ntroduction to Rel Anlysis (Mth 315) Spring 2005 Lecture Notes Mrtin Bohner Author ddress: Version from April 20, 2005 Deprtment of Mthemtics nd Sttistics, University of Missouri Roll, Roll, Missouri 65409-0020 E-mil ddress: bohner@umr.edu URL: http://www.umr.edu/~bohner
Contents Chpter 1. The Riemnn Stieltjes ntegrl 1 1.1. Functions of Bounded Vrition 1 1.2. The Totl Vrition Function 1 1.3. Riemnn Stieltjes Sums nd ntegrls 2 1.4. Nondecresing ntegrtors 3 Chpter 2. Sequences nd Series of Functions 5 2.1. Uniform Convergence 5 2.2. Properties of the Limit Function 5 2.3. Equicontinuous Fmilies of Functions 6 2.4. Weierstrß Approximtion Theorem 7 Chpter 3. Some Specil Functions 9 3.1. Power Series 9 3.2. Exponentil, Logrithmic, nd Trigonometric Functions 10 3.3. Fourier Series 10 3.4. The Gmm Function 12 Chpter 4. The Lebesgue ntegrl 13 4.1. The Lebesgue Mesure 13 4.2. Mesurble Functions 14 4.3. Summble Functions 15 4.4. ntegrble Functions 16 4.5. The Spces L p 17 4.6. Signed Mesures 18 iii
iv CONTENTS
CHAPTER 1 The Riemnn Stieltjes ntegrl 1.1. Functions of Bounded Vrition Definition 1.1. Let, b R with < b. A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1 <... < x n 1 < x n = b. The set of ll prtitions of [, b] is denoted by P = P[, b]. f P P, then the norm of P = {x 0, x 1,..., x n } is defined by P = sup x i, where x i = x i x i 1, 1 i n. 1 i n Definition 1.2. Let f : [, b] R be function. We put (P, f) = n k=1 The totl vrition of f on [, b] is defined s f(x k ) f(x k 1 ) for P = {x 0,..., x n } P. b f = sup (P, f). P P f b f <, then f is sid to be of bounded vrition on [, b]. We write f BV[, b]. Exmple 1.3. f f is nondecresing on [, b], then f BV[, b]. Theorem 1.4. f f B[, b], then f BV[, b]. Theorem 1.5. BV[, b] B[, b]. 1.2. The Totl Vrition Function Lemm 1.6. BV[, b] BV[, x] for ll x (, b). Definition 1.7. For f BV[, b] we define the totl vrition function v f : [, b] R by v f (x) = x f for ll x [, b]. Lemm 1.8. f f BV[, b], then v f is nondecresing on [, b]. Lemm 1.9. f f BV[, b], then v f f is nondecresing on [, b]. Theorem 1.10. f BV[, b] iff f = g h with on [, b] nondecresing functions g nd h. 1
2 1. THE REMANN STELTJES NTEGRAL 1.3. Riemnn Stieltjes Sums nd ntegrls Definition 1.11. Let f, g : [, b] R be functions. Let P = {x 0,..., x n } P[, b] nd ξ = (ξ 1,..., ξ n ) such tht Then x k 1 ξ k x k for ll 1 k n. S(P, ξ, f, g) = n f(ξ k ) [g(x k ) g(x k 1 )] k=1 is clled Riemnn Stieltjes sum for f with respect to g. The function f is clled Riemnn Stieltjes integrble with respect to g over [, b], we write f R(g), if there exists number J with the following property: ε > 0 δ > 0 P P, P < δ : S(P, ξ, f, g) J < ε (independent of ξ). n this cse we write b fdg = J, nd J is clled the Riemnn Stieltjes integrl of f with respect to g over [, b]. The function f is lso clled integrnd (function) while g is clled integrtor (function). Theorem 1.12 (Fundmentl nequlity). f f B[, b], g BV[, b], nd f R(g), then b b fdg f g, where f = sup x b f(x). Exmple 1.13. f g(x) = x for ll x [, b], then f R(g) iff f R[, b]. { 0 if x t Exmple 1.14. f C[, b] nd g(x) = p if t < x b. Theorem 1.15. Let f R[, b] nd g C[, b]. Then f R(g) Exmple 1.16. 1 0 xd(x2 ) = 2/3. nd b fdg = Theorem 1.17. f f R(g) R(h), then f R(g + h) nd fd(g + h) = b fg. fdg + fdh. f f R(h) nd g R(h), then f + g R(h) nd (f + g)dh = fdh + gdh. f f R(g) nd ρ R, then ρf R(g), f R(ρg), nd (ρf)dg = fd(ρg) = ρ fdg. Lemm 1.18. f f R(g) on [, b] nd if c (, b), then f R(g) on [, c].
1.4. NONDECREASNG NTEGRATORS 3 Remrk 1.19. Similrly, the ssumptions of Lemm 1.18 lso imply f R(g) on [c, b]. Also, we mke the definition b fdg = fdg if < b. Theorem 1.20. f f R(g) on [, b] nd if c (, b), then b c fdg + b c fdg = Theorem 1.21. f f R(g), then g R(f), nd b fdg + Exmple 1.22. 2 xd x = 5/2. 1 b b fdg. gdf = f(b)g(b) f()g(). Theorem 1.23 (Min Existence Theorem). f f C[, b] nd g BV[, b], then f R(g). 1.4. Nondecresing ntegrtors Throughout this section we let f B[, b] nd α be nondecresing function on [, b]. Definition 1.24. f P P, then we define the lower nd upper sums L nd U by n L(P, f, α) = m k [α(x k ) α(x k 1 )], m k = min f(x) x k 1 x x k nd U(P, f, α) = k=1 n k=1 M k [α(x k ) α(x k 1 )], M k = mx x k 1 x x k f(x). We lso define the lower nd upper Riemnn Stieltjes integrls by b fdα = sup L(P, f, α) P P Lemm 1.25. b fdα, b fdα R. nd Theorem 1.26. f P, P P with P P, then b fdα = inf U(P, f, α). P P L(P, f, α) L(P, f, α) nd U(P, f, α) U(P, f, α). Theorem 1.27. b fdα b fdα. Theorem 1.28. f f C[, b] nd α is nondecresing on [, b], then f R(α) on [, b].
4 1. THE REMANN STELTJES NTEGRAL
CHAPTER 2 Sequences nd Series of Functions 2.1. Uniform Convergence Definition 2.1. Let {f n } n N be funtions defined on E R. Suppose {f n } converges for ll x E. Then f defined by f(x) = lim n f n(x) for x E is clled the limit function of {f n }. We lso sy tht f n f pointwise on E. f f n = n k=1 g k for functions g k, k N, then f is lso clled the sum of the series n k=1 g k, write k=1 g k. Exmple 2.2. (i) f n (x) = 4x + x 2 /n, x R. (ii) f n (x) = x n, x [0, 1]. (iii) f n (x) = lim m [cos(n!πx)] 2m, x R. (iv) f n (x) = sin(nx) n, x R. 2n 2 x if 0 x 1 2n 1 (v) f n (x) = 2n(1 nx) if 2n x 1 n 0 if 1 n x 1. Definition 2.3. We sy tht {f n } n N converges uniformly on E to function f if ε > 0 N N n N x E : f n (x) f(x) ε. f f n = n k=1 g k, we lso sy tht the series k=1 g k converges uniformly provided {f n } converges uniformly. Exmple 2.4. (i) f n (x) = x n, x [0, 1/2]. (ii) f n (x) = x n, x [0, 1]. Theorem 2.5 (Cuchy Criterion). The sequence {f n } n N converges uniformly on E iff ε > 0 N N m, n N x E : f n (x) f m (x) ε. Theorem 2.6 (Weierstrß M-Test). Suppose {g k } k N stisfies g k (x) M k x E k N nd M k converges. Then k=1 g k converges uniformly. k=1 2.2. Properties of the Limit Function Theorem 2.7. Suppose f n f uniformly on E. Let x be limit point of E nd suppose A n = lim t x f n (t) exists for ll n N. 5
6 2. SEQUENCES AND SERES OF FUNCTONS Then {A n } n N converges nd lim A n = lim f(t). n t x Theorem 2.8. f f n re continuous on E for ll n N nd f n f uniformly on E, then f is continuous on E. Corollry 2.9. f g k re continuous on E for ll k N nd k=1 g k converges unifomly on E, then k=1 g k is continuous on E. Definition 2.10. Let be metric spce. By C() we denote the spce of ll complexvlued, continuous, nd bounded functions on. The supnorm of f C() is defined by f = sup f(x) for f C(). x Theorem 2.11. (C(), d(f, g) = f g ) is complete metric spce. Theorem 2.12. Let α be nondecresing on [, b]. Suppose f n R(α) for ll n N nd f n f uniformly on [, b]. Then f R(α) on [, b] nd b fdα = lim n b f n (x)dα. Corollry 2.13. Let α be nondecresing on [, b]. Suppose g k R(α) on [, b] for ll k N nd k=1 g k converges uniformly on [, b]. Then b b g k dα = g k dα. k=1 Theorem 2.14. Let f n be differentible functions on [, b] for ll n N such tht {f n (x 0 )} n N converges for some x 0 [, b]. f {f n} n N converges uniformly on [, b], then {f n } n N converges uniformly on [, b], sy to f, nd k=1 f (x) = lim n f n(x) for ll x [, b]. Corollry 2.15. Suppose g k re differentible on [, b] for ll k N nd k=1 g k is uniformly convergent on [, b]. f k=1 g k(x 0 ) converges for some point x 0 [, b], then k=1 g k is uniformly convergent on [, b], nd ( ) g k = g k. k=1 k=1 2.3. Equicontinuous Fmilies of Functions Exmple 2.16. f n (x) = x 2 x 2 +(1 nx) 2, 0 x 1, n N. Definition 2.17. A fmily F of functions defined on E is sid to be equicontinuous on E if ε > 0 δ > 0 ( x, y E : x y < δ) f F : f(x) f(y) < ε. Theorem 2.18. Suppose K is compct, f n C(K) for ll n N, nd {f n } n N converges uniformly on K. Then the fmily F = {f n : n N} is equicontinuous. Definition 2.19. The sequence {f n } n N is clled pointwise bounded if there exists function φ such tht f n (x) < φ(x) for ll n N. t is clled uniformly bounded if there exists number M such tht f n M for ll n N.
2.4. WEERSTRASS APPROMATON THEOREM 7 Theorem 2.20. A pointwise bounded sequence {f n } n N on countble set E hs subsequence {f nk } k N such tht {f nk (x)} k N converges for ll x E. Theorem 2.21 (Arzelà Ascoli). Suppose K is compct nd {f n } n N C(K) is pointwise bounded nd equicontinuous. Then {f n } is uniformly bounded on K nd contins subsequence which is uniformly convergent on K. 2.4. Weierstrß Approximtion Theorem Theorem 2.22 (Weierstrß Approximtion Theorem). Let f C[, b]. Then there exists sequence of polynomils {P n } n N with P n f uniformly on [, b].
8 2. SEQUENCES AND SERES OF FUNCTONS
CHAPTER 3 Some Specil Functions 3.1. Power Series Definition 3.1. A function f is sid to be represented by power series round provided f(x) = c n (x ) n for some c n, n N 0. Such n f is clled nlytic. Theorem 3.2. f n=0 c n x n converges for x < R, n=0 then it converges uniformly on [ R + ε, R ε] for ll ε > 0. Also, f defined by f(x) = c n x n for x < R n=0 is continuous nd differentible on ( R, R) with f (x) = (n + 1)c n+1 x n for x < R. n=0 Corollry 3.3. Under the hypotheses of Theorem 3.2, f (n) exists for ll n N nd f (k) (x) = n(n 1)... (n k + 1)c n x n k holds on ( R, R). n prticulr, n=k f (n) (0) = n!c n for ll n N 0. Theorem 3.4 (Abel s Theorem). Suppose n=0 c n converges. Put f(x) = c n x n for x ( 1, 1). Then n=0 lim f(x) = x 1 c n. Theorem 3.5. Suppose n=0 n, n=0 b n, nd n=0 c n converge to A, B, nd C, where c n = n k=0 kb n k. Then C = AB. 9 n=0
10 3. SOME SPECAL FUNCTONS Theorem 3.6. Given double sequence { ij } i,j N. f ij = b i for ll i N nd then j=1 i=1 j=1 j=1 i=1 b i converges, i=1 ij = ij. Theorem 3.7 (Tylor s Theorem). Suppose n=0 c nx n converges in ( R, R) nd put f(x) = n=0 c nx n. Then f (n) () f(x) = (x ) n for x < R. n! n=0 3.2. Exponentil, Logrithmic, nd Trigonometric Functions Definition 3.8. We define the exponentil function by z k E(z) = for ll z C. k! k=0 Remrk 3.9. n this remrk, some properties of the exponentil function re discussed. Definition 3.10. We define the trigonometric functions by C(x) = E(ix) + E( ix) 2 nd S(x) = E(ix) E( ix). 2i Remrk 3.11. n this remrk, some properties of trigonometric functions re discussed. 3.3. Fourier Series Definition 3.12. A trigonometric polynomil is sum N f(x) = 0 + ( n cos(nx) + b n sin(nx)), where k, b k C. n=1 A trigonometric series is series f(x) = 0 + ( n cos(nx) + b n sin(nx)). n=1 Remrk 3.13. n this remrk, some properties of trigonometric series re discussed. Definition 3.14. A sequence {φ n } n N is clled n orthogonl system of functions on [, b] if f, in ddition φ n, φ m := b φ n (x)φ m (x)dx = 0 for ll m n. φ n 2 2 := φ n, φ n = 1 for ll n N, then {φ n } is clled orthonorml on [, b]. f {φ n } is orthonorml on [, b], then c n = f, φ n for ll n N
3.3. FOURER SERES 11 re clled the Fourier coefficients of function f reltive to {φ n }, nd f(x) c n φ n (x) is clled the Fourier series of f. n=1 Theorem 3.15. Let {φ n } be orthonorml on [, b]. Let n s n (f; x) := s n (x) := c m φ m (x), where f(x) nd put t n (x) = m=1 n γ m φ m (x) with γ m C. m=1 Then f s n 2 2 f t n 2 2 with equlity if γ m = c m for ll m N. c m φ m (x), Theorem 3.16 (Bessel s nequlity). f {φ n } is orthonorml on [, b] nd if f(x) n=1 c nφ n (x), then c n 2 f 2 2. n prticulr, lim n c n = 0. n=1 Definition 3.17. The Dirichlet kernel is defined by N D N (x) = e inx for ll x R. n= N Remrk 3.18. n this remrk, some properties of the Dirichlet kernel re discussed. Theorem 3.19 (Locliztion Theorem). f, for some x R, there exist δ > 0 nd M < with f(x + t) f(x) M t for ll t ( δ, δ), then lim s N(f; x) = f(x). N Corollry 3.20. f f(x) = 0 for ll x in some intervl J, then s N (f; x) 0 s N for ll x J. Theorem 3.21 (Prsevl s Formul). f f nd g hve period 2π nd re Riemnn integrble, f(x) c n e inx nd g(x) γ n e inx, then nd n= 1 π f(x)g(x)dx = 2π π lim N n= c n γ n, n= m=1 1 π f(x) 2 dx = 2π π 1 f(x) s N (f; x) 2 dx = 0. 2π n= c n 2
12 3. SOME SPECAL FUNCTONS 3.4. The Gmm Function Definition 3.22. For x (0, ), we define the Gmm function s Γ(x) = 0 t x 1 e t dt. Theorem 3.23. The Gmm function stisfies the following. (i) Γ(x + 1) = xγ(x) for ll x (0, ); (ii) Γ(n + 1) = n! for ll n N; (iii) log Γ is convex on (0, ). Theorem 3.24. f f is positive function on (0, ) such tht f(x + 1) = xf(x) for ll x (0, ), f(1) = 1, log f is convex, then f(x) = Γ(x) for ll x (0, ). Theorem 3.25. We hve n!n x Γ(x) = lim n x(x + 1)... (x + n). Definition 3.26. For x > 0 nd y > 0, we define the Bet function by B(x, y) = Theorem 3.27. f x > 0 nd y > 0, then 1 0 t x 1 (1 t) y 1 dt. B(x, y) = Γ(x)Γ(y) Γ(x + y). Exmple 3.28. Γ(1/2) = π nd e s2 ds = π. Theorem 3.29 (Stirling s Formul). We hve lim x Γ(x + 1) ( x e ) x 2πx = 1.
CHAPTER 4 The Lebesgue ntegrl 4.1. The Lebesgue Mesure Exmple 4.1. n this exmple, the method of the Lebesgue integrl is discussed. Definition 4.2. Let = 1.. N, b = b 1.. b N R N. We write b if i b i for ll 1 i N. The set [, b] = {x R N : x b} is clled closed intervl. The volume of = [, b] is defined by N = (b i i ). i=1 Similrly we define intervls (, b], [, b), nd (, b), nd their volumes re defined to be [, b], too. Lemm 4.3. f, k, J k denote intervls in R N, then (i) J = J ; (ii) = n j=1 j = = n j=1 j ; (iii) n j=1 J j = n j=1 J j. Definition 4.4. We define the outer mesure of ny set A R N by µ (A) = inf j : A j nd j re closed intervls for ll j N. j=1 j=1 Exmple 4.5. For A = {} we hve µ (A) = 0. Lemm 4.6. The outer mesure µ is (i) monotone: A B = µ (A) µ (B); (ii) subdditive: µ ( k=1 A k) k=1 µ (A k ). Exmple 4.7. Ech countble set C hs µ (C) = 0. Lemm 4.8. f is n intervl, then µ () =. Lemm 4.9. (i) f d(a, B) > 0, then µ (A B) = µ (A) + µ (B); (ii) if A k k nd {k o} k N re pirwise disjoint, then µ ( k=1 A k) = k=1 µ (A k ). Theorem 4.10. We hve A R N ε > 0 O A : O open nd µ (O) < µ (A) + ε. 13
14 4. THE LEBESGUE NTEGRAL Definition 4.11. A set A R N is clled Lebesgue mesurble (or L-mesurble) if ε > 0 O A : O open nd µ (O \ A) ε. f A is L-mesurble, then µ(a) = µ (A) is clled its L-mesure. Theorem 4.12. The countble union of L-mesurble sets is L-mesurble. Theorem 4.13 (Exmples of L-mesurble Sets). Open, compct, closed, nd sets with outer mesure zero re L-mesurble. Theorem 4.14. f A is L-mesurble, then B = A c cn be written s B = N F k, k=1 where µ (N) = 0 nd F k re closed for ll k N. Definition 4.15. For set A we define the power set P(A) by P(A) = {B : B A}. Definition 4.16. A P() is clled σ-lgebr in provided (i) A; (ii) A A = A c A; (iii) A k A k N = k=1 A k A. Theorem 4.17. The collection of ll L-mesurble subsets of R N is σ-lgebr. Theorem 4.18. A R N is L-mesurble iff for ll ε > 0 there is closed set F A with µ (A \ F ) < ε. Definition 4.19. A triple (, A, µ) is clled mesure spce if (i) ; (ii) A is σ-lgebr on ; (iii) µ is nonnegtive nd σ-dditive function on A with µ( ) = 0. The spce is clled complete if ech subset B A with µ(a) = 0 stisfies µ(b) = 0. Theorem 4.20. (R N, A, µ) is complete mesure spce, where A is the set of ll L- mesurble sets, nd µ is the L-mesure. 4.2. Mesurble Functions Throughout this section, (, A, µ) is mesure spce. Definition 4.21. A function f : R is clled mesurble if for ech R, is mesurble (i.e., is n element of A). (f ) := {x : f(x) } Exmple 4.22 (Exmples of Mesurble Functions). (i) Constnt functions re mesurble. (ii) Chrcteristic functions K E re mesurble iff E is mesurble. (iii) f = R nd µ = µ L, then continuous nd monotone functions re mesurble. Theorem 4.23. f f nd g re mesurble, then re mesurble. (f < g), (f g), nd (f = g)
4.3. SUMMABLE FUNCTONS 15 Theorem 4.24. f c R, f nd g re mesurble, then re mesurble. cf, f + g, fg, f, f + = sup(f, 0), nd f = sup( f, 0) Theorem 4.25. f f n : R re mesurble for ll n N, then re mesurble. sup f n, n N inf f n, n N lim sup f n, nd lim inf f n n n Definition 4.26. We sy tht two functions f, g : R re equl lmost everywhere nd write f g, if there exists N A with µ(n) = 0 nd {x : f(x) g(x)} N. Theorem 4.27. f f, g : R, f is mesurble, nd f g, then g is mesurble, too, provided the mesure spce is complete. 4.3. Summble Functions Definition 4.28. A mesurble function f : [0, ] is clled summble. Nottion 4.29. For summble function f, we introduce the following nottion: A 0 (f) = {x : f(x) = 0}, A (f) = {x : f(x) = }, { k 1 A nk (f) = x : 2 n < f(x) k } 2 n, s n (f) = k=1 k 1 2 n µ (A nk(f)) + µ (A (f)). Theorem 4.30. Let f nd g be summble. Then for ll n N, (i) s n (f) s n+1 (f); (ii) f g = s n (f) s n (g); (iii) f g = s n (f) = s n (g). Definition 4.31. f f is summble, then we define fdµ = lim s n(f). n f B is mesurble, then we define fdµ = fk B dµ. B Theorem 4.32. f f nd g re summble with f g, then fdµ gdµ. Theorem 4.33. f f nd g re summble with f g, then fdµ = gdµ. Exmple 4.34. Let c 0 nd A be mesurble. Then cdµ = cµ(a). A
16 4. THE LEBESGUE NTEGRAL Theorem 4.35. f f is summble, then f 0 fdµ = 0. Theorem 4.36. Let (, A, µ) be mesure spce nd f be summble on. Define ν(a) = fdµ. Then (, A, ν) is mesure spce. Theorem 4.37. f f is summble nd c 0, then (cf)dµ = c fdµ. Theorem 4.38 (Beppo Levi; Monotone Convergence Theorem). Let f nd f n be summble for ll n N such tht f n is monotoniclly incresing to f, i.e., f n (x) f n+1 (x) for ll n N nd f n (x) f(x), n, for ll x. Then lim n A f n dµ = fdµ. Theorem 4.39. f f nd g re summble, then (f + g)dµ = fdµ + gdµ. Theorem 4.40 (Ftou Lemm). f f n re summble for ll n N, then lim inf f ndµ lim inf f n dµ. n n 4.4. ntegrble Functions f f : R is mesurble, then f + nd f re mesurble by Theorem 4.24 nd hence summble. f f dµ <, then becuse of f + f, f f, nd monotonicity, we hve f + dµ < nd f dµ <. Definition 4.41. A mesurble function f : R is clled integrble if f dµ <, we write f L(), nd then we define fdµ = f + dµ f dµ nd A fdµ = fk A dµ if fk A is integrble. Lemm 4.42. (i) f f is integrble, then µ(a ( f )) = 0; (ii) if f is integrble nd f g, then so is g (in complete mesure spce); (iii) if f nd g re integrble nd f g, then fdµ = gdµ; (iv) if f : R is integrble nd f = g h such tht g, h 0 re integrble, then fdµ = gdµ hdµ. Theorem 4.43. The integrl is homogeneous nd liner.
4.5. THE SPACES L p 17 Theorem 4.44 (Lebesgue; Dominted Convergence Theorem). Suppose f n : R re integrble for ll n N. f f n f, n (pointwise) nd if there exists n integrble function g : [0, ] with then f n (x) g(x) for ll x nd ll n N, fdµ = lim f n dµ. n Theorem 4.45. Let be n rbitrry intervl, suppose f(x, ) is Lebesgue integrble on for ech x [, b], nd define the Lebesgue integrl F (x) := f(x, y)dy. (i) f for ech y the function f(, y) is continuous on [, b] nd if there exists g L() such tht f(x, y) g(y) for ll x [, b] nd ll y, then F is continuous on [, b]. (ii) f for ech y the function f(, y) is differentible with respect to x nd if there exists g L() such tht f(x, y) x g(y) for ll x [, b] nd ll y, then F is differentible on [, b], nd we hve the formul F f(x, y) (x) = dy. x Theorem 4.46. f f : [, b] R is Riemnn integrble, then it is lso Lebesgue integrble, nd the two integrls re the sme. Theorem 4.47 (Arzelà). f f n R[, b] converge pointwise to f R[, b] nd re uniformly bounded, i.e., f n (x) M for ll x [, b] nd ll n N, then the Riemnn integrl stisfies b lim n f n dx = b fdx. 4.5. The Spces L p Definition 4.48. Let p 1. Let be n intervl. We define the spce L p () s the set of ll mesurble functions with the property f p L(). Theorem 4.49. L p () is liner spce. Theorem 4.50 (Hölder). Suppose p, q > 1 stisfy 1/p + 1/q = 1. g L q (). Then fg L 1 () nd ( ) 1/p ( 1/q fgdx f p dx g dx) q. Let f L p () nd
18 4. THE LEBESGUE NTEGRAL Theorem 4.51 (Minkowski). For f, g L p () we hve ( 1/p ( 1/p ( 1/p f + g dx) p f dx) p + g dx) p. Definition 4.52. For ech f L p () we define ( 1/p f p := f dx) p. Lemm 4.53. Suppose g k L p () for ll k N such tht n=1 g n p converges. Then n=1 g n converges to function s L p () in the L p -sense. Theorem 4.54. L p () is Bnch spce. Remrk 4.55. n this remrk, some connections to probbility theory re discussed. Definition 4.56. We sy tht sequence of complex functions {φ n } is n orthonorml set of functions on { 0 if n m φ n φ m dx = 1 if n = m. f f L 2 () nd if then we write c n = fφ n dx for n N, f c n φ n. n=1 Theorem 4.57 (Riesz Fischer). Let {φ n } be orthonorml on. Suppose n=1 c n 2 converges nd put s n = n k=1 c kφ k. Then there exists function f L 2 () such tht {s n } converges to f in the L 2 -sense, nd f c n φ n. n=1 Definition 4.58. An orthonorml set {φ n } is sid to be complete if, for f L 2 (), the equtions fφ n dx = 0 for ll n N imply tht f = 0. Theorem 4.59 (Prsevl). Let {φ n } be complete orthonorml set. f f L 2 () nd if f c n φ n, then n=1 f 2 dx = c n 2. n=1 4.6. Signed Mesures