B2 ( 19 ) Lebesgue ( ) ( ) 0 This note is c 2007 by Setsuo Taniguchi. It may be used for personal or classroom purposes, but not for commercia

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1 B2 ( 19) Lebesgue ( ) ( ) 0 This note is c 2007 by Setsuo Taniguchi. It may be used for personal or classroom purposes, but not for commercial purposes.

2

3 i Riemann f n : [0, 1] R 1, x = k (1 m n, 0 k m), f n (x) = m 0, f(x) = { 1, x [0, 1] Q, 0, x [0, 1] \ Q, f n (x) f(x) f n Riemann 1 0 f n (x)dx = 0 0 = lim 1 0 f n (x)dx = 1 0 f(x)dx f Riemann 1 0 f(x)dx Riemann Riemann {(x, y) R 2 x 2 + y 2 1, y 0} 1 Riemann x2 dx n 1 lim n 1 (k/n)2 1 n (x ) Riemann y (x ) Riemann (x ) y = x sin(1/x)? Lebesgue

4 CONTENTS ii Riemann L- () Lebesuge Contents 1 Riemann 1 2 () 13 3 Lebesgue Hölder Minkowski L p 49 8 Fubini 56

5 1 RIEMANN 1 1. Riemann Riemann < a < b < Def 1.1. (i) a = a 0 < a 1 < < a n 1 < a n = b = {a 0,..., a n } [a, b] = max{a j+1 a j 0 j n 1} (ii) = {a 0,..., a n } f : [a, b] R n 1 S(f; ) = inf{f(x) x [a j, a j+1 ]} (a j+1 a j ), j=0 n 1 S(f; ) = sup{f(x) x [a j, a j+1 ]} (a j+1 a j ) j=0 S(f) = sup{s(f; ) [a, b] } S(f) = inf{s(f; ) [a, b] } (iii) S(f) = S(f) f [a, b] Riemann S(f) = S(f) b a f(x)dx f [a, b] Riemann Thm 1.2 (Darboux). f : [a, b] R { n } n 0 n 0 lim S(f; n) = S(f), lim S(f; n) = S(f). f Riemann lim S(f; n) = lim S(f; n) = b a f(x)dx. Example 1.3. (1) f : [a, b] R f Riemann (2) n N Q n = {k/m k Z, m N, m n} f : [a, b] R { 1, x [a, b] \ Q n, f(x) = 0, x [a, b] Q n,

6 1 RIEMANN 2 f Riemann (3) f : [a, b] R { 1, x [a, b] \ Q, f(x) = 0, x [a, b] Q α < β inf{f(x) x [α, β]} = 0 sup{f(x) x [α, β]} = 1 S(f; ) = 0, S(f; ) = b a f Riemann Def 1.4 (). (i) A R { λ } 0(A) = inf (b j a j ) a j b j, j = 1, 2,..., (a j, b j ] A j=1 j=1 A (a, a] = (ii) λ 0(A) = 0 A R Prop 1.5. (i) A B λ 0(A) λ 0(B) A ( ) (ii) A 1, A 2, R λ 0 j=1 A j j=1 λ 0(A j ) A j j=1 A j Proof. (i) j=1 (a j, b j ] B j=1 (a j, b j ] A (ii) λ 0(A j ) = j λ 0(A j ) < ( j) ε > 0 a j,k < b j,k (a j,k, b j,k ] A j, k=1 (b j,k a j,k ) λ 0(A j ) + ε 2 j k=1 (a j,k, b j,k ] A j, j,k=1 j=1 (b j,k a j,k ) λ 0(A j ) + ε. j,k=1 j=1 λ 0 ε 0 ( ) A j λ 0(A j ) + ε. j=1 j=1

7 1 RIEMANN 3 Thm 1.6 (Lebesgue). f : [a, b] R f x [a, b] J(f; x) { } J(f; x) = inf f(y) inf f(y) I R x I [a, b] y I sup y I N f = {x [a, b] J(f; x) 0} f Riemann λ 0(N f ) = 0 Proof. Def 1.7. (i) f : [a, b] R a = a 0 < a 1 < < a n 1 < a n = b {a j } n j=0 {c j} n 1 j=0 f(x) = c j, x (a j, a j+1 ), j = 0, 1,..., n 1 (ii) f : [a, b] R L- {f n } N f(x) = lim f n(x), x / N f n f a.e. Prop 1.8. f, g L- (i) α, β R αf +βg L-(αf +βg)(x) = αf(x) + βg(x) (ii) fg L- (fg)(x) = f(x)g(x) (iii) max{f, g} min{f, g} L- Proof. (max{f, g})(x) = max{f(x), g(x)} (min{f, g})(x) = min{f(x), g(x)} Lem 1.9. f : [a, b] R L- (i) f {f n } sup n N,x f n (x) < f n f a.e. b lim a f n (x)dx f dλ

8 1 RIEMANN 4 (ii) F n = max{ n, min{f, n}} F n dλ < (1.1) Proof. sup n N lim F n dλ 6 f dλ Def (1.1) L- f : [a, b] R L- Lem 1.9 f dλ f L- Remark. L- Lem 1.9 (i) ( 6 ); (i) f {f n } sup n N,x f n (x) < f n f a.e. (ii) {f n } sup n N,x f n (x) < f n f a.e. lim b a f n (x)dx Thm (i) f, g : [a, b] R L-α, β R (a) αf + βg L- (αf + βg) dλ = α f dλ + β g dλ (b) f g a.e. ( N x / N f(x) g(x) ) f dλ g dλ (ii) f : [a, b] R Riemann L- b f dλ = f(x)dx a Proof. (i) f, g (a) f, g Lem 1.9 {f n } {g n} αf n + βg n αf + βg a.e. sup αf n (x) + βg n (x) < n N,x

9 1 RIEMANN 5 L- Riemann b (αf + βg) dλ = lim (αf n + βg n )(x)dx a { b = lim α f n (x)dx + β a = α f dλ + β g dλ. (b) H n = max{f n, g n } h n = min{f n, g n } b a } g n (x)dx H n f a.e., h n g a.e., sup H n (x) <, sup h n (x) <. n N,x n N,x L- Riemann b b f dλ = lim H n (x)dx lim h n (x)dx = a (ii) Q1.9 f n f n a g dλ. f n f a.e., sup f n (x) sup f(x) <. n N,x x f L-L- Thm 1.2 b b f dλ = lim f n (x)dx = f(x)dx. a a Thm 1.12 (Lebesgue ). f n, f : [a, b] R, n = 1, 2,..., L-f n f a.e. L- Φ 0 f n Φ a.e. ; lim f n dλ = f dλ. Proof. 6 Cor α < β f : (α, β) [a, b] (t, x) f(t, x) R t f(t, ) L- (i) (Lebesgue ()) x [a, b] (α, β) t f(t, x) R L- Φ 0 sup t (α,β) f(t, ) Φ a.e. t f(t, ) dλ ; lim f(t, ) dλ = f(t 0, ) dλ, t t 0 t 0 (α, β).

10 1 RIEMANN 6 (ii) (Lebesgue ()) (a) x [a, b] (α, β) t f(t, x) R C 1 (b) t (α, β) f(t, ) L- (c) L- Φ 0 sup f t (α,β) t (t, ) Φ a.e. t f(t, ) dλ f(t, ) dλ = t f (t, ) dλ. t Proof. (i) t 0 (α, β) {t n } (α, β) t n t 0 f n (x) = f(t n, x) f n Φ a.e. f n f(t 0, ) Thm 1.12 f(t n, ) dλ = f n dλ f(t 0, ) dλ (n ) {t n } (ii) t 0 (α, β) h < min{t 0 α, β t 0 } h R f(t 0 + h, x) f(t 0, x), (h 0), g(h, x) = h f t (t 0, x), (h = 0), (i) L- f : R C ; f f = u + iv (i 2 = 1) u, v L-f L-f L- f dλ = u dλ + i v dλ Thm 1.12 Cor L- Example r > 0 f r : [0, π] C f r (x) = exp(ire ix ) sin x Cauchy 0 x dx = π 2 π f r (x)dx Riemann f r dλ L- 0 r r 0 sin x 0 (x [0, π]) exp(ire ix ) = exp( r sin x) 1 x [0, π] ( ) [0,π] lim f r(x) = 1 (x [0, π]), r 0 lim f r(x) = 0 (x (0, π)) r

11 1 RIEMANN 7 f r 0 a.e. ( ) Φ(x) 1 Thm 1.12 π f r dλ = 1 dλ = 1dx = π, lim r 0 lim r [0,π] [0,π] f r dλ = [0,π] [0,π] 0 dλ = 0 π 0 0dx = 0 Q1.1. a R λ 0({a}) = 0 A λ 0(A) = 0 { Q1.2. λ 0 (A) = inf j=1 (b j a j ) a j b j, j = 1, 2,..., } j=1 (a j, b j ) A λ 0(A) = λ 0 (A) ( A R) Q1.3. [a + ε, b ε] (ε > 0) Q 1.2 λ 0((a, b)) b a Q1.4. a < b, R A {(a, b), (a, b], [a, b), [a, b]} λ 0(A) = b a. Q1.5. f : [a, b] R J(f; x) Thm 1.6 (i) x 0 [a, b] J(f; x 0 ) = 0 f x = x 0 (ii) c 0 {x [a, b] J(f; x) c} Q1.6. f : [a, b] R N f = {x [a, b] J(f; x) > 0} λ 0(N f ) = 0 (i) ε > 0 (a 1, b 1 ),..., (a n, b n ) n j=1 (b j a j ) < ε {x [a, b] J(f; x) ε} n j=1 (a j, b j ) (ii) ε > 0 a c 1 < d 1 c 2 < d 2... d n 1 c m < d m b sup x [ck,d k ] f(x) inf x [ck,d k ] f(x) < ε (k = 1,..., m) I \ m k=1 [c k, d k ] n j=1 (a j, b j ) (iii) f Riemann Q1.7. Thm 1.6 N f n N N f,n = {x [a, b] J(f; x) 1/n} f Riemann (i) ε > 0 0 = a 0 < a 1 < < a N 1 < a N = 1

12 1 RIEMANN 8 o i = sup x [ai,a i+1 ] f(x) inf x [ai,a i+1] f(x) I n = {i o i 1/n} i I n (a i+1 a i ) < ε (ii) N f,n i I n [a i, a i+1 ] {a 0,..., a N } (iii) λ 0(N f,n ) = 0 (iv) λ 0(N f ) = 0 Q1.8. Prop 1.8 Q1.9. f : [a, b] R Riemann f n (x) = f(a + (b a)k2 n ) x [a + (b a)k2 n, a + (b a)(k + 1)2 n ) k = 0,..., 2 n Thm 1.6 f n f a.e. Q1.10. Cor 1.13(ii) Q1.11. U R n f : U [a, b] R (a) t U f(t, ) : [a, b] R L-(b) x [a, b] U t f(t, x) R C 1 (c) L- Φ 0 sup f (t, ) Φ a.e.(i = 1,..., n) t U t i t = (t 1,..., t n ) i = 1,..., n t U f/ t i (t, ) L- f f(t, ) dλ = (t, ) dλ. t i t i Q1.12. f L-L- f n : [a, b] R 0 f n f n+1 f (a) f L- sup n N f n dλ < (Hint: a n,m = min{f n, m} dλ = lim lim a n,m = lim m lim a n,m m ) (b) f L- lim f n dλ = fdλ

13 1 RIEMANN 9 Q1.13. L- f n, f : [a, b] R sup{ f n (x), f(x) x [a, b], n N} < f n f a.e. lim f n dλ = f dλ Q1.14. f n : [a, b] R L- x [a, b] f n(x) < f(x) = f n(x) g(x) = f n(x) f, g : [a, b] R L- f n dλ f dλ = f n dλ Q1.15. L- f n : [a, b] R L- f f n dλ = f dλ lim Q1.16. f n, f : [a, b] [0, ) n = 1, 2,... L- f n f n+1 f lim f n dλ = f dλ Q1.17. Riemann 1 ( ) x (i) lim exp 0 n + x 2 dx 3 ( x 2 ) + 1 (ii) lim n sin dx n (iii) lim n 2{ 1 e (x/n)2} dx Q1.18. f : [a, b] R f 0 Riemann (i) n > 2/(b a) f n : [a, b] R { f(x), x [a + (1/n), b (1/n)], f n (x) = 0,, 0 f n (x) f n+1 (x) f(x) (x (a, b)) (ii) f L- f dλ f Riemann

14 1 RIEMANN 10 Q1.19. L- f : [a, b] R C > 0 0 < α < 1 f(x) C min{(x a), (b x)} α ( x (a, b)) f L- Q1.20. f : [a, b] R L- f(x) min{x a, b x} ( x [a, b]) (i) g n (x) = e nf(x) L- (ii) lim g n (iii) lim e nf dλ = 0 Q1.21. f : [a, b] R e f L- (i) f n L- (ii) e f L- (iii) e f 1 dλ = f n dλ n! Q1.22. f : [a, b] R f < f L- n=0 (i) f n L- 1 (ii) 1 f L- 1 (iii) 1 f dλ = n=0 f n dλ Q1.23. f : [a, b] R L-g : R R C y R f y (x) = g(x y)f(x) (x [a, b]) (i) f y : [a, b] R L- (ii) F (y) = f y dλ F : R R C (iii) a 0,..., a n R n j=0 a jg (j) = 0 n a j F (j) = 0 j=0 Q1.24. (Laplace ) f : [a, b] R L- y R f y (x) = e yx f(x) (x [a, b]) (i) f y : [a, b] R L-

15 1 RIEMANN 11 (ii) F (y) = f y dλ F : R R C (iii) f C n f (j) (a) = f (j) (b) = 0 (0 j n 1) a 0,..., a n R L- g : [a, b] R n a j f (j) (x) = g(x), x [a, b] j=0 n a j y j F (y) = j=0 g y dλ Q1.25. (Fourier ) f : [a, b] R L- y R f y (x) = e iyx f(x) (x [a, b]) (i) f y : [a, b] C L- (ii) F (y) = f y dλ F : R R C (iii) f C n f (j) (a) = f (j) (b) = 0 (0 j n 1) a 0,..., a n R L- g : [a, b] R n a j f (j) (x) = g(x), x [a, b] j=0 n a j ( iy) j F (y) = j=0 g y dλ Q1.26. (Fourier-Laplace ) f : [a, b] R L- z C f z (x) = e zx f(x) (x [a, b]) (i) f z : [a, b] C L- (ii) F (z) = f z dλ (z C) F (iii) f n (x) = x n f(x) F (x) = n=0 z n f n dλ. n!

16 1 RIEMANN 12 Q1.27. (Cauchy ) f : C C r > 0 z < r z C f z (x) = irf(reix )e ix re ix, x [0, 2π] z (i) f z : [0, 2π] C L- (ii) F (z) = [0,2π] f z dλ {z C z < r} n F (n) Q1.28. (Beta ) α, β C + = {z C Re z > 0} B(α, β) = 1 Riemann 0 x α 1 (1 x) β 1 dx (i) Riemann (ii) C 2 + (α, β) B(α, β)

17 2 () () Def 2.1 (). E 2 2 (i) E 3 (a), E (b) A E A c := \ A E (c) A, B E A B E (ii) E σ (c) (c ) A 1, A 2, E j=1 A j E Def 2.2 ( ). (i) E µ : E [0, ] µ A, B E, A B = = µ(a B) = µ(a) + µ(b) = µ (ii) E σ µ : E [0, ] µ ( ) A j E, A i A j = (i j) = µ A j = µ(a j ) = µ σ (iii) E σ µ (, E, µ) j=1 Thm 2.3 ( ). µ σ E j=1 (i) A n E A n A n+1 n = 1, 2,... µ lim µ(a n ) ( A n ) = (ii) A n E A n ( A n+1 n = 1, 2,... µ(a 1 ) < ) µ = lim µ(a n ) A n Proof. (i) A 0 = B n = A n \ A n 1 {B n ; n N} k B n = A k B n = A n µ σ ( ) ( ) µ A n = µ B n = µ(b n ) = lim k k µ(b n ) = lim k µ ( k B n ) = lim k µ(a k).

18 2 () 14 (ii) A 1 \ A n A 1 \ A n+1 (i) ( µ A 1 \ ) ( ) A n = µ {A 1 \ A n } = lim µ(a 1 \ A n ). ( µ ) ( A n + µ A 1 \ ) A n = µ(a 1 ), µ(a n ) + µ(a 1 \ A n ) = µ(a 1 ), µ(a 1 ) < ( µ ) ( A n = µ(a 1 ) µ A 1 \ ) A n = lim {µ(a 1) µ(a 1 \ A n )} = lim µ(a n). Def 2.4 (). E µ { µ (A) = inf µ(b j ) B j E, j=1 µ µ j=1 } B j A, A Thm 2.5 (Caratheodory ). E 0 µ µ A 1, A 2, E 0 A i A j = (i j) j=1 A j E 0 µ ( j=1 A j) = j=1 µ(a j) E = {B µ (G) = µ (B G) + µ (B c G), G } (i) E σ (ii) µ E (iii) E 0 E µ (A) = µ(a) ( A E 0 ) Proof. (1) µ (A) = µ(a), A E 0. (2.1) ) A E 0 A A µ (A) µ(a) ε > 0 A j E 0 j=1 A j A j=1 µ(a j) µ (A) + ε

19 2 () 15 { ( j 1 )} C j = A A j \ k=1 A k C j E 0 j=1 C j = A C i C j = µ(a) = µ(c j ) µ(a j ) µ (A) + ε. j=1 j=1 ε 0 µ(a) µ (A) (2.1) /// (2) ( ) µ B j µ (B j ), B 1, B 2, 2. (2.2) j=1 j=1 ) ε > 0 A n,k E 0 k=1 A n,k B n k=1 µ(a n,k) < µ (B n ) + ε2 n n,k=1 A n,k B n ( ) µ B n n,k=1 ε 0 (2.2) /// { µ(a n,k ) µ (B n ) + ε2 n} µ (B n ) + ε. (3) E B 1,..., B n E ( n µ ( Bk G )) = k=1 n µ (B k G), G. (2.3) k=1 ) µ ( ) = 0, E E B E B c E B 1, B 2 E B 1 B 2 E ( Q2.1 ) B 1, B 2 E µ ((B 1 B 2 ) c G) + µ ((B 1 B 2 ) G) = µ ((B c 1 B c 2) G) + µ ((B 1 B 2 ) G) = µ (B 2 {(B c 1 B c 2) G}) + µ (B c 2 {(B c 1 B c 2) G}) + µ ((B 1 B 2 ) G) = µ (B 2 B c 1 G) + µ (B c 2 G) + µ ((B 1 B 2 ) G) = µ (B 2 G) + µ (B c 2 G) = µ (G) B 1 B 2 E (2.3) n = 2 B 1, B 2 E B 1 B 2 = G B 1 E µ ((B 1 B 2 ) G) = µ (B 1 {(B 1 B 2 ) G}) + µ (B c 1 {(B 1 B 2 ) G}) = µ (B 1 G) + µ (B 2 G).///

20 2 () 16 (4) E σ µ E ) B n E B = B n G G = (B G) (B c G) (2.2) µ (G) µ (B G) + µ (B c G). (2.4) M n = n k=1 B k (3) M n E (2.3) µ (G) = µ (M n G) + µ (M c n G). M n B M n c B c µ (Mn c G) µ (B c G)(2.3) µ (M n G) = n k=1 µ (B k G) n µ (G) µ (B k G) + µ (B c G). k=1 n (2.2) k=1 µ (B k G) µ (B G) µ (G) µ (B k G) + µ (B c G) µ (B G) + µ (B c G) k=1 (2.4) µ (G) = µ (B k G) + µ (B c G) = µ (B G) + µ (B c G). k=1 B E G = B σ /// (5) E 0 E ) A E 0 G A n E 0 A n G A G (A A n ), A c G (A c A n ) µ(a n ) = {µ(a A n ) + µ(a c A n )} µ (A G) + µ (A c G). {A n } µ (G) µ (A G) + µ (A c G). µ A E µ (G) = µ (A G) + µ (A c G). Cor 2.6. E 0 µ A 1, A 2, E 0 j=1 A j E 0 µ ( j=1 A j) = j=1 µ(a j) σ[e 0 ] µ µ(a) = µ(a) ( A E 0 )

21 2 () 17 Q2.1. (i) {, } 2 σ (ii) A J = {,, A, A c } J (iii) E Def2.1 (a) (b) E A, B E A B E (iv) E σ Q2.2. (i) F, G 2 σ F G σ (ii) Λ α Λ σ F α 2 α Λ F α σ (iii) A 2 A σ ( σ σ[a] A σ ) Q2.3. = R 2 E = {A R 1 A R 1 } G = {R 1 B B R 1 } (i) E, G σ (ii) E G σ? Q2.4. (i) µ µ( ) = 0 (ii) (iii) µ A, B E A B µ(a) µ(b) µ(b) < µ(b \ A) = µ(b) µ(a) Q2.5. µ A, B E µ(a B) µ(a) ( + µ(b) ) µ A j E µ j=1 A j j=1 µ(a j) ( ) Q2.6. M N = {1, 2,..., M} E = 2 µ(a) = #A/M (A E) µ Q2.7. = N E = 2 a n > 0 (n N) µ(a) = a n A (n), A E A (n) = 1 (n A) = 0 (n / A) µ Q2.8. Cor2.6 Q2.9. (, E, µ) A 1, A 2, E ( µ(a n) < ) µ m=1 n=m A n = 0

22 2 () 18 Q2.10. (i) µ µ µ (B) = 0 A B µ (A) = 0 µ (B) = 0 G µ (G) = µ (B G) + µ (B c G). ( ) (ii) µ (A n ) = 0 µ A n = 0 Q2.11. n N, > 3 = {1, 2,..., n} N = {1, 2} E 0 = {, N, B, N B B {3,..., n}} µ(a) = #(A {3,..., n}) (A E 0 ) (i) E 0 σ µ E 0 (ii) Thm2.5 E 0 E E = 2 E 0 E ( E 0 σ E ) Q2.12. = {0, 1, 2,... } { (i) 0 A #A E 0 = A c } <, (ii) 0 / A #A < µ 0 (E) = #E (= ) (i) E 0 µ 0 (ii) µ 0(A) (iii) Thm2.5 E 0 E E = 2 (iv) α > 0 µ α (A) = α A (0) + #(A \ {0}) (, E, µ α ) µ α (E) = µ 0 (E) ( E E 0 ) Q2.13. E 0 µ µ Caratheodory (Thm 2.5) (, E, µ ) µ k E 0 µ( k ) < k=1 k = ν (, σ(e 0 )) ν(e) = µ(e) ( E E 0 ) ν(a) = µ (A) ( A σ(e 0 ))

23 3 LEBESGUE Lebesgue N N Def 3.1. (i) I R N a i b i i = 1,..., N I = N i=1 (a i, b i ] (a, a] = (, ] = R (ii) J R N I k k = 1,..., n J = n k=1 I k F 0 ; F 0 = { J = n } I k I i I j = (i j), I k. k=1 (iii) I I = N i=1 (b i a i ) J n λ 0 (J) = I k. ( = ) k=1 Lem 3.2. I I k k = 1,..., n I = n k=1 I k I I = λ 0 (I) J λ 0 (J) Proof. Prop 3.3. (i) F 0 (ii) λ 0 F 0 Proof. (i) = N i=1 (0, 0] RN = N i=1 (, ], RN F 0 A F 0 A c F 0 A, B F 0 A B F 0 (ii) A 1,..., A n Def 3.4. (i) λ 0 λ 0 { λ 0(A) = inf λ 0 (J n ) J n F 0, A J n }, A R N. (ii) F = {A } R N λ 0 (A G) + λ 0(A c G) = λ 0(G), G R N λ = λ 0 F Thm 3.5. F σ F 0 F λ F λ(j) = λ 0 (J) ( J F 0 ) (F Lebesgue λ R N Lebesgue )

24 3 LEBESGUE 20 Proof. F n F 0 F = F n F 0 λ 0 (F ) = λ 0 (F n ) (3.1) Caratheodory λ 0 (F ) < (3.1) H n = F \ n k=1 F k H n λ 0 (Prop.3.3) λ 0 (F ) = λ 0 (H n ) + n λ 0 (F k ) λ 0 (H n ) 0 (3.1) λ 0 (H n ) 0 λ 0 (H n ) λ 0 (H n+1 ) 0 ε > 0 λ 0 (H n ) 2ε ( n N) Q2.7 J k F 0, H k J k H k λ 0 (H k \ J k ) < ε2 k (B B ) n ) ( n ) λ 0 (H n \ J k = λ 0 {H n \ J k } k=1 k=1 ( n ) λ 0 {H k \ J k } k=1 k=1 n λ 0 (H k \ J k ) < k=1 ( n ) λ 0 J k = λ 0 (H n ) λ 0 (H n \ k=1 n ) J k > ε. k=1 n ε2 k ε. n k=1 J k n k=1 J k { J 1 J 1 n } J 1 k=1 J k n = 1, 2,... k=1 J k H n λ 0 (F ) = Q2.7 K m F, K m F 0 λ 0 (K m ) m λ 0 (F K m ) < F K m = (F n K m ) (3.1) m m λ 0 (K m ) = λ 0 (F n K m ) λ 0 (F n ). λ 0 (F n ) = = λ 0 (F ) λ 0 (F ) = (3.1) Thm 3.6 (). A F x R N A + x = {y + x y A} A + x F λ(a + x) = λ(a) k=1

25 3 LEBESGUE 21 Proof. λ 0 (J + x) = λ 0 (J) G R N λ 0(G + x) = λ 0(G) A + x F λ(a + x) = λ(a) A + x F (A + x) c = A c + x λ 0((A + x) G) + λ 0((A + x) c G) A + x F Thm 3.7. = λ 0({A (G + ( x))} + x) + λ 0({A c G + ( x)} + x) = λ 0(A (G + ( x))) + λ 0(A c G + ( x)) = λ 0(G). (i) A R N λ 0(A) = 0 A F λ(a) = 0 (ii) A R N A F A F Proof. (i) Q2.10 (ii) A J n F 0 A = J n Thm3.5 A F A R N \ A Thm3.5 A F Thm 3.8. (i) λ 0(A) = inf{λ(g) G G A} ( A R N ) (ii) A F, ε > 0 λ(g \ A) < ε G A (iii) A F, ε > 0 λ(a \ F ) < ε F A Proof. (i) A G λ 0(A) λ 0(G) λ 0(A) inf{λ(g) } ε > 0 λ 0(A) = λ 0 (A) (Q3.6) J n F 0 J n A λ 0(J 0 ) λ 0(A) + ε J n ( λ ) λ(jn) λ(j n ) = λ 0 (J n ) λ 0(A) + ε. J n inf{λ(g) } λ 0(A) + ε ε 0 inf{λ(g) } λ 0(A) (ii) B n = {x R N x < n} C n = A (B n \ B n 1 ) C n F Def3.4 λ 0(C n ) = λ(c n ) λ(c n ) < (i) G n C n λ(g n ) < λ(c n ) + ε2 n G = G n G G A G \ A (G n \ C n ) λ(g \ A) λ(g n \ C n ) < ε2 n = ε. (iii) A c (ii) G A c λ(g \ A c ) < ε G \ A c = G A = A \ G c A G c G c

26 3 LEBESGUE 22 Q3.1. Lem3.2 Q3.2. Prop3.3 Q3.3. (i) a R N A = {a} λ 0(A) = 0 (ii) a n R N A = {a 1, a 2,... } λ 0(A) = 0 (iii) Λ α Λ λ 0(A α ) = 0 ) λ0( α Λ A α > 0 Q3.4. (i) A R N λ 0(A) < (ii) 0 < λ 0(A) < Q3.5. K 0 = [0, 1] K 1 = [0, 1] \ (1/3, 2/3) K 2 = K 1 \ {(1/9, 2/9) (7/9, 8/9) K 3 = K 2 \ {(1/27, 2/27) (7/27, 8/27) (19/27, 20/27) (25/27, 26/27)} K n K n 1 1/3 K = n=0 K n (i) K (ii) λ 0(K) = 0 Q3.6. A R N λ 0 (A) = inf { λ 0(J n ) J n F 0, J n A } B B λ 0(A) = λ 0 (A) Q3.7. J F 0, λ 0 (J) < ε > 0 K F 0 K J λ 0 (J \ K) < ε K K λ 0 (J) = m > 0 K F 0 K J λ 0 (K) m Q3.8. J F 0 λ 0(J) = λ 0 (J) Q3.9. A J n F 0 A = J n Q3.10. A F {O n } E F λ(e) = 0 E O n A = O n \ E Q3.11. A F λ(a) = sup{λ(f ) F A, F } {F n } E F λ(e) = 0 A = E F n Q3.12. A F ε > 0 λ(r N \ F ) < ε F R N F A A

27 3 LEBESGUE 23 Q3.13. A R N ε > 0 O F F A O λ(o \ F ) < ε A F Q3.14. α > 0 T : R N R N T x = αx A R N T A = {T x x A} λ 0(T A) = α N λ 0(A) Q3.15. α > 0 T : R N R N T x = αx (x R N ) A F T A = {T x x A} T A F λ(t A) = α N λ(a) Q3.16. U : R N R N ( ) A R N UA = {Ua a A} (i) x R N, α > 0 I(x, α) = {y R N 0 < y i x i < α, i = 1,... N} λ(u(i(x, α))) = λ(u(i(0, 1))λ(I(x, α)) (ii) O UO F λ(uo) = λ(u(i(0, 1))λ(O) (iii) λ(u(i(0, 1)) = 1 (iv) A F λ(ua) = λ(a) Q3.17. N = 1 A / F A R (i) x y Q x y (ii) R/ = Λ α Λ x α (0, 1] α = {x R x x α } A = {x α α Λ} r Q (0, 1] A r = {x (0, 1] x = a + r or x = a + r 1 ( a A)} r s Q (0, 1] = A r A s =, (0, 1] = r Q (0,1] (iii) A F λ(a r ) = λ(a) (ii) Q3.18. Q [0, 1] = {q 1, q 2,..., } a n > 0 n = 1, 2,... (q n a n, q n + a n ) [0, 1] Lebesgue A r

28 Def 4.1. (i) (, E, µ) f : [, ] a [, ] [f > a] = {x f(x) > a}, [f < a] = {x f(x) < a}, [f a] = {x f(x) a}, [f a] = {x f(x) a}, [f = a] = {x f(x) = a} f : [, ] (E-) [f > a] E ( a [, ]) (ii) = R N F Lebesgue F- Lebesgue (, E, µ) Prop 4.2. f : [, ] (i) (a) f E-(b) [f a] E ( a [, ]) (c) [f < a] E ( a [, ]) (d) [f a] E ( a [, ]) (ii) f [f = a] E ( a [, ]) (iii) f, g : [, ] [f > g] := {x f(x) > g(x)} E [f g] := {x f(x) g(x)} E [f = g] := {x f(x) = g(x)} E Proof. (i) [f a] = \ [f > a], [f < a] = [f a] = \ [f < a], [f > a] = [f a (1/n)], [f a + (1/n)]. (a) (b) (c) (d) (a) (ii) [f = a] = [f a] [f a] (i) (iii) [f > g] = q Q [f > q] [g < q] (i) [f > g] E [f g] E f, g [g f] E [f = g] E Thm 4.3. (i) f : [, ] A E f A (x) = f(x) (x A) = 0 (x / A) f A (ii) f n : [, ] (n = 1, 2,... ) sup n N f n (x) inf n N f n (x) lim sup n N f n (x) lim inf n N f n (x)

29 4 25 (iii) f n : [, ] (n = 1, 2,... ) lim f(x) = f n(x), (lim sup f n (x) = lim inf f n(x) ), 0, () (iv) f, g [ f = ] = [ g = ] = α, β R αf + βg ( : ) (v) fg fg Proof. (i) a < 0 [f A > a] = A c [f > a] E a > 0 [f A [ > a] = [f > a] \] A E (ii) sup n N f n > a = [f n > a] sup n N f n Def 4.4. f : R a 1,..., a n R A 1,..., A n E f(x) = n k=1 a k Ak (x) ( x ) B (x) = 1 (x B) = 0 (x / B) CF Thm 4.5. (i) g CF (ii) f : [0, ] {f n } CF 0 f n (x) f n+1 (x) lim f n (x) = f(x) ( x R N ) Proof. (i) (ii) E(n, k) = [k2 n f < (k + 1)2 n ] f n (x) = k2 n E(n,k) (x) + n [f n] n2 n k=0 f n CF 0 f n f n+1 f n (x) f(x) Lebesgue ; Lebesgue (R N, F, λ) Thm 4.6 (Egorov ). f n : R N R Lebesgue E F λ(e) < x E lim f n (x) = f(x) ε > 0 F E ; (a) λ(e \ F ) < ε (b) {f n } F f Proof. ε > 0 p, n N E p,n = {x E f k (x) f(x) < 2 p } k=n

30 4 26 f n (x) f(x) ( x E) E p,n E p,n+1 E(n ) λ(e p,n ) λ(e) {n(p)} N λ(e p,n(p) ) > λ(e) ε2 p 1 H p = E \ E p,n(p), H = E \ p=1 H p H, H p E ( ) λ(e \ H) = λ H p λ(h p ) ε2 p 1 = ε/2. p=1 p=1 x H p N x / H p H p x E p,n(p) ( p) f n (x) f(x) < 2 p ( n n(p), p N) sup f n (x) f(x) 0 (n ). x H Thm.3.8(ii) F H λ(h \ F ) < ε/2 F Thm 4.7 (Lusin ). f : R N R Lebesgue ε > 0 K λ(r N \ K) < ε K f Proof. f 0 Thm4.5 0 f n f n+1 f f n CF ε > 0 Q2.12 λ(r N \ F n ) < ε2 n 1 F n R N F n f n F = F n F F f n λ(r N \ F ) p=1 λ(r N \ F n ) < ε/2. L k = F {n 1 x < k} Thm4.6 K k L k λ(l k \K k ) < ε2 k 1 K k {f n } K = K n K n λ(r N \ K) = λ(r N \ F ) + λ(f \ K) = λ(r N \ F ) + + λ(l k \ K k ) < ε K ( ) (, E, µ) R N Lebesgue (R N, F, λ) k=1

31 4 27 Q4.1. (i) f : (0, ) g(x) = 1/f(x) (ii) f : [, ) (a) f (b) G R G f = {x f(x) G} (c) F R F f = {x f(x) F } (iii) f : R g : R R g f : R Q4.2. f n : [, ] (n = 1, 2,... ) inf n N f n (x) lim sup n N f n (x) lim inf n N f n (x) Q4.3. f n : [, ] (n = 1, 2,... ) {x lim f n (x) } { lim f n (x), (lim f n (x) ), f(x) = 0, () lim f n (x) ± Q4.4. (i) f α R αf (ii) f, g αf + βg (iii) f n : [0, ] f n Q4.5. f, g : R (i) (f ± g) 2 (ii) fg Q4.6. f CF a 1,..., a n R A 1,..., A n F a i a j A i A j = (i j) n j=1 A j = f = n j=1 a j Aj f CF Q4.7. (i) f : R N R Lebesgue (ii) f : R N R a (a R N ) a f(x) = lim h 0 {f(x + ha) f(x)}/h ( : ) Q4.8. (i) g : R N [, ] Lebesgue f : R N [, ] E f = {x R N f(x) g(x)} λ 0(E f ) = 0 f (ii) Q2.11 (, E, µ) (i)

32 4 28 Q4.9. O = {G R N } E = σ[o] (Q2.2 ) (i) E F (ii) µ(a) = λ(a) A E (R N, E, µ) (iii) Lebesgue f : R N R E- g : R N R [f g] F λ([f g]) = 0 Q4.10. f : R N R, CF ε > 0 λ(r N \ F ) < ε F R N F f Q4.11. K k R N K k {k 1 x < k} K = k=1 K k (i) K (ii) f : K R K k K Q4.12. A R N, F λ(r N \ E) = 0 G n G n+1 A E = {x lim Gn (x) = A (x)} Q4.13. A R N, F λ(r N \ E) = 0 J n F 0 E = {x lim Jn (x) = A (x)} Q4.14. (i) F R N f : R N R F g(x) = f(x) (x F ) = 0 (x / F ) g (ii) F n F n+1 R N λ(f c n) < 1/n f : R N R F n f (Lusin ) Q4.15. F R N F = λ(f ) > 0 (i) x F x n / F ε n > 0 x n x B(x n, ε n ) F = B(y, r) = {z R N y z < r} (ii) E F λ(e) = 0 y R N r > 0 B(y, r) \ E (iii) E F λ(e) = 0 F R N \ E (Lusin ε > 0 ) Q4.16. N = 1 K n [0, 1] K 1 = [0, 1]\( , ) K n = 2 n j=1 [an j, bn j ] (bn j < an j+1 ) K n+1 = 2 j=1{ n [a n j, b n j ] \ ( an j +bn j , an j +bn j n ) } F = n+1 K n F F = λ(f ) = 1/2

33 Def 5.1. (, E, µ) (i) f 0 f CF, f(x) = n a j Aj (x), j=1 x (a j 0, A j E, j = 1,..., n, A i A j = (i j)) f(x)µ(dx) (x fdµ ) n f(x)µ(dx) = a j µ(a j ). j=1 fdµ < f (µ-) (ii) f : [0, ] f(x)µ(dx) ( fdµ ) { f(x)µ(dx) = sup g(x)µ(dx) }. g CF, 0 g f fdµ < f (µ-) (iii) f : [, ] f + (x) = max{0, f(x)} f (x) = max{0, f(x)} f ± dµ < f (µ-) f(x)µ(dx) ( fdµ ) f(x)µ(dx) = f + (x)µ(dx) f (x)µ(dx). (iv) E F f : [, ] (a) f 0 (b) f(x)µ(dx) = f(x) E (x)µ(dx) E Def 5.2. x P (x) E E µ(e) = 0 x / E P (x) P P a.e. Thm 5.3. (i) () f, g : [, ] a, b R a + by (af + bg)dµ = a fdµ + b gdµ. (5.1)

34 5 30 A B = A, B E fdµ = fdµ + A B A B fdµ. (ii) () f, g : [, ] f g a.e. fdµ gdµ. f = g a.e. f, g fdµ gdµ f g dµ. (iii) f f < a.e. Lem 5.4. (i) f 0 f CF fdµ f (ii) f, g CF a, b 0 af + bg (5.1) (iii) f, g CF f g a.e. fdµ gdµ f = g a.e. (iv) f = n j=1 a j Aj CF n j=1 { a j µ(a j ) = sup gdµ }. g CF, 0 g f Proof. (i) f = n j=1 a j Aj = m k=1 b k Bk (a j, b k R A j, B k E) n j=1 A j = m k=1 B k = A j B k C 1,..., C p E {C 1,..., C p } = { D j {A j, A c } j}, 1 j n, D 1 D n+m D j {B j, Bj c }, n + 1 j n + m C i C j = (i j) j, k i(j, 1) < < i(j, s(j)) ĩ(k, 1) < < ĩ(k, r(k)) A j = s(j) l=1 s(j) C i(j,l), B k = r(k) l=1 Aj = Ci(j,l), Bk = l=1 Cĩ(k,l) r(k) Cĩ(k,l) l=1

35 5 31 n j=1 a j Aj = m k=1 b k Bk j: l s.t. i(j,l)=q a j = µ n a j µ(a j ) = j=1 = n j=1 p ( q=1 s(j) a j l=1 µ(c i(j,l) ) = (k,l):ĩ(k,l)=q k: l s.t. ĩ(k,l)=q p ( q=1 b k )µ(c q ) = b k, (j,l):i(j,l)=q m k=1 r(k) b k l=1 1 q p a j )µ(c q ) µ(cĩ(k,l) ) = m b k µ(b j ). k=1 (ii) f, g CF A i A j n i=1 A i = A 1,..., A n F = f = n a i Ai, g = i=1 n b i Ai (iii) (ii) f, g f g a.e. a i < b i i µ(a i ) = 0 fdµ = n a i µ(a i ) = i=1 i=1 a i µ(a i ) b i µ(a i ) i;a i b i i;a i b i n = b i µ(a i ) = i=1 gdµ. a i b i i a i = b i f = g (iv) (iii) Lem 5.5. (i) f n, f CF f n, f 0 f n f n+1 lim f n f fdµ lim f ndµ (ii) f n, g n CF f n, g n 0 f n f n+1 g n g n+1 lim f n = lim g n lim f ndµ = lim g ndµ (iii) f n CF f n 0 f n f n+1 f = lim f n fdµ = lim f ndµ (iv) f, g : [0, ] a, b 0 (5.1) (v) f, g f g a.e. fdµ gdµ f = g a.e.

36 5 32 (vi) f f dµ < f g g f g Proof. (i) f = m k=1 a k Ak (A i A j = ) ε > 0 A k,n = {x A k f n (x) (1 ε)a k } A k,n A k,n+1 A k (n ) f n (1 ε) m k=1 a k Ak,n Lem 5.4 m a k µ(a k,n ) lim f n dµ (1 ε) lim inf = (1 ε) k=1 m a k µ(a k ) = (1 ε) k=1 fdµ. ε 0 (ii) lim f n g m ( m). (i) lim f ndµ g mdµ m lim f ndµ lim g ndµ (iii) f ndµ fdµ g n CF 0 g n f lim g ndµ = fdµ h n = max{f n, g 1,..., g n } h n CF, 0 h n h n+1 f (ii) 5.4 lim f n dµ = lim h n dµ lim g n dµ = fdµ. (iv) D n,k (f) = {x k2 n f(x) < (k + 1)2 n } E n (f) = {x f(x) n} f n = n2 n 1 k=0 k2 n Dn,k (f) + n En(f), g n = n2 n 1 k=0 k2 n Dn,k (g) + n En(g). (a) f n, g n CF, 0 (b) f n f n+1 f g n g n+1 g (c) af n + bg n CF, 0 (d) af n + bg n af n+1 + bg n+1 af + bg (iii) (af + bg)dµ = lim (af n + bg n )dµ = lim a f n dµ + b g n dµ = a fdµ + b gdµ. (v) (iv) f n, g n f g a.e. f n g n a.e. Lem 5.4 f ndµ g ndµ n fdµ gdµ A n = {f g 1/n} f g (1/n) An 0 = (f g)dµ 1 n A n dµ = 1 n µ(a n). µ(a n ) = 0 ( ) 0 µ A n µ(a n ) = 0.

37 5 33 µ ( {f g > 0} ) = 0. f g a.e. f g a.e. f = g a.e. (vi) f = f + + f (iv) (v) Lem 5.6. (i) f, g a, b R af + bg (5.1) (ii) f, g f g a.e. fdµ gdµ. f = g a.e. (iii) f f < a.e. Proof. (ii) (iii) (ii) f g 0 a.e. (i) fdµ gdµ = (f g)dµ 0. Lem 5.5(v) (iii) E n = {x f(x) n} f n En ( n) (ii) Lem µ(e n ) = En dµ ( f /n)dµ 0 (n ). ( µ({ f = }) = µ E n ) = lim µ(e n) = 0. f < a.e. (i) af + bg a f + b g 5.5(iv,vi) a > 0 (af) ± = af ± a 0 (af) ± = a f afdµ = a fdµ. (f + g) + (f + g) = f + g = f + f + g + g (f + g) + f + g = (f + g) + f + + g +. Lem 5.5 (f + g) + dµ + f dµ + g dµ = {(f + g) + + f + g }dµ = {(f + g) + f + + g + }dµ = (f + g) dµ + f + dµ + g + dµ.

38 5 34 (f + g)dµ = (f + g) + dµ (f + g) dµ = f + dµ + g + dµ f dµ g dµ = fdµ + gdµ. Lebesgue Lebesgue (R N, F, λ) Lebesgue Lebesgue Thm 5.7. f : R N [, ] ε > 0 g : R N R sup{ x : g(x) 0} < ( ) f dλ g dλ f g dλ < ε. R N R R N N Proof. f = E E F E n = {x E : x n} λ(e) < λ(e \ E n ) 0 n N f En dλ = λ(e \ E n ) < ε/2. R N Theorem 3.8 F E n G E n λ(g \ F ) < ε/4 F E n F α = dist(f, R N \ G) > 0 () g(x) = max{1 (2dist(x, F )/α), 0} g : R N R 0 g 1 g(x) = 1 (x F ) = 0 (x / G) En g 2 G\F R N En g dλ 2λ(E \ E n ) < ε/2. f g dλ R N f En dλ + R N En g dλ < ε. R N f 0 f n = k=0 k2 n [k2 n f<(k+1)2 n ] 0 f n f n+1 f Lemma 5.5 f n dλ fdλ. R N R N n 0 (f f n ) dλ < ε/2. R N

39 5 35 g k : R N R ε g k [k2 n f<(k+1)2 n ] dλ < R 4n max{k, 1} N g = n2 n k=0 k2 n g k n2 n f n g dλ 2 n ε R 4n (n2n + 1)2 n ε 4n < ε/2. N k=0 f g dλ f f n dλ + f n g dλ < ε. R N R N R N f ; f = f + f Thm 5.8. f : R N [, ] y R N f y (x) = f(x + y) f y f y dλ = f dλ. R N R N ( ) ± Proof. fy = f ± ( + y) Lebesgue f 0 E(n, k) = [k2 n f < (k +1)2 n ] f n = k=0 k2 n E(n,k) 0 (f n ) y (f n+1 ) y f( +y) (f n ) y = k=0 k2 n E(n,k)+y (f n ) y CF Lem 5.5 Theorem 3.6 f y dλ = lim (f n ) y dλ = lim R N R N = lim n2 n k=0 n2 n k=0 k2 n λ(e(n, k)) = lim k2 n λ(e(n, k) + y) f n dλ = fdλ. R N R N (, E, µ) R N Lebesgue Q5.1. = N E = 2 N a n 0 (n N) µ(a) = n A a n (i) (, E, µ) (ii) f : R f f(n)a n (iii) f : R fdµ = f(n)a n

40 5 36 Q5.2. a n > 0 A n E (n N) f(x) = a n An (x) (i) 0 f(x) ( x ) (ii) f n = n j=1 a j Aj f ndµ (iii) fdµ = a nµ(a n ) Q5.3. f : [, ] a > 2 µ([ f n]) n a ( n N) µ([ f 1]) < (i) g = n [n 1 f <n] g (ii) f g a.e. f Q5.4. f : [0, ] k N µ([f n]) n k ( n N) a > 0 exp[ af] Q5.5. f : [0, ] α > 0 µ([ f n]) e αn ( n N) a > 0 exp[ a f 2 ] Q5.6. K R N F R N K F = dist(k, F ) > 0 K, F K F = dist(k, F ) = 0 Q5.7. f : R N R sup x R N (1 + x ) N+1 f(x) < f Lebesgue Q5.8. N = 1 f : R [, ] f = n [n,n+(1/n 3 )) (i) f Lebesgue (ii) lim sup x f(x) = Q5.9. α > 0 T : R N R N T x = αx f : R N [, ] (i) g(x) = f(t (x)) g (ii) R N gdλ = α N R N fdλ Q5.10. f : R N [, ] (i) ε > 0 g C(R N ) sup f(x + y) g(x + y) λ(dx) < ε. y R N R N (ii) lim y 0 R N f(x + y) f(x) λ(dx) = 0.

41 5 37 Q5.11. U : R N R N f : R N [, ] f U (x) = f(ux) f U dλ = fdλ. R N R N Q5.12. N = 1 Lebesgue f : R [0, ) (x y f(x) f(y)) g(x) = f(x) (x 0) = 0 (x < 0) h = k=0 f(k) [k,k+1) (i) n=0 f(n) < h, g (ii) f(n) = gdλ = R (iii) α > 0 f(x) = (1 + x 2 ) α/2 (x 0) = 0 (x < 0) f(x)dλ(x) < α > 1 R Q5.13. N = 1 g(x) = {(3π/4)+2nπ} (x [(π/4)+2nπ, (3π/4)+ 2nπ), n = 0, 1,... ) = 0 () (i) R gdλ = (ii) f(x) = (sin x)/x (x > 0) = 0 (x 0)

42 Thm 6.1 ( ). (, E, µ) f, f n : [, ] 0 f n f n+1 f a.e. f n dµ = fdµ. = lim Proof. (i) E E µ( \ E) = 0 0 f n (x) f n+1 (x) f(x) ( x E) f 0 = 0 (f m f m 1 ) E 0 0 g m,n g m,n+1 (f m f m 1 ) E (n ) g m,n CF, 0 h n = n m=1 g m,n h n CF, 0 h n h n+1 k g m,n h n m=1 n (f m f m 1 ) E f n E (k n). m=1 h n f E Lemma 5.5(iii) fdµ = f E dµ = lim h n dµ lim inf f n E dµ f n E dµ f E dµ = lim sup fdµ. Thm 6.2 (Fatou ). f n : [, ] f n 0 a.e. lim inf ndµ lim inf f n dµ. Proof. g n = inf{f m m n} 0 g n g n+1 lim inf k f k a.e. lim inf f ndµ = lim g n dµ. g n f n g ndµ f ndµ lim inf f ndµ lim inf f n dµ. Thm 6.3 (Lebesgue ). f n, Φ 0 f n Φ a.s. f n f a.s. f f n dµ = fdµ. lim

43 6 39 Proof. f Φ a.e. f 0 (±f n ) + Φ ±f + Φ a.e. Fatou ± fdµ = (±f + Φ)dµ Φdµ lim inf (±f n + Φ)dµ Φdµ = lim inf ± f n dµ. Cor 6.4 (). f n : [, ] n = 1, 2,... (a) f n 0 a.e., (b) f n f n dµ = f n dµ. f n dµ <. Proof. (a) f n (= ) (b) (a) f n dµ = f n dµ <. Thm 5.3(iii) f n < a.e. f n m f n f n Lebesgue Cor 6.5 (Lebesgue ()). d N U R d f : U R (a) t U f(t, ) (b) x U t f(t, x) R N (c) Φ t U f(t, ) Φ a.e. U t R N f(t, x) dλ(x) Proof. t 0 U t n, s n t 0 lim f(s n, x)µ(dx) = lim inf lim f(t n, x)µ(dx) = lim sup t t 0 t t 0 f(t, x)µ(dx), f(t, x)µ(dx)

44 6 40 Lebesgue f(s n, x)µ(dx) = lim lim inf t t 0 R N f(t 0, x)µ(dx) = lim f(t n, x)µ(dx) f(t, x)µ(dx) = lim sup f(t, x)µ(dx) = f(t 0, x)µ(dx). t t 0 R N Cor 6.6 (Lebesgue ()). a < b f : (a, b) R (a) t (a, b) f(t, ) : R (b) x (a, b) t f(t, x) R f (c) Φ : [0, ] sup t (a,b) t (t, ) Φ a.e. t (a, b) f (t, ) (a, b) t t f(t, x)µ(dx) Proof. t f(t, x)µ(dx) = t f(t, x) f (t, x)µ(dx). (6.1) t f(t, x) = lim n{f(t + (1/n), x) f(t, x)}. t f(t, ) (c) t t 0 (a, b) f(t + t 0, x) f(t 0, x), t (a t 0, b t 0 ), t 0, h(t, x) = t t f(t 0, x), t = 0, (a t 0, b t 0 ) t h(t, x) Taylor t 0 h(t, x) = 1 t+t0 f t (s, x)ds t Φ(x), a.e. x. t 0 h(t, ) Φ a.e. Corollary 6.5 { } 1 f(t + t 0, x)µ(dx) f(t 0, x)µ(dx) = h(t, x)µ(dx) t f h(t 0, x)µ(dx) = t (t 0, x)µ(dx) (t 0).

45 6 41 t f(t, x)µ(dx) (6.1) Lebesgue Thm 6.7 (Riemann ). < a i < b i < i = 1,..., ND = N i=1 [a i, b i ] f : D R Riemann f D Lebesgue b1 a 1 bn a N f(x 1,..., x N )dx 1... dx N = Riemann D ( ) fdλ = f D dλ R N. Proof. f D sup x D f(x) D f D Lebesgue n N k = (k 1,..., k N ) {0, 1,..., } N N [ I(n, k) = ai + k i (b i a i )2 n, a i + (k i + 1)(b i a i )2 n), i=1 m n,k = inf{f(x) : x I(n, k)}, h n (x) = 2 n 1 k=0 m n,k I(n,k), H n (x) = M n,k = sup{f(x) : x I(n, k)}, 2 n 1 k=0 M n,k I(n,k), h n, H n CF f Riemann Darboux R N h n dλ = R N H n dλ = 2 n 1 k=0 b1 bn a 1 2 n 1 k=0 b1 bn a 1 m n,k 2 nn N i=1 (b i a i ) a N f(x 1,..., x N )dx 1... dx N, M n,k 2 nn N i=1 (b i a i ) a N f(x 1,..., x N )dx 1... dx N. (6.2) h n h n+1 H n H n+1 h = lim h n H = lim H n h, H Lebesgue h n, H n sup x D f(x) D Lebesgue (6.2) b1 bn h dλ = lim h n dλ = f(x 1,..., x N )dx 1... dx N R N R N a 1 a N = lim H n dλ = H dλ. R N R N h n f D H n h f D H h = f D = H a.e. (Thm 5.3) b1 bn f(x 1,..., x N )dx 1... dx N = f D dλ. a 1 a N R N

46 6 42 Thm 6.8. N = 1 < a < b < f : [a, b] R f : R R f(x) = f(x) (x [a, b]) = 0 (x / [a, b]) (i) f L- f Lebesgue (ii) f L- f Lebesgue f dλ = f dλ R Proof. (1) f : [a, b] R L- {f n } E [a, b] λ 0(E) = 0 lim f n (x) = f(x) (x / E) f n f a.e. lim sup f n = f a.e. f n CF f Lebesgue (2) f L- f f { sup x f(x) } Lebesgue f Lebesgue M = sup n N,x f n (x) < {f n } f n f a.e. b a f n(x)dx fdλ b a f n(x)dx = R f ndλ f n M Lebesgue sup n N f dλ = lim b f min{n, f }dλ <, fdλ = lim a f n (x)dx = lim f n dλ = f dλ. R R f dλ = lim min{n, f }dλ R R = lim min{n, f }dλ = lim R max{ n, min{f, n}} dλ. min{n, f }dλ <. f Lebesgue max{ n, min{f, n}} f max{ n, min{f, n}} f

47 6 43 Lebesgue f dλ = lim max{ n, min{f, n}} dλ = lim = lim (3) f Lebesgue R R max{ n, min{f, n}} dλ max{ n, min{f, n}} dλ = (3-1) Lebesgue h, g h g d(h, g) = 1 + h g dλ R R fdλ. 0 d(h, g) <, d(h, g) = d(g, h), d(h, g) d(h, k) + d(k, g) a + b 1 + a + b a 1 + a + b, a, b R. 1 + b (3-2) Lebesgue h n, h d(h n, h) 0 {h nj } j=1 lim j h nj = h a.e. n j < n j+1 d(h nj, h) < 4 j A j = {x [a, b] h(x) h nj (x) 2 j } [0, ) y y 1+y 2 j j A j h h n j 1 + h h nj. λ(a j ) j 2 j d(h nj, h) 2 (j 1). A = m=1 j=m A j ( ) 0 λ(a) λ A j λ(a j ) 0 (m ) j=m λ(a) = 0x [a, b]\a m N j m h nj (x) h(x) < 2 j lim j h nj (x) = h(x) lim j h nj = h a.e. (3-3) j=m y 1 Lebesgue 1 + y lim d(max{ n, min{f, n}}, f) = 0

48 6 44 min{f, n} Lebesgue Thm 5.7 g n,m lim m R max{ n, min{f, n}} g n,m dλ = 0 y y 1 + y lim d(max{ n, min{f, n}}, g n,m) = 0. m g n,m,i (x) = g n,m ([2 i x]2 i ) lim i g n,m,i = g n,m y 1 Lebesgue 1 + y lim d(g n,m,i, g n,m ) = 0. i {g n,m,i n, m, i = 1, 2,... } {g n } lim d(g n, f) = 0 (3-4) (3-3) g j (3-2) {g nj } g nj f a.e. g nj [a, b] f L- (4) f Lebesgue (3) f L- f max{n, f } L-(2) sup n N max{n, f }dλ = sup n N f L- R max{n, f }dλ R f dλ <. (, E, µ) R N Lebesgue Q6.1. U R n f : U R (a) t U f(t, ) : R (b) x U t f(t, x) R C 1 (c) Φ : [0, ] t U f (t, ) Φ a.e. t = (t 1,..., t n ) t i

49 6 45 i = 1,..., n t U f/ t i (t, ) f f(t, x)µ(dx) = (t, x)µ(dx). t i t i Q6.2. a 1, a 2,... a n < = N E = 2 N µ(a) = a n A (n) (, E, µ) f : [, ] fdµ = a n f(n) Lebesgue h(z) = a n z n, z D = {z C z < 1} h (z) = a n nz n 1, z D. Q6.3. a n 0 a n < b n R b n a n ( n) b n Lebesgue Q6.4. E E µ(e) < f n : [, ] f n f n+1 f inf{f 1 (x) x E} > lim E f ndµ = fdµ E Q6.5. E E µ(e) < f n, f : R sup{ f n (x) x E, n N} < f n f a.e. f n dµ = fdµ lim E Q6.6. E E µ(e) < f n : R E f f E lim E f ndµ = E fdµ Q6.7. (i) f n : [0, ] n = 1, 2,... f n f n+1 f lim f ndµ = fdµ (ii) f n E

50 6 46 Q6.8. f n, f : [, ] sup { f n dµ n N } < f n f a.e. f Q6.9. f n, f : [, ] sup { f n (x) x, n N } < f n f a.e. g : [, ] lim f ngdµ = fgdµ Q6.10. g : [, ] f n, f f n (x) f(x) ( x ) f n [ fn >M]gdµ 0 (M ) sup n N (M) (i) M > 0 lim f n f (M) gdλ = 0 M, g(x) M, g (M) (x) = g(x), g(x) < M, M, g(x) M (ii) [ f > M] m=n [ f m > M] f [ f >M] lim inf f n [ fn >M] (iii) lim M f [ f >M]gdµ = 0 (iv) fg (v) lim f f n gdµ = 0 (vi) lim f ngdµ = fgdµ Q6.11. f : R N R Lebesgue f(x) x 2 (i) g(x) = e x 2 g (ii) lim e nf(x) (iii) lim R N e nf(x) λ(dx) = 0 Q6.12. (i) f : R R f 0 Riemann f(x)dx 0 f (0, ) Lebesgue (0, ) fdλ = 0 f(x)dx (ii) f 0

51 6 47 Q6.13. f : R N R k N sup{ f(x) /(1 + x ) k x R n } < I(y) = R N f(x + y)e x λ(dx) (i) (1 + x + y ) k 2 k (1 + x ) k + 2 k y k (ii) sup{ f(x + y) /(1 + x ) k x R N, y n} < (iii) g(x) = (1 + x ) k e x g (iv) I : R N R Q6.14. N = 1 f : R R Lebesgue n = 0, 1, 2,... φ n (x) = x n f(x) Lebesgue (i) y 0 x e yx f(x) [0, ) (x) Lebesgue (ii) y 0 L(y) = R e yx f(x) [0, ) (x)λ(dx) L [0, ) C Q6.15. N = 1 f : R C u : R R v : R R f Lebesgue R fdλ = R udλ+i R vdλ (i 2 = 1) g : R R a R x e ax g(x) (i) z C x e zx g(x) (ii) F (z) = R ezx g(x)λ(dx) (z C) F (iii) F (z) = n=0 z n x n g(x)λ(dx). n! R Q6.16. (Gamma ) z C + = {z C Re z > 0} Γ(z) = Riemann 0 e x x z 1 dx (i) Riemann (ii) C + z Γ(z) ( 1) k 1 (iii) z 0, 1, 2,... Γ(z) = k! z + k + k=0 1 e x x z 1 dx (iv) Γ(n + 1) = n! Γ(n ) = (2n)! π (n = 0, 1, 2,... ) 2 2n n!

52 6 48 Q6.17. N = 1 f : R R n = 0, 1,... φ n (x) = x n f(x) ˇf(ξ) = e iξx f(x)λ(dx) (i 2 = 1) (i) ˇf C (R) (ii) ˇf (n) (ξ) ( ˇf n ) ˇ φ n (iii) a 0,, a n C R n ξ j a j ˇφj (ξ) = 0, j=0 ξ R ˇf (n) Q6.18. t R I(t) = R eitx e x2 /2 λ(dx) (i 2 = 1) (i) I(t) (ii) I C (iii) I I Q6.19. N = 1 g : R R C n 0 k n δ > 0 sup y δ g (k) ( + y) : R R g (k) k f I(y) = f(x + y)g(x)λ(dx) R (i) I(y) = f(x)g(x y)λ(dx) R (ii) I(y) C n Q6.20. N = 1 x R t > 0 g(x, t) = (2πt) 1/2 exp[ x 2 /(2t)] (i) g t = 1 2 g 2 x 2 (ii) t, R > 0, n = 0, 1, 2,... C > 0 y C x n g(x y, t) g(x y, 2t) ( x R) (iii) f : R R u(x, t) = f(x y)g(y, t)λ(dy) R u(x, t) R (0, ) C u t = 1 2 u 2 x 2 Q6.21. u (i) u(x, t) = R f(x t1/2 y)g(y, 1)λ(dy) (ii) lim t 0 u(x, t) = f(x)

53 7 HÖLDER MINKOWSKI LP Hölder Minkowski L p Def 7.1. (, E, µ) p > 0 { } L p (µ) = f : [, ] f f(x) p µ(dx) < Thm 7.2 (Hölder ). p, q > 1 1 p + 1 q = 1 f L p (µ) g L q (µ) fg dµ { fg dµ } 1/p { 1/q f p dµ g dµ} q. Lem 7.3 (Young ). a, b 0 p, q > 1 1 p + 1 q = 1 Proof. ab ap p + bq q. ab = 0 ab 0 ϕ(t) = tp p + 1 q t (t 0) ϕ (t) = t p 1 1 ϕ t = 1 0 t tp p + 1 q. t = ab q/p b q q (q/p) = 1 Proof (Thm 7.2). Thm 5.3 n f = f p dµ n g = g p dµ n f = 0 f = 0 a.e. n g = 0 g = 0 a.e. n f n g 0 a = f(x) /n 1/p f b = g(x) /n 1/q g Young f(x)g(x) 1 n 1/p f n 1/q p g f(x) p + 1 q n f g(x) q n g. 1 n 1/p f n 1/q g fg dµ 1 p f p dµ n f + 1 q g q dµ n g = 1 p + 1 q = 1. n 1/p f n 1/q g

54 7 HÖLDER MINKOWSKI LP 50 Thm 7.4 (Minkowski ). p 1 { 1/p { f + g dµ} p 1/p { 1/p f dµ} p + g dµ} p f, g L p (µ). Proof. p = 1 f + g f + g p > 1 f + g p dµ = 0 f + g p dµ > 0 (a + b) p 2 p (a p + b p ) (a, b 0) 0 < f + g p dµ ( f + g ) p dµ <. q = p/(p 1) Hölder ( f + g ) p dµ = f ( f + g ) p 1 dµ + { } 1/p { f p dµ { + } 1/p { g p dµ ( f + g ) (p 1)q dµ g ( f + g ) p 1 dµ } 1/q 1/q ( f + g ) dµ} (p 1)q. (p 1)q = p f + g p ( f + g ) p { ( f + g ) p dµ } 1/p 0 { 1/p { f + g dµ} p { ( f + g ) p dµ } 1/p 1/p { 1/p f dµ} p + g dµ} p. Thm 7.5 (). p 1 f n L p (µ) (n = 1, 2,... ) f n f m p dµ = 0 (7.1) lim n,m lim f n f p dµ = 0 (7.2) f L p (µ) Proof. { f L p (µ) (7.2) Minkowski 1/p { f n f m dµ} p f n f p dµ} 1/p + { f m f p dµ} 1/p 0 (n, m )

55 7 HÖLDER MINKOWSKI LP 51 (7.1) (7.1) f n(1) < < n(k) < n(k + 1) <... f n f n(k) p dµ < 2 pk, n n(k) (7.3) Minkowski { ( m p 1/p m { f n(k+1) f n(k) ) dµ} f n(k+1) f n(k) p dµ k=1 k=1 () m (7.3) { ( k=1 p } 1/p f n(k+1) f n(k) ) dµ { k=1 f n(k+1) f n(k) p dµ} 1/p } 1/p 2 k < A E µ(a) = 0 k=1 f n(k+1)(x) f n(k) (x) < ( x / A) f n(1) (x) + (f n(k+1) (x) f n(k) (x)), x / A, f(x) = k=1 0, x A (7.2) f n(k) (x) f(x) ( x / A) ( f f n(k) p j=k+1 ) p f n(j+1) f n(j) k=1 ( p f n(j+1) f n(j) ) 2 p { f n(1) p + f p } j=1 Lebesgue f n(k) f p dµ = 0. (7.4) lim k Minkowski (7.3) n n(k) { 1/p { f n f dµ} p { 2 k + 1/p { f n f n(k) dµ} p + f n(k) f p dµ} 1/p. } 1/p f n(k) f p dµ

56 7 HÖLDER MINKOWSKI LP 52 { 1/p { 1/p 0 lim sup f n f dµ} p 2 k + f n(k) f dµ} p. k (7.4) { 1/p lim sup f n f dµ} p = 0. lim f n f p dµ = 0. (, E, µ) R N Lebesgue Q7.1. (i) f, g L 2 (µ) fg (ii) µ() < f L 2 (µ) f (iii) f L 2 (µ) µ f Q7.2. p 1,..., p n > 1 n i=1 (1/p i) = 1 f 1,..., f n : [, ] f 1... f n dµ n ( i=1 ) 1/pi f i pi dµ Q7.3. p 1 f n L p (µ) S n = n i=1 f i (i) S n L p (µ) (ii) i=1( f i p dµ) 1/p < S L p (µ) S n S p dµ 0 Q7.4. (N, 2 N, µ) µ(a) = #A = A Hölder Minkowski a n, b n R n = 1, 2,...

57 7 HÖLDER MINKOWSKI LP 53 ( ) 1/p ( ) 1/q a n b n a n p b n q ( ) 1/r ( ) 1/r ( ) 1/r a n + b n r a n r + b n r. p, q > 1 (1/p) + (1/q) = 1 r 1 Q7.5. C 0 (R N ) f : R N R A > 0 x > A f(x) = 0 C 0 (R N ) L p (λ) f L p (λ) ε > 0 R N f g p dλ < ε g C 0 (R N ) Q7.6. p, q > 1 1 p + 1 q = 1 f Lp (λ) g L q (λ) R N y f(x + y)g(x)λ(dx) R N Q7.7. H L 2 (µ) (a) a, b R f, g H af + bg H (b) f n H f n f 2 dµ 0 f H f L 2 c > 0 { } c 2 = inf f g 2 dµ g H (i) f n H lim f n f 2 dµ = c 2 (ii) (i) f n ( lim n,m lim n,m lim n,m f f 2 n + f m 1/2 2 dµ) = c, (f f n )(f f m )dµ = c 2, (f n f m ) 2 dµ = 0 (iii) f H f f 2 dµ = c 2 (iv) (f f)gdµ = 0 ( g H)

58 7 HÖLDER MINKOWSKI LP 54 Q7.8. f L 2 (µ) { 1/2 { f dµ} 2 = sup fgdµ g L2 (µ), } g 2 dµ 1. Q7.9. < a < b < f : R R Lebesgue [a, b] f : [a, b] R M = sup x f(x) (i) f L p (λ) ( p 1) (ii) ε > 0 A ε = {x [a, b] f(x) M ε} A ε F λ(a ε ) > 0 R f p dλ (M ε) p λ(a ε ) ( ) 1/p (iii) R f p dλ M (p ) Q7.10. f n L 2 (µ) f nf m dµ = δ nm ( {f n } ) a2 n < {a n } S N = N a nf n S L 2 (µ) S N S 2 dµ 0 (N ) Sf ndµ = a n (n = 1, 2,... ) ( S a nf n ) Q7.11. {f n } f L2 (µ) Bessel ( 2 ff n dµ) f 2 dµ Q7.12. {f n } (a) f L 2 (µ) ff ndµ = 0 (n = 1, 2,... ) f = 0 a.e. (b) f L 2 (µ) ( f = ( Q7.10 ) k=1 ) ff n dµ f n. (c) f L 2 (µ) Parseval ( 2 ff n dµ) = f 2 dµ (d) f L 2 (µ) ε > 0 N N a n R N f a n f n 2dµ < ε

59 7 HÖLDER MINKOWSKI LP 55 Q7.13. (Legendre ) n = 0, 1, 2,... x [ 1, 1] [n/2] d n P n (x) = 1 2 n n! dx n (x2 1) n ( 1) k (2n 2k)! (i) P n (x) = 2 n k!(n k)!(n 2k)! xn 2k k=0 (ii) (R 1, F) µ µ(a) = λ(a [ 1, 1]) P n P m dµ = 0 (n m) Q7.14. (Hermite ) n = 0, 1, 2,... x R [n/2] H n (x) = ( 1) n e x2 dn 2 dx n e x ( 1) k n! (i) H n (x) = k!(n 2k)! (2x)n 2k k=0 (ii) (R 1, F) µ µ(a) = e x2 λ(dx) R A H n H m dµ(dx) = 0 (n m) Q7.15. (Laguerre ) α > 1 n = 0, 1, 2,... x > 0 (i) P n (x) = n k=0 L α n(x) = e x x α n! Γ(n + α + 1) Γ(k + α + 1 d n dx n ( e x x n+α) ( x) k k!(n k)! (ii) (R 1, F) µ µ(a) = A (0, ) xα e x λ(dx) R L α nl α mdµ(dx) = 0 (n m) R

60 8 FUBINI Fubini Def 8.1. (, E, µ ) (Y, E Y, µ Y ) { n E 0 = k=1 } E k F k E k E, F k E Y, k = 1,..., n, n N σ(e 0 ) E E Y σ Thm 8.2 (). k E, Y k E Y µ ( k ) < µ Y (Y k ) < ( k) k=1 k = k=1 Y k = Y E 0 µ 0 µ 0 (E F ) = µ (E)µ Y (F ), E E, F E Y ( Y, E E Y ) µ µ(e) = µ 0 (E) ( E E 0 ) µ µ µ Y Lem 8.3. B E 0 B (Y ) x = {y Y (x, y) B}, B () y = {x (x, y) B}, x, y Y (i) B (Y ) x E Y B () y (ii) x µ Y (B (Y ) x (iii) µ 0 (B) = µ Y (B (Y ) x E ) y µ (B () ) y )µ (dx) = Y µ (B () )µ Y (dy). Proof. E i E i = F i F j = (i j) E i E, F i E Y I {(i, j) 1 i, j n} B = E i F j (i,j) I 1 i n I(i) = {j (i, j) I} { B x (Y ) j I(i) = F j, x E i,, x / n i=1 E i, B (Y ) x E Y ( x) µ Y (B (Y ) x ) = n Ei (x) µ Y (F j ). i=1 x µ Y (B x (Y ) ) µ Y (B (Y ) x )µ (dx) = y j I(i) n µ (E i ) µ Y (F j ) = µ(b). i=1 j I(i)

61 8 FUBINI 57 B () y Proof(Thm 8.2). Caratheodory (Thm 2.5) A j E 0 A i A j = A = j=1 A j E 0 µ 0 (A) = µ(a j ) (8.1) µ 0 (A) < B n = A \ n j=1 A j j=1 n 1 µ 0 (A) = µ(a j ) + µ(b n ). j=1 µ 0 (B n ) 0 B n Lem 8.3 B n = (i,j) I(n) E n i F n j µ (Ej n) > 0, µ Y (Fj n ) > 0 ( j, n) ) J = {Ej n µ (Ej n) = 0} J Y = {Fj n µ Y (Fj n) = 0} ( B 0 = E Y E J ) ( E J Y E B 0 E 0 µ 0 (B 0 ) = 0 B n \B 0 B n /// µ 0 (B 1 ) = µ (E i )µ Y (F j ) Lem 8.3 µ 0 (B n ) = (i,j) I µ(e i ) <, µ(f j ) < (i, j) I(1). (8.2) µ Y ([B n ] x (Y ) )µ (dx). x [B n ] (Y x ) [B n+1 ] (Y x ) (n ) Lem 8.3 µ Y ([B n ] x ) µ Y ([B 1 ] x ) = E 1 i (x)µ Y (Fj 1 ). (8.3) (i,j) I(1) (8.2) µ Y ([B 1 ] (Y x ) ) < µ Y ([B n ] x ) 0 ( x) (8.2) (8.3) Lebesgue µ 0 (B n ) = µ Y ([B n ] (Y x ) )µ (dx) 0 (n ). )

62 8 FUBINI 58 µ 0 (A) < (8.1) µ 0 (A) = Z k = k Y k µ 0 (A Z k ) < µ 0 (A Z k ) = µ 0 (A j Z k ) µ 0 (A j ). j=0 j=0 µ 0 (A Z k ) A = n j=1 E j F j ((E i F i ) (E j F j ) = ) µ 0 (A) = n µ (E j )µ Y (F j ) j=1 j µ (E j )µ Y (F j ) = µ 0 (A Z k ) µ 0 ((E j F j ) Z k ) µ (E j k )µ Y (F j Y k ) µ (E j )µ Y (F j ) = (k ). Thm 8.4 (Fubini ()). (, E, µ ) (Y, E Y, µ Y ) k E, Y k E Y µ ( k ) < µ Y (Y k ) < ( k) k=1 k = k=1 Y k = Y E E Y - f : Y [, ] (a) (b) (F1) f (Y ) x Y f x (Y ) : Y y f(x, y) [, ] E Y - x (y)µ Y (dy) E - (F2) f y () : x f(x, y) [, ] E - y (x)µ (dx) E Y - f () y (F3) fd(µ µ Y ) = Y = Y ( ( Y ) f x (Y ) (y)µ Y (dy) µ (dx) ) f y () (x)µ (dx) µ Y (dy). (8.4) Lem 8.5. f n : Y [, ] (F1) (F3) (a) 0 f n (x, y) f n+1 (x, y) f(x, y) ( (x, y) Y ) (b) Φ : R, L 1 (µ ) Φ Y : Y R, L 1 (µ Y ) f n (x, y) Φ (x)φ Y (y) lim f n (x, y) = f(x, y) ( (x, y) Y )

63 8 FUBINI 59 f (F1) (F3) Proof. f (Y ) x (a) 0 [f n ] (Y ) x [f n+1 ] (Y ) x f (Y ) x. 0 E Y - 0 [f n ] (Y x ) dµ Y [f n+1 ] (Y x ) dµ Y Y Y Y f (Y ) x dµ Y. Y f x (Y ) dµ Y E - fd(µ µ Y ) = lim f n d(µ µ Y ) f () y Y = lim = lim (b) Y ( ( Y Y [f n ] (Y ) x ) (y)µ Y (dy) ) f x (Y ) (y)µ Y (dy) µ (dx) µ (dx). lim [f n] (Y x ) (y) = f x (Y ) (y), y Y, [f n ] (Y x ) (y) Φ (x)φ Y (y). f (Y ) x 0 E Y - Lebesgue [f n ] (Y x ) dµ Y f x (Y ) dµ Y. Y Y f x (Y ) dµ Y E - [f n ] (Y x ) dµ Y Φ (x) Φ Y dµ Y Y Y Lebesgue fd(µ µ Y ) = lim f n d(µ µ Y ) Y Y ( ) = lim [f n ] (Y x ) (y)µ Y (dy) µ (dx) Y ( ) = lim f x (Y ) (y)µ Y (dy) µ (dx). f () y Lem 8.6. M 2 Y E 0 M 2 Y Y

64 8 FUBINI 60 (a) A 1, A 2, M A 1 A 2... j=1 A j M (b) A 1, A 2, M A 1 A 2... j=1 A j M E E Y M Proof. Λ = {N 2 Y E 0 N, N (a) (b) } M 0 = N N Λ M 0 Λ M 0 M M 0 (a) M 0 σ E E Y M 0 M M = {A M 0 A c M 0 } E 0 M C D C c D c M (a) (b) M = M 0 A M 0 A c M 0 A, B M 0 A B M 0 M A = {E M 0 A E M 0 } A E 0 E 0 M A ( ) ( ) E j A = (E j A), E j A = (E j A) j=1 j=1 M A Λ M 0 M A M 0 = M A A E 0 B M 0 A B M 0 A M 0 E 0 M A M 0 = M A M 0 Lem 8.7. A E E Y f = A (F1) (F3) Proof. Z k = k Y k 0 A Zk A Zk+1 A Lem 8.5 f = A Zk (F1) (F3) k M = {A E E Y A Zk (F1) (F3) } j=1 j=1 0 A Zk (x, y) k (x) Yk (y), (x, y) Y Lem 8.5 M Lem 8.6 (a) (b) E 0 A E 0 A Z k E 0 Lem 8.3 E 0 (F1) (F3) E 0 M Lem 8.6 E E Y M Lem 8.8. Thm 8.4 Proof. f 0 f n = n2 n k=0 k2 n [k2 n f<(k+1)2 n ] 0 f n f n+1 f Lem 8.5 f f = f + f Lebesgue N = d + e F N, F d, F e λ N, λ d, λ e R N, R d, R e

65 8 FUBINI 61 Lebesgue Lebesgue Fubini Thm 8.9 (Fubini (Lebesgue )). F N - f : R N [, ] (a) (b) (FL1) f x (e) : R e y f(x, y) [, ] F d - λ d -a.e. x R d x f (e) R e x (y)λ e (dy) F d - (FL2) f y (d) : R d x f(x, y) [, ] F d - λ e -a.e. y R d y f (e) R d y (x)λ d (dx) F e - (FL3) fdλ N = R N = R d R e ( ( f x (e) R e ) (y)λ e (dy) ) f y (d) (x)λ d (dx) R d λ d (dx) λ e (dy). (8.5) Lem F d F e F N λ d λ e (A) = λ N (A) ( A F d F e ) Proof. { n F 0 = j=1 } E j F j E j F d, F j F e, n N F 0 F N, λ d λ e (A) = λ N (A), A F 0 (8.6) G R d G R e In d R d In e R e G = i=1 Id i G = j=1 Ie j G G = i,j=1 I d i I e j F N, λ N (G G ) = λ d (Ii d )λ e (Ij e ) = λ e (G G ). i,j=1 G i R d G j Re ( i=1 ) ( G i j=1 G j ) ( n = i=1 ) ( n G i j=1 G j ) F 0

66 8 FUBINI 62 k N [( ) ( )]) λ (( k, N k) N G i k i=1 ( = lim λn ( k, k) N ( = lim λd ( k, k) d = λ d (( k, k) d λ N ([( i=1 i=1 ) ( G i j=1 j=1 [( n n i=1 i=1 G j ) ( G i j=1 G n j G i ) λ e (( k, k) e G i ) λ e (( k, k) e G j )]) ( = λ d i=1 E R d λ d (E) = 0 I (n) i E i=1 I (n) i, i=1 i=1 G i )]) n i=1 ). G i G i ) λ e ( R d I (n) i < 1/n, n N, i=1 ) G i ). E [ k, k) e i=1 I (n) i [ k, k) e, i=1 I (n) i [ k, k) e < k e /n, n N, λ N (E [ k, k) e ) = 0 k λ N 0 (E R e ) = 0 B F e E B F N λ N (E B) = 0 E R e λ d (E ) = 0 A E d A E E N λ N (A E ) = 0 A F d B F e G n R d, A G n R e B λ d( G n \ A ) = 0 λ e( G n \ B ) = 0 (Thm 3.8) ( ( ) ([( ( )] ) A B G n ), λ N G n ) \(A B) = 0. G n A B F N λ N (A B) = λ N (( = λ d ( ) ( G n G n ) λ e ( G n G n )) G n ) = λ d (A)λ e (B).

67 8 FUBINI 63 (8.6) M = {A F d F e A F N, λ N (A) = λ d λ e (A)} (8.6) F 0 M A j M A j A j+1 j=1 A j MA j A j+1 j=1 A j F N A j ( k, k) N λ N (A) = λ d λ e (A) Lem 8.6 F d F e M Lem A F N f = A (FL1) (FL3) Proof. H n A H = H n E = A\H λ N (E) = 0 (Thm 3.8) A = H + E E E R N \ H n F 0 ( Lem 8.10 ) Lem 8.10 R N \H n F d F e H n F d F e H F d F e Thm 8.4 H (FL1) (FL3) () E (FL1) (FL3) G n E F = G n λ N (F ) = 0 (Thm 3.8) G n F d F e F F d F e Thm 8.4 F (FL1) (FL3) Lem 8.3 F (x, y) = F R e x E Re x Fx Re, Ey Rd Fy Rd x R d, y R e. (8.7) (y) = (x) λ N (F ) = 0 Fy Rd e λ e (Fx R )λ d (dx) = R d λ d -a.e. x R d λ e (F Re x λ d (F Rd y λ d (E Rd y d λ d (Fy R )λ e (dy) = 0 R e ) = 0 (8.7) λ d -a.e. x R d λ e (E Re x ) = 0 λ d -a.e. x R d Ex Re F d E (FL1) (FL3) Ey Rd Lem Thm 8.9 Proof. Lem 8.8 ) = 0 λ e -a.e. y R e ) = 0 λ e -a.e. y R e F e λ d -a.e. y R e (, E, µ ) (Y, E Y, µ Y ) R N Lebesgue Q8.1. f : [, ] g : Y [, ] h(x, y) = f(x)g(y) h : Y [, ] Y hd(µ µ Y ) = fdµ Y gdµ Y

68 8 FUBINI 64 Q8.2. f : [, ] g : Y [, ] µ ([f 0]), µ Y ([g 0]) > 0 h(x, y) = f(x)g(y) Y f, g µ µ Y Q8.3. f : [, ] g : Y [, ] h(x, y) = f(x) + g(y) (i) µ () < h µ µ Y f µ (ii) a Y µ Y (A) = A (a) µ Y µ () = Q8.4. f : Y [, ] µ µ Y -a.e. f 0 µ Y ({y f(x, y) 0}) > 0 x Q8.5. f : Y [, ] µ µ Y -a.e. f 0 A E, B E Y µ (A) > 0, µ Y (B) > 0 µ -a.e. x A µ Y ({y B f(x, y) < 0}) = 0 µ Y -a.e. y B µ ({x f(x, y) < 0}) = 0 Q8.6. d, e N N = d + e B(R N ) = B(R d ) B(R e ) Q8.7. N = 2d d = e A F d f : R N (x, y) f(x, y) = A (x + y) (i) F G F (x + y), G (x + y) R N (ii) F n R d n = 1, 2,... G n R d n = 1, 2,... F n F n+1 A G n+1 G n λ d (A \ F ) = λ d (G \ A) = 0 ( F = F n, G = G n) F (x + y), G (x + y) R N (iii) R N F (x + y) G (x + y) d(λ d λ d )(x, y) = 0 (iv) A (x + y) Q8.8. N = 2d f, g L 1 (λ d ) h(x, y) = f(x y)g(y), k(x, y) = f(y)g(x y) h, g L 1 (λ N ) R d h(x, y)dλ d (y) = R d k(x, y)dλ d (y) R N hdλ N = R N kdλ N Q8.9. N = 2d p > 1 f L p (λ d ), g L 1 (λ d ) h(x, y) = f(x y)g(y) x = (x 1,..., x d ) max{ x 1,..., x d } > 1/2 f(x) = 0 h L 1 (λ N ) h dλ N ( f p dλ d) 1/p g dλ d R N R d R d

69 8 FUBINI 65 Q8.10. K L 2 (µ µ Y ) f L 2 (µ Y ) T f(x) = K(x, y)f(y)µ Y (dy), Y x (i) T f(x) T f L 2 (µ ) (ii) f n, f L 2 (µ Y ) g L 2 (µ Y ) lim g(f n f)dµ Y = Y 0 lim T f n T f 2 dµ = 0 ( : sup f n 2 dµ Y < n N Y 1 ) Q8.11. f, g : R R h(x) = R f(x y)g(y)dy f(ξ) = R f(x)e iξx λ 1 (dx) h L 1 (λ 1 ) ĥ(ξ) = f(ξ)ĝ(ξ) ( ξ) () Q8.12. S = {f : R C sup x R { x k f (m) (x) } <, k, m Z, 0} (i) S L 1 (λ 1 ) (ii) f S (iii) f, g S R f(ξ)g(ξ)e iξx λ 1 (dξ) = R f(x + y)ĝ(y)λ1 (dy) (iv) ε > 0 g(ξ) = e ε2 ξ 2 /2 ĝ(y) = (2π) 1/2 ε 1 e y2 /(2ε 2) (v) R f(ξ)e iξx λ 1 (dξ) = (2π) 1/2 f(x) Q8.13. = Y = [0, 1) B = B Y = {A [0, 1) A F 1 } µ (A) = µ Y (A) = λ 1 ([0, 1) A) f(x, y) = (x 2 y 2 )/(x 2 + y 2 ) 2 ((x, y) [0, 1) 2 ) (i) (, E, µ ) (ii) f ± (iii) Y f ± d(µ µ Y ) = (iv) ( Y f(x, y)µ Y (dy) ) µ (dx) ( Y f(x, y)µ (dx) ) µ y (dy) Fubini

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II Brown Brown II 16 12 5 1 Brown 3 1.1..................................... 3 1.2 Brown............................... 5 1.3................................... 8 1.4 Markov.................................... 1 1.5

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