,...). We say that {fn} converges 2 2 to f in L (µ) if kfn −fk → 0. We say that {fn} is a Cauchy sequence in L (µ) if for every > 0 there exists an integer N such that n ≥ N, m ≥ N implies kfn −fmk < We will now give a proof that every Cauchy sequence in L2(µ) converges - that is, that L2(µ) is complete.

2 2 Theorem 3.51. (L (µ) is complete.) If {fn} is a Cauchy sequence in L (µ), then 2 2 there exists a function f ∈ L (µ) such that {fn} converges to f in L (µ).

Proof. By deﬁnition, there exists a sequence {nk}(k ∈ N) such that 1 (3.52) kf n ) − f n )k < (k ∈ ) ( k ( k+1 2k N 8 ANNIE WANG

Choose a function g ∈ L2(µ). By the Schwarz inequality, Z g (3.53) |g(f(nk) − f(nk+1)| dµ ≤ kf(nk) − f(nk+1kkgk ≤ k X 2 Adding the inequalities Z g (3.54) |g(f(nk) − f(nk+1)| dµ ≤ k X 2 for each k, we obtain ∞ X Z (3.55) |g(f(nk) − f(nk+1)| dµ ≤ kgk k=1 X By Theorem 3.15 we may interchange the summation and integration above. It follows that ∞ X (3.56) |g(x)| |f(nk)(x) − f(nk+1(x)| < +∞ k=1 almost everywhere on X (since its integral is also ﬁnite). Then ∞ X (3.57) |f(nk)(x) − f(nk+1(x)| < +∞ k=1 almost everywhere on X. If the series above diverged on a set E of positive measure, then we could take g(x) to be nonzero on a subset of E of positive measure, contradicting (3.56). Since the kth partial sum of the telescoping series ∞ X (3.58) |fnk+1 (x) − fnk (x)| < +∞ k=1 which converges almost everywhere on X, is

(3.59) fnk+1 (x) − fn1 (x) we see that

(3.60) f(x) = lim fn (x) k→∞ k deﬁnes f(x) for almost all x ∈ X. Now let > 0 be given and choose an integer N such that n ≥ N, m ≥ N implies kfn − fmk ≤ . By Fatou’s theorem, if nk > N then

(3.61) kf − fn k ≤ lim inf fn − fn (x)k ≤ k k→∞ i k ∈ ∈ Thus f − fnk ∈ L (µ). Since f = (f − fnk ) + fnk , we see that f ∈ L (µ) also. Since was arbitrary,

(3.62) lim kf − fn k = 0. k→∞ k Finally, the inequality

(3.63) kf − fnk ≤ kf − fnk k + kfnk − fnk ∈ shows that fn converges to f in L (µ), since by taking n and nk large enough each of the terms on the right can be made arbitrarily small. LEBESGUE MEASURE AND L2 SPACE. 9

In addition, by defining the inner product for L2 of two functions f and g on a measure space X with Z (3.64) hf, gi = fgdµ X L2(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, L2 is an infinite-dimensional vector space. The space L2 is unique among Lp spaces as a Hilbert space. Hilbert spaces have many useful properties. In particular, its similarity to Eu- clidean space makes possible the use of geometric notions such as distance and orthogonality. The Pythagorean identity also holds true in L2. In addition, Hilbert spaces, and in particular the L2 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 con- cepts; 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 REU program possible.

References [1] Rudin, Walter. Principles of Mathematical Analysis. McHraw-Hilll, Inc. 1974. [2] Taylor, Michael E. Measure Theory and Integration. American Mathematical Society. 2006. [3] Tung, Diana. Lebesgue Integration and Measure:Its Properties and Implications. http://www. cims.nyu.edu/vigrenew/ug_research/DianaTung05.pdf [4] Bieren, Herman J. Introduction to Hilbert Spaces. http://econ.la.psu.edu/~hbierens/ HILBERT.PDF