Peano-Jordan-Borel Lebesgue Archimedes Gallilei, Pascal, Torricelli, Fermat Newton Leibniz Cauchy Daniell (1963) ( 4,200) 1

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1 Peao-Jorda-Borel Lebesgue Archimedes Gallilei, Pascal, Torricelli, Fermat Newto Leibiz Cauchy Daiell (1963) ( 4,200) 1

2 (1965) ( 2,600) (1966) (1972) ( 2,700) (1980) (1995) (2000) ( 2,500) (2000) ( 3,200) (2004) ( 3,600) (2005) ( 4,000) (2006) ( 4,000) P. Halmos, Measure Theory, Va Nostrad (1950) L.H. Loomis, A Itroductio to Abstract Harmoic Aalysis, Va Nostrad (1953) N. Bourbaki, Itégratio, Herma (1965) W. Rudi, Real ad Complex Aalysis, Academic Press (1970) H.L. Royde, Real Aalysis, 3rd ed., Pretice-Hall (1988) F. Riesz ad B. Nagy, Fuctioal Aalysis, Dover (1990) [ ] [ ] [ ] (F. Riesz ) [ ] Bourbaki ( Daiel ) [ ] W.H. Youg [ ] Riesz [ ] [ ] [ ] [ ] Fubii Riesz-Fisher [ ] [ ] [ ] [ ] 2

3 [Halmos] [Loomis] [ ] [ ] [Rudi] Daiell [Bourbaki] Riesz-Rado Daiell [Rudi] [Royde] Daiell Postscript 43 3

4 N 0 Z = {0, ±1, ±2,... }, R =, R + = [0, + ), R = [, ], C =. a b = max{a, b}, a b = mi{a, b}, a a, a a. f : X R [a < f < b] = {x X; a < f(x) < b}. X [f] = [f 0], [f 0] = {x X; f(x) 0} 1 a b = max{a, b}, a b = mi{a, b} a 1 a 2 a a 1 a = max{a 1,... a }, a 1 a = mi{a 1,... a }. A sup A, if A A A A A sup A sup A = + A if A = ± y = arcta x x = ± y = ±π/2 R (exteded real lie) R = [, + ] A sup A, if A R A B = sup(a) sup(b), if(a) if(b). sup =, if = + {a } 1 sup{a ; 1} sup{a ; 2}... 4

5 lim sup a {a } (upper limit) (lower limit) lim if a lim if{a k ; k } R 1.1. {a } lim if a lim sup a a = lim a lim if a = a = lim sup for a R. {a } a j a k (j k) (icreasig sequece) a j a k (j k) (decreasig sequece) {a } a a a {a } a a a. a j < a k (j < k) {a i } i I a i [0, + ] i I {a i } i I (summable) (sum) a i = a i 0 ( a i ) 0 R i I i I i I I I = j J I j j J {a i } i Ij { i I j a i } j J a i = a i. i I j J i Ij 1. i I a i < + {i I; a i 0} [a, b] [a, b] : a = x 0 < x 1 < < x = b (mesh) = mi{x 1 x 0, x 2 x 1,..., x x 1 } f i = sup{f(x); x [x i 1, x i ]}, f i = if{f(x); x [x i 1, x i ]} S(f, ) = f i (x i x i 1 ), S(f, ) = f i (x i x i 1 ). i=1 i=1 5

6 1.2., S(f, ) S(f, ) S(f, ) S(f, ). S(f) = if{s(f, ); }, S(f) = sup{s(f, ); } Darboux (upper ad lower itegrals) S(f) S(f) 1.3. f : [a, b] R S(f) = S(f) b a f(x) dx f (itegral). Riema (1857) Darboux (1875) 2 Riema f 1.4. f : [a, b] R b a f(x) dx = lim 0 i=1 f(x j )(x j x j 1 ). (improper itegral) f(x) dx < + R 0 f(x) dx = + R lim R 0 f(x) dx 1.5. (i) (ii) (i) 0 si x 1 + x 2 dx. 6

7 (ii) 0 si x 1 + x dx (Bolzao). {a } 1 N N k k {a k } k R K K K 2.3. (X, d) K (compact) X a X B r (a) r > 0 (locally compact) R. (metric space) (metric) d : X X [0, + ) (i) d(x, y) = 0 x = y. (ii) d(x, y) = d(y, x). (iii) [ ] d(x, y) d(x, z) + d(z, y) (i) X = R d(x, y) = j=1 x j y j 2. (ii) * X B(X) d(f, g) = sup{ f(x) g(x) ; x X}. (iii) * X d(x, y) = { 0 if x = y, 1 if x y. X x δ x B(X) 2.5. * (X, d) sup{d(x, y); x, y X} < + M > 0 7

8 M d (K, d K ) X = K N d X (x, y) = d K(x, y ). 2. * K X 3. * K = {0, 1,..., p 1} K N t [0, 1] p 2.6. * X (i) A, B X A B (ii) K X 4. * (X, d) B r (a) = {x X; d(x, a) < r}, B r (a) = {x X; d(x, a) r} a X r > 0 (ope ball) (closed ball). B r (a) B r (a) B r (a) B r (a) (iii) 5. * a X B r (a) r > 0 A d(x, A) = if{d(x, a); a A} x X A X (distace) d(x, A) = 0 x A d(x, A) d(y, A) d(x, y), x d(x, A) 6. x, y X 2.7. * r > 0 K K r = {x X; d(x, K) r} K X K r r > 0 Proof. r > 0, x K, B r (x) is compact K {x } 1 B 1/ (x ) x x K B r (x) r > 0 d(x, x) r/2 1/ r/2 B 1/ (x ) B r/2 (x ) B r (x) 8

9 B 1/ (x ) B r (x) r > 0 K r/2 y K r/2 x K d(x, y ) 2r/3 x x K d(x, x) r/3 d(y, x) d(x, y ) + d(x, x) r B r (x) {y } K r/2 {a } 1 X a X (coverge) lim d(a, a) = 0 a {a } (limit poit) a = lim a Cauchy lim m, d(a m, a ) = 0. ɛ > 0, N, m, N, d(a m, a ) ɛ 7. * lim a = a, lim a = a a = a Cauchy (complete) R (completio) R Q (i) x X, ɛ > 0, δ > 0, y B δ (x) = f(x) f(y) ɛ. (ii) a, b R, [a < f < b] {x X; a < f(x) < b} 8. (i) (ii) f, g : X R Φ : R 2 R Φ(f, g) : x Φ(f(x), g(x)) f + g, fg, f g, f g 9. (a, b) a b, a b f (support) [f] [f 0] f(x) = 0 (x [f]) 10. [f + g] [f g] [f g] [f] [g], [fg] [f] [g]. (i) (ii) R f, g [fg] [f] [g] 11. f f(x) = 0 (x F ) F 9

10 X X C c (X) C c (X) f, g C c (X) = f g, f g, fg C c (X) 12 (F. Riesz). * (X, d) F h : F [0, + ) { h(x) if x F, f(x) = { } d(x, F ) sup h(y) d(x,y) ; y F if x F X 2.8. (X, d) f : X R δ > 0 f (the degree of uiform cotiuity) C f (δ) = sup{ f(x) f(y) ; d(x, y) δ} 13. f : R R M = sup{ f (x) ; x R} C f (δ) Mδ. 2.9 (Heie). f : X R (uiform cotiuity) Proof. δ = 1/ lim C f (δ) = 0. δ 0 ɛ > 0, δ > 0, x, y X, d(x, y) δ, f(x) f(y) > ɛ. x, y X, d(x, y ) 1, f(x ) f(y ) ɛ. {x } 1 x a y a f f(x ) f(y ) ɛ lim f(x ) = f(a) = lim f(y ) 14. R R (rectagular solid) [a, b] = [a 1, b 1 ] [a, b ] f : [a, b] R f(x) dx [a, b] [a,b] S(f, ) S(f, ) C f ( ) (b 1 a 1 )... (b a ) 10

11 15. * [a, b] f C c (R ) [f] [a, b] f(x) dx = f(x) dx R [a,b] [a, b] (i) C c (R ) f R f(x) dx (ii) f 0 R f(x) dx 0. (iii) y R (iv) T : R R f(x + y) dx = R R f(t x) dx = 1 det(t ) f(x) dx. R f(x) dx. R 16. * [a, b] R Φ : [a, b] R f : [a, b] R lim 0 j=1 f(x j )(Φ(x j ) Φ(x j 1 )) = b a f(t)dφ(t) Stieltjes X A (idicator fuctio) 1 A (x) = { 1 if x A, 0 otherwise. A R - A A A = 1 A (x) dx * X {x i } i 1 {r i } i 1 U = i 1 B ri (x i ) F X = R {x i } r i = r/2 i U i (x i r i, x i + r i ) = i=1 r = 2r 2i 1 11

12 3 X f : X R {f : X R} 1 x X, lim f (x) = f(x) f f (coverge poit-wise) f {f } (limit fuctio) {f } x {f (x)} (mootoe sequece) {f } f f f f f {f } f f : X R f = sup{ f(x) ; x X} [0, + ] f f < + f f (coverge uiformly) lim f f = 0 f (x) f(x) f f. lim ( a 1 p + + a p ) 1/p = a 1 a p f : [a, b] R f(x) dx (b 1 a 1 ) (b a ) f. [a,b] 3.2. f f (uiformly) lim [a,b] f (x) dx = [a,b] f(x) dx * X f (Dii). K {f } 1 x K, f (x) 0 lim f = 0 Proof. f 0 r > 0, N 1, N, f > r. 12

13 1 < 2 <... f j > r (j 1) f j > r x j X, f j (x j ) > r {x j } j 1 x j x X m 1 j 1 j m f m (x) = f m (x) f m (x ) + f m (x ) f m (x) f m (x j ) + f j (x j ) > f m (x) f m (x j ) + r. f m x j x (j ) f m (x) r m f : R m R f 0 lim R f (x) dx = 0. m 3.7. * f : X R (lower semicotiuous) (i) lim x = x lim if f(x ) f(x). (ii) x X, ɛ > 0, δ > 0, d(x, y) δ = f(y) f(x) ɛ. (iii) a R [f > a] f (upper semicotious) f 17. * 3.8. * Proof. f f α R [f > α] = 1 [f > α] [a,b], 1 (a,b], 1 [a,b), 1 (a,b) 3.9 (Baire). * X f : X (, ] {f : X R} 1 f (x) = if{f(x ) + d(x, x ); x X} (i) f (x) f (y) d(x, y) (x, y X). f Lipschitz (ii) f f f ( ). Proof. (i) x, y X ɛ > 0, x X, f(x ) + d(x, x ) f (x) + ɛ. f (x) f (y) f (x) (f(x ) + d(y, x )) d(y, x ) + d(x, x) ɛ d(x, y) ɛ. 13

14 ɛ > 0 f (x) f (y) d(x, y). x, y (ii) f f f f(x) x = a ɛ > 0, δ > 0, d(x, a) δ = f(x) f(a) ɛ. if{f(x) + d(x, a); d(x, a) δ}. > 0 δ + if z X f(z) f(a) ɛ d(x, a) δ = f(x) + d(a, x) if f(z) + δ f(a) ɛ. z X f (a) f(a) ɛ lim f f if f(a) ɛ if f(z). δ 19. * (ii) f(a) < + f(a) = + 4 X L f, g L = f g, f g L X (vector lattice) (f g)(x) = max{f(x), g(x)}, (f g)(x) = mi{f(x), g(x)}. L L + = {f L; f 0} 4.1. (i) R C c (R ). X C c (X). (ii) * N 1 X = {1, 2,..., N} N L. (iii) * X f [f 0] L. (iv) * S = {x = (x 0, x 1,..., x ) R +1 ; (x 0 ) 2 + (x 1 ) (x ) 2 = 1} C(S ). 20. f = f 0 f 0 L. L = L + L + 14

15 21. X L (i) L (ii) f L f 0 L. (iii) f L f L L I : L R L (Daiell itegral) (i) [Liearity] I(αf + βg) = αi(f) + βi(g), α, β R, f, g L. (ii) [Positivity] f 0 = I(f) 0. (iii) [Cotiuity] f 0 = I(f ) 0. L I (L, I) (itegratio system) 4.3. Dii (i) f C c (R ) I(f) = f(x) dx. R (ii) * {p 1,..., p N } I(f) = k 1,...,k f(k 1,..., k, )p k1... p k. (iii) * X L = C c (X) I(f) = x X f(x) (iv) * L = C(S ) I(f) = 0< x 1 f ( ) x dx x 22. * Φ : R R L = C c (R) Stieltjes I(f) = f(t)dφ(t) f g = I(f) I(g). f f = I(f ) I(f). 15

16 25. * L + f {h } 1 f =1 h = I(f) I(h ) =1 h L 26. * X C c (X) f 0 K = [f 1 ] g(x) = 1 1 Nd(x, K) g g K = 1 N 2.7 g C c (X) 0 f f g 4.4. X L L = {f : X (, + ]; a sequecef L, f f}, L = {f : X [, + ); a sequecef L, f f} L + = {f L ; f 0} (i) L = L L L L. (ii) α, β R +, f, g L = αf + βg, f g, f g L. (iii) α, β R +, f, g L = αf + βg, f g, f g L.. (i) f(x) = ± 0f(x) = (ii) L L ± L = C c (R ) A R (i) 1 A L A (ii) 1 A L A 28. f(x) = x/(x 2 + 1) C c (R) C c (R) 29. * L = C c (R ) L = {f : R (, ]; f } (i) 1 h C c (R ) (ii) f L Baire fh m f,m C c (R ) f,m fh (iii) f m = 1 m f,m f m f 30. * X L L L C(X) X L = C c (X) L L = L 16

17 4.7. L f, g lim f lim g lim f, lim g L lim I(f ) lim I(g ) Proof. f m lim g f m = lim f m g. (f m f m g 0) m 4.8. I : L (, + ] I(f m ) = lim I(f m g ) lim I(g ). I (f) = lim I(f ), f f, f L I : L [, + ) I (f) = lim I(f ), f f, f L 31. (well-defied) 4.9. L = C c (R ) I I (1 (a,b) ) = (b 1 a 1 )... (b a ) = I (1 [a,b] ). 32. * Φ : R R Stieltjes I : C c (R) R I (1 (a,b) ) = Φ(b 0) Φ(a + 0), I (1 [a,b] ) = Φ(b + 0) Φ(a 0) (i) I ( f) = I (f) for f L ( L = L (ii) I, I I f L L (f) = I(f) = I (f). I (0) = I (0) = 0 (iii) I, I α, β R + f, g L f, g L I (αf + βg) = αi (f) + βi (g) I I (iv) f, g L, f g I (f) I (g). Proof. (iv) f f, g g f g f g = g 17

18 4.11. f f L + {f L + } 1 f = =1 f I (f) = I(f ) = f L + f L + 1 I f = I (f ). 1 1 Proof. {f,m L + } f m f,m I (f ) = m I(f,m) f = m, f,m L + ( ) I f = I(f,m ) = ( ) I(f,m ) = I (f ) m, m L = C c (R) Q = {q } 1 ɛ > 0 A = 1(q ɛ/2, q + ɛ/2 ) R 1 A L. 1 A 1 1 (q ɛ/2,q +ɛ/2 ) I (1 A ) =1 2ɛ 2 = 2ɛ f : X R (upper itegral) (lower itegral) I(f) = if{i (g); g L, f g}, I(f) = sup{i (g); g L, g f} R = [, + ] if( ) = +, sup( ) = f g g L I(f) = Q R 1 Q Dirichlet I(1 Q ) = (i) 1 Q (x) = lim m lim (cos(πm!x)) 2 18

19 (ii) Darboux S(1 Q [a,b] ) = 0, S(1 Q [a,b] ) = b a 5.3. (i) I(f) = I( f) for ay f. (ii) I(λf) = λi(f) for 0 λ < +. I(0) = 0 (iii) f g I(f) I(g). (iv) f + g f(x) = ± g(x) = x X I(f + g) I(f) + I(g). (v) I(f) I(f). (vi) f L L I(f) = I(f). f L f L I (f) I (f) Proof. (i) (iv) (v) (iv) g = f (i) (ii) I(0) = 0 (vi) I(f) = I (f) (f L ) I(f) = Ī(f) (f L ) f L I(f) = I(f) = I(f). f L f f f L I (f) = lim I(f ) = lim I(f ) I(f). f L I(f) = I (f) I(f) = I(f). 34. (i) (iv) 5.4. f : X R (Lebesgue itegrable) I(f) = I(f) R L 1 f L 1 I(f) = I(f) R I(f) 4.10 (ii) 5.3 (vi) L = C c (R ) I L 1 L 1 (R ) 35. [a, b] f : [a, b] R I(f) = b a f(t) dt. ɛ > 0, S(f) ɛ I(f) I(f) S(f) + ɛ f : X R ɛ, f + L, f L, f f f +, I (f + ) I (f ) ɛ. f f f + I (f + ) I (f ) = I (f + f ) 0 0 f ± I (f ) I(f), I (f + ) I(f). Proof. I (f ) I(f) I(f) I (f + ) 19

20 5.6. (i) L 1 X L L (ii) I : L 1 R I(f) = I (f) = I (f) (f L L ). I : L 1 R I : L R Proof. f, g L 1 f +, g + L f, g L f f f +, g g g + f + g f + g f + + g + I (f + + g + ) I (f + g ) = (I (f + ) I (f )) + (I (g + ) I (g )) f + g L 1 I(f + g) = I(f) + I(g). λ > 0 λf λf λf + I (λf + ) I (λf ) = λ(i (f + ) I (f )) λf L 1 I(λf) = λi(f). f + f f ( f + L, f L ) I ( f ) I ( f + ) = I (f + ) I (f ) f L 1 I( f) = I(f). L 1 I L 1 f L 1 = f 0 L 1 f 0 f 0 f f + 0 f 0 f + f 0 I (f + 0) I (f 0) = I (f + 0 f 0) I (f + f ) f 0 I(f) = I(f 0) I (f + 0) 0 I(f) 0 f L L f f f + f ± L 5.3 (vi) I(f) = I(f) [I(f ), I(f + )] 5.7. f : R R f L 1 R lim f(t) dt < + R + R I(f) = R lim f(t) dt R + R f 0 f f f C c (R) + f (t) = f(t) ( t R ) R f (t) dt f(t) dt R 1 20

21 si t t R si t dt lim dt = π R R t 36. α > 0 0 ( ) 1 si t α dt 37. * ɛ > 0, f L 1, g L, I( f g ) ɛ X (L, I), Y (M, J) φ : X Y M φ L I(f φ) = J(f) (f M) M 1 φ L 1 I(f φ) = J(f) (f M 1 ) Proof. M φ L, I (f φ) = J 5.9. (i) φ : X Y L = M φ L 1 = M 1 φ I(f) = J(f φ) (f M 1 ) (ii) X (L, I), (M, J) L M, J L = I (M, J) (L, I) L 1 M 1 M 1 L 1 I A R C c (A) C c (R ) L 1 (A) L 1 (A) L 1 (R ) L 1 (A) L 1 (R ) 1 A = (a, b) b f(x) dx = a R f(x)1 A(x) dx. 8 A f(x) dx = A f(x)1 A (x) dx f L 1 (R ) y R f(x + y) x R f(x + y) dx = f(x) dx. R R 38. f L 1 (R ) λ > 0 f(λx) dx = λ R f(x) dx R 21

22 6 6.1 (subadditivity of upper itegral). f : X [0, + ] f = =1 f (f 0) I(f) I(f ). =1 Proof. I(f ) = + I(f ) < + ( 1) ɛ > 0 g L + f g, I(g ) = I (g ) I(f ) + ɛ 2 f g 4.12 g L + I ( g ) = I (g ) ( ) I(f) I g = I (g ) I(f ) + ɛ 2 = I(f ) + ɛ 6.2 (Mootoe Covergece Theorem). f L 1 f : X R f lim I(f ) < + I(f) = lim I(f ). Proof. I(f ) = I(f ) I(f) lim I(f ) = + I(f) = + f L 1. lim I(f ) < + f f 0 = =1 (f f 1 ) I(f f 0 ) I(f f 1 ) = I(f f 1 ) = (I(f ) I(f 1 )) = lim I(f ) I(f 0 ). =1 =1 =1 I(f) I(f 0 ) + I(f f 0 ) = I(f 0 ) + I(f f 0 ) lim I(f ) f f f L 1 I(f ) = I(f ) I(f) lim I(f ) I(f) I(f) lim I(f ) 6.3. I L 1 f L 1 f 0 I(f ) 0. (L, I) (L 1, I) 22

23 6.4 (Domiated Covergece Theorem). f L 1 g L 1 f g ( 1) if 1 f, sup 1 f, lim if f, lim sup f I(lim if f ) lim if I(f ) lim sup I(f ) I(lim sup f ) f = lim f f L 1 Proof. m I(f) = lim I(f ). g if f f m f f m f sup f g m m f m f if m f, f m f sup f m I if m f, sup m f L 1 I( if f ) = lim I(f m f ) lim I(f m ) I(f ) = if I(f m ) m I( sup f ) = lim I(f m f ) lim I(f m ) I(f ) = sup I(f ). m m I(g) I( if f ) if I(f ) sup I(f ) I( sup f ) I(g) m m m m lim if f, lim sup f L 1 I(g) I(lim if f ) lim if I(f ) lim sup I(f ) I(lim sup f ) I(g). domiated covergece theorem 6.5. (i) t 0 e x2 +tx dx = =0 t! 0 x e x2 dx. (ii) t > t > 0 d dt 0 =1 e tx2 dx = ( 1) x 2 e tx2 dx. 1 t = 1 x t 1 Γ(t) 0 e x 1 dx 0 23

24 40. f(x) (0 x 1) lim 0 f ( x ) e x dx 41. f L 1 (R ) f(x + y) f(x) dx y R 42. f L 1 (R ) R g fg 43. f L 1 (R) a > 0 a a f a (x) = f(x t)e at2 dt = f(t)e a(t x)2 dt π π f a lim f a(x) = 0, x ± f a (x) dx f(x) dx 7 L 1 L X M R X (mootoe class) M f g f(x) = lim f (x) g(x) = lim g (x) f, g M 7.2. S R X M S S M M M M S (the mootoe class geerated by S) M(S) Proof. S. S R X S S 1, S 1 S 2, S 3, S 4,... S M ( = 1, 2,... ). M S = S S (S ) 1, (S ) 2, (S ) 3, (S ) 4,... 24

25 (ordiary umber) 7.3. X L L M(L) Proof. f, g M(L) = f + g M(L) (i) [L + M(L) M(L)] f L L M(L) (ii) [M(L) + M(L) M(L)] g M(L) {g M(L); f + g M(L)} {f M(L); f + g M(L)} L M(L) (lim sup lim if) L 1 M(L) I : L 1 R (itegratio system) 45. X L, X L φ : X X L φ L M(L ) φ M(L) 7.5. (mootoe-complete) f : X R f L, f f lim I(f ) < + f L I(f ) I(f) L 1 M(L) L I : L R (mootoe-completio) f L, g L [f, g] = {h : X R; f h g} (i) M(L) [f, g] = M(L [f, g]) (ii) (L, I) f, g L M(L) [f, g] = L [f, g]. (iii) M(L) = [f, g] M(L). f M(L) f L, f + L f f f + f L,g L M(L) + = M(L + ) 25

26 Proof. f g (i) f f, g g (f, g L) M = {h M(L); (f h) g M([f, g] L)} (f h) g = f (h g) M h L (f h) g = lim lim (f m h) g ([f, g] L) m L M M = M(L) f h g (h M) h = (f h) g M([f, g] L) M(L) [f, g] M(L [f, g]). (ii) L [f, g] (i) M(L) [f, g] = M(L [f, g]) = L [f, g]. (iii) h M(L), f L, g L, f h g h 4.12 L M(L) h M(L) + h g g L (i) h M(L) [0, g] = M(L [0, g]) M(L + ) M(L + ) M(L) + M(L) + L X L + M(L)+ = M(L + ) (i) L = C c (R ) 7.7. (L, I) M(L) + = L+ L1 M(L) = L Proof. f M(L) + f f (f M(L) + ) (iii) f g g L L h sup g = lim g 1 g 1 L f h h h h L + f = f h (ii) f M(L) [0, h ] = L [0, h ] L + f f f L +. f L 1 M(L) (±f) 0 L 1 M(L) + L + g, h L + g (f 0), h ( f) 0 I(g ) I(f 0) < +, I(h ) I(( f) 0) < + 26

27 f 0 = lim g, L + f = f 0 ( f) 0 L. ( f) 0 = lim h 7.8. (L, I) M(L) + = (L1 M(L)) +. f M(L)+ I(f) = I(f) f f f L 1 M(L) + f I(f ) I(f) = I(f) 47. M(L) = (L 1 M(L)) 7.9. I M(L) + I(f) = lim I(f ), f f, f (L 1 M(L)) +. L 1 M(L) + L+ I. f I(f) = I(f) ± I(f) = I(f) = ± f α,β { x α e x β if 0 < x, f α,β (x) = + if x = 0 f M(C c (R)) + I(f) = { 2 β Γ ( ) 1 α β if α < 1, + if α (i) A X L- (L-measurable) 1 A M(L) L- M(L) (ii) L X M(L) 1 X M(L) σ- (σ-fiite) (L, I) L σ- σ X (i) X K C c (X)- (ii) C c (X) σ- X = 1 X (X X σ- (σ-compact) Proof. (i) 1 K 1/ ( 2.7) 1 1 (d(, K)) 1 K = lim (1 1 d(, K)) 27

28 K C c (X) (ii) (i) M = {f M(L) + ; [f 0] 1 K } {K } f M C c (X) + M = M(L + ) = M(L) + 1 X M(L) + [1 X ] = X = 1 K X = R C c (X) σ- σ L σ- M(L) σ- (σ-boolea algebra) (i), X M(L). (ii) {A } 1 M(L) = 1 A, (iii) A M(L) = X \ A M(L). 48. A M(L). 1. σ- σ- (σ-field) σ- (σ-rig) 8.5. L = C c (R ) M(L) 8.6. A M(L) I- (I-measure) A I [0, + ] A I = I(1 A ) A I = { I(1 A ) if 1 A is itegrable, + otherwise. L = C c (R ) I- (Lebesgue measure) I A 8.7. [a, b] R [a, b] = (b 1 a 1 )... (b a ). 49. (i) (i) T T (A) = det(t ) A 8.8. I- (i) I = 0. 28

29 (ii) A = A = A I = A I. 1 = σ- B 2 X [0, + ] µ (measure) (i) µ( ) = 0, (ii) {A } 1 B A m A = (m ) µ A = µ(a ) 1 σ- µ X = 1 X (µ(x ) < + ) σ- µ(x) < + (fiite measure) µ(x) = 1 (probability measure) X σ- B 2 X B µ (X, B, µ) (measure space) =1. 2 X X (power set) X 50. A B A µ(a 1 ) < + µ(a ) σ- L I M(L) σ- [ 7.6 (iii) ] X σ- B f : X R (i) a R, [f > a] B. (ii) a R, [f a] B. (iii) a R, [f < a] B. (iv) a R, [f a] B. {x X; f(x) > a} = [f > a] σ- B 2 X f : X R B- (B-measurable) (radom variable) f : X R {f(x); x X} (simple fuctio) S S M(S) B- Proof. f f(x) = {a 1 < < a m } A i [f = a i ] = [f a i ] [f a i ] B m f = a i 1 Ai, i=1 i A i 29

30 g S g = i,j A i B j b j 1 Bj, j=1 j B j f + g = i,j (a i + b j ) 1 Ai B j, fg = i,j a i b j 1 Ai B j, f g = i,j (a i b j ) 1 Ai B j, f g = i,j (a i b j ) 1 Ai B j M M {f } f f = [f a] = 1[f a] B, f f = [f a] = 1[f a] B. M(S) M. f M m if f(x) > m, f m, (x) = m + 2mj/ if m + 2m(j 1)/ < f(x) m + 2mj/ for 1 j, m if f(x) m f m, (x) f(x) 1 if m < f(x) m lim f m,(x) = f(x) if m < f(x) m. lim lim f m,(x) = f(x) m for ay x X M M(S) 53. B- {f : X R} 1 {x X; lim f (x) } B µ : B [0, + ] f µ([f = a]) < + (0 a R) µ- µ- S µ µ σ- M(S) = M(S µ ) f S µ f = i=1 a i 1 Ai (µ(a i ) < + ) I µ (f) = a i µ(a i ) i=1 I µ : S µ R σ- I µ (S µ, I µ ) µ f(x) µ(dx) 30

31 A B f(x) µ(dx) = 1 A (x)f(x) µ(dx) A 54. * f S µ f 0 ɛ > 0, I(f ) = I(1 [f ɛ]f ) + I(1 [f >ɛ]f ) ɛi(f 1 ) + f 1 µ([f > ɛ]) [f > ɛ] µ lim I(f ) ɛi(f 1 ) ( ɛ > 0) X σ- (L, I) f : X R M(L)- f ± = (±f) 0 M(L) +. f M(L) + Proof. f ± = 0 (±f) M(L) + I(f) = lim r 1+0 = r 0 r [r < f r +1 ]. I 1 X ((f ± f ± r)) 1 [f±>r] (σ- 1 X M(L) ) [f ± > r] M(L) [f > a] M(L) (a R) [f > a] M(L) ( a R) h = f ± [h > r] M(L) (0 r < + ) r > 1 Z [r < h r +1 ] M(L) h r r 1 [r <h r +1 ] M(L) + = h r h (r 1) h M(L) + (h = f ±) 55. h r (x) = r r < h(x) r +1 (i) lim r 1 h r (x) = h(x) (x X). (ii) 1 < s < r h r h s (L, I) (X, M(L), µ) (S µ, I µ ) (L, I) (S µ, I µ ) M(S µ ) = M(L) (i) M(L)- f f α M(L) + (α > 0) (ii) M(L) f, g M(L) 4fg = (f + g) 2 (f g) 2 M(L). 31

32 56. (i) I µ L 1 L 1 (X, µ) * C c (R ) Proof. µ C = µ((0, 1] (0, 1]) µ m = 1, 2,... µ((0, 1/2 m ] (0, 1/2 m ]) = C 2 m = (0, 1/2m ] (0, 1/2 m ] f f(x) µ(dx) = C f(x) dx g C c (R ) f C c (R ) 57 (Chebyshev s iequality). f(x) α dx r α µ([ f r]). X * R T Z T = Q T = Z + θz (θ Q). R/T W W [0, 1) (wild set) W (W + t + Z) [0, 1) = W (t T ) Z t < + 1 (W + t + Z) [0, 1) = (W + t 1) [0, t ) (W + t ) [t, 1) (W + t + Z) [0, 1) = (W + t) [ + 1, t + 1) + (W + t) [t, + 1) = W + t = W R = ṫ T/Z (W + t + Z) [0, 1) = ṫ T/Z (W + t + Z) [0, 1) 1 = ṫ T/Z W

33 f : X R I( f ) < I((±f) 0) < I(f) = I(f 0) I(( f) 0) R ξ (expectatio) (mea) ξ f : X C Rf If f I(f) = I(Rf) + ii(if) f : X C f f f I(f) I( f ) Proof. I(f) = I(f) e iθ I(f) = I(Rf) cos θ + I(If) si θ = I ( (Rf) cos θ + (If) si θ ) I( f ) 9.3. e x2 +itx dx = πe t2 /4 59. f : R C a > 0 x R a f a (x) = f(x t)e at2 dt π C 9.4. f : X R I( f ) = 0 (ull fuctio) A (ull set) 1 A 1 A M(L) I(1 A ) = 0 N(I) 9.5. (i) R (ii) Cator (iii) R 60. R 33

34 9.6. (i) N(I). (ii) N N(I) (iii) {N } 1 =1 N (i) f M(L) [f 0] M(L) I( f ) = 0 (ii) f M(L) + I(f) < + [f = + ] M(L) Proof. (i) f (1 [f 0] ) f ( f ) 1 = 1 [f 0] I( f ) = lim I( f (1 [f 0]) lim I(1 [f 0]) I(1 [f 0] ) = lim I(( f ) 1) lim I( f ). (ii) A = [f = + ] 1 A f ( = 1, 2,... ) A I I(f) (i) N(I) M(L) f N(I), g M(L) = fg N(I). (ii) f, g L 1 I( f g ) = 0 N f(x) = g(x) (x N) (0,1] f(x) dx, [0,1] f(x) dx 9.9. f, g : X R [f g] f = g (a.e.) a.e. almost everywhere N 1, N 2 f j : X \ N j R (j = 1, 2) f 1 + f 2 X \ (N 1 N 2 ) f 1 (x) + f 2 (x)

35 9.10. Dirichlet 63. f g L 1 M(L) f = g (a.e) L 1 L 1 M(L) f(x) = ± x {f } 1 I( f ) < + 1 N (i) X \ N f (ii) x X \ N f(x) < +. f(x) = f (x) (x N) f I(f) = I(f ) Proof. f j N j f j (x) = f j (x) (x j N j ) 0 f j f M + 1 I f = I( f ) = I( f ) < f (x) = + 1 x N 0 N = 0 N j x N f(x) = 1 f (x) = 1 f (x) {q ; 1} f(x) = 1 e 3 (x q ) 2 35

36 x R f(x) dx = π 3 < +. (i) L 1 L f 1 = I( f ) (ii) L 1 = L 1 L L 1 L f L 1 f ± L 1 L N f(x) = f + (x) f (x) (x N) Proof. (ii) L f f I(f f) 1/2 f,m L f,m f (m ), I(f f,m ) 1 2 m+ 0 I(f,m f,m 1 ) = I(f f,m 1 ) I(f f,m ) 1 2 m = 2 2 m+ m, 1 (f 1,m f 1,m 1 ) lim f = f 0 + (f f 1 ) 1 = f 0,0 + (f 0,m f 0,m 1 ) +,0 f 1,0 ) m 1 1(f + (f,m f,m 1 ) (f 1,m f 1,m 1 ) m, 1 m, 1 I( f,0 f 1,0 ) I( f,0 f ) + I( f f 1 ) + I( f 1 f 1,0 ) 6 2 f 0,0 = 0 f 0,0 0 ( f 0,0 ), f,0 f 1,0 = 0 (f,0 f 1,0 ) 0 (f 1,0 f,0 ) f(x) = lim f (x) (a.e. x X) (ii) f ± L 1 L f + = 0 f 0,0 + (f 0,m f 0,m 1 ) + 0 (f,0 f 1,0 ) + (f,m f,m 1 ) m 1 1 m, 1 f = 0 ( f 0,0 ) + 0 (f 1,0 f,0 ) + (f 1,m f 1,m 1 ) 1 m, 1 (i) f L 1 lim m lim sup I( f m f ) = 0. 36

37 { k } k 1 I( f k f k+1 ) 1/2 k f = lim k f k = f 0 f f k f f 0 + (f k f k+1 ) k=0 f k f k+1 L 1 k=0 0 = lim I( f f k ) = lim I( f f ). Riesz L 1 L = {f; f L, f f, sup I(f ) < + }, 1 I(f ) I(f) (ii) L 1 f = f + f I(f) = I(f + ) I(f ) 1 [Riesz-Nagy] [ ] [ ] 9.14 ( (the law of large umbers)). * (X, µ) {ξ } 1 (i) { ξ } 1 Cesaro (ii) σ 2 = (ξ ξ ) 2 {σ} 2 1 Cesaro (iii) {ξ (x) ξ } 1 {ξ } Cesaro ξ 1 (x) + + ξ (x) ξ ξ lim = lim (µ-a.e. x X). Proof. η = ξ ξ η 1 (x) + + η (x) lim = 0 (µ-a.e. x X). 1 (η 1 (x) + + η (x)) 2 µ(dx) = 1 1 j,k η j (x)η k (x) µ(dx) = σ σ 2 σ σ 2 sup < + 1 ( ) 2 η1 (x) + + η k 2(x) µ(dx) < +. k=1 k 2 k=1 1 k 4 (η 1(x) + + η k 2(x)) 2 < + 37 (µ-a.e. x X),

38 η 1 (x) + + η lim 2(x) m m 2 = 0 (µ-a.e. x X) 1 m 2 < (m + 1) 2 m 1 η 1 (x) + + η (x) η 1 (x) + + η (x) m 2 η 1 (x) + + η m 2(x) m 2 + η m 2 +1(x) + + η (x) m 2 η 1 (x) + + η m 2(x) m 2 + η m 2 +1(x) + + η (m+1) 2(x) m 2 η 1 (x) + + η m 2(x) m 2 + (m + 1)2 m 2 m 2 sup η k (x) k 1 (m ) (Borel s ormal umber theorem). * N 2 X = {0, 1,..., N 1} N {p j = 1/N} 0 j N 1 X µ N X = {0, 1,..., N 1} N x = (x k ) k 1 x = N k x k [0, 1] d 1, d 2,..., d m 1 {0, 1,..., N 1}, d m {0, 1,..., N 2} x k = d k (1 k m) ξ (j) k ξ (j) k = 1/N N 1 d N m d m x N 1 d N m+1 d m 1 + N m (d m + 1). : X {0, 1} (0 j N 1, k 1) ξ (j) k (x) = f (j) k f (j ) k = (ξ (j) k 1/N)(ξ (j ) k 1/N) = { 1 if x k = j, 0 otherwise k=1 1 N if k k, 2 1 N if k = k, j = j, 0 if k = k, j j. 0 if k = k, N 1 N if k = k, j = j, 2 1 N if k = k, j j. 2 j {0, 1,..., N 1} {ξ (j) k } k 1 lim ξ (j) 1 (x) + + ξ(j) (x) = 1 N 65. * N = 10 (µ-a.e. x X) 38

39 10 σ- π : Ω X Ω σ- F X (L, µ) x X π 1 (x) (L x, µ x ) f F (i) x X, f π 1 (x) L x (ii) f(ω) µ x (dω) x L π 1 (x) (F, (L, µ), {(L x, µ x )}) (fibered itegratio system) I(f) = µ(dx) µ x (dω) f(ω) X π 1 (x) F σ- (L X, I X ), (L Y, I Y ) µ X, µ Y S X, S Y { } S X S Y = f i g i ; f i S X, g i S Y i=1 f g : X Y R (f g)(x, y) = f(x)g(y) (i) S X S Y X Y (ii) f S X S Y x X Y f(x, y) S Y I Y Y f(x, y) µ Y (dy) x S X X Y (iii) f S X S Y ( ) f(x, y) µ Y (dy) µ X (dx) = X Y Y ( X ) f(x, y) µ X (dx) µ Y (dy). (iv) I(f) I : S X S Y R S X S Y 66. L X L Y (repeated itegratio system) X, Y * X, Y L X = C c (X), L Y = C c (Y ) C c (X Y ) M(S X S Y ). 39

40 Proof. X Y d(x, y; x, y ) = max{d X (x, x ), d Y (y, y )} B r (x, y) = B r (x) B r (y) f C c (X Y ) K r > 0, x 1, x 2,..., x K, r > 0, fiite F K, K B r (x j ). j=1 K B r (y). y F K {x j } j 1 x 1 K, x 2 B r (x 1 ), x 3 B r (x 1 ) B r (x 2 ),... d(x i, x j ) r (1 i < j) {x j } K f ɛ > 0, δ > 0, x 1,..., x K, x K, i 1, d(x, x i ) < δ, f(x) f(x i ) ɛ. B i = B δ (x j ) X = (B 1 B c 1) (B B c ) X = 2 j=2 A j (B 1 B ) c 2 j 2 a j = x i (x i A j ) { f(a j ) if x A j (2 j 2 ), f ɛ (x) = 0 if x (B 1 B ) c f ɛ S X S Y f f ɛ ɛ (π : Ω X, F, (L, µ), {(L x, µ x )}) (i) f M(F ) + x X f π 1 (x) M(L x ) + f(ω) µ x (dω) π 1 (x) x M(L) + I(f) = X ( f(ω) µ x (dω) π 1 (x) ) µ(dx). (ii) f M(F ) F 1 N X x X \ N f π 1 (x) M(L x ) L 1 x X \ N x f(ω) µ x (dω) 40 π 1 (x)

41 M(L) L 1 ( ) I(f) = f(ω) µ x (dω) µ(dx). X π 1 (x) Proof. (i) F σ- 1 Ω M(F ) + φ F + 1 Ω φ 7.6 (iii) φ F + φ φ ϕ = 1 Ω φ ϕ M(F ) F 1 ϕ 1 Ω [0, ϕ ] M M F [0, ϕ ] ϕ φ φ F M M(F ) [0, ϕ ] f M(F ) + f = f ϕ f M(F ) [0, ϕ ] (i) f f f (ii) f = f 0 ( f) 0 f M(L) + (i) ( ) 9.7 (ii) f(ω) µ x (dω) π 1 (x) µ(dx) = I(f) < Ω = R 2, X = R, X = R, π : Ω (t, x) t X, F = C c (Ω), L = C c (X), π 1 (t) = {t} R = R L t = C c ({t} R) = C c (R), { 1 µ t (dx) = 4πt e x2 /4t dx if t > 0, δ(x) otherwise, I(f) = dt R f(t, x) µ t(dx) (Lebesgue-Fubii-Toelli). F = S X S Y I (i) M(L X ) M(L Y ) = M(S X ) M(S Y ) M(F ) (ii) f M(F ) + x X f(x, ) M(L Y ) + M(L X ) + I(f) = X Y X µ X (dx) µ Y (dy)f(x, y). Y Y f(x, y) µ Y (dy) x (iii) f M(F ) F 1 N X X x X \ N X f(x, ) M(L Y ) L 1 Y f(x, y) µ Y (dy) x M(L X ) L 1 X Y I(f) = X Y X µ X (dx) µ Y (dy)f(x, y). Y 41

42 10.6 ( ). f L 1 (R m+ ) N R m x R m \ N f(x, ) L 1 (R ) R m \ N x f(x, y) dy R L 1 (R m ) ( ) f(x, y) dxdy = f(x, y) dy m+ R R m R e t e tx2 dtdx = e t e tx2 dxdt = C 0 0 dx. 1 x dx = π 2, e t 1 t dt = 2C 2. C = 0 e y2 dy = e t 1 t dt. 67. t > * α R, β > 0 0 R e x β t x α dx = 0 e xy si x dxdy tx si x e x dx = π 2 arcta t ( ) βγ(/2) Γ α β if α <, + if α. { 2π / dxdy < + x y α α 70. f : R C a > 0 a f a (x) = f(x t)e at2 dt π lim x ± f a (x) = 0 lim f(x) f a (x) dx = 0. a + 42

43 11 Postscript Dieudoe Rado-Nikodym 43