27 6 2 1 2 2 5 3 8 4 13 5 16 6 19 7 23 8 27 N Z = {, ±1, ±2,... }, R =, R + = [, + ), 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 ], [f ] = {x X; f(x) } X A (idicator fuctio) { 1 if x A, 1 A (x) = otherwise. 1
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}... 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 2
. a j < a k (j < k) {a i } i I a i = sup{ a i ; F I } [, + ] i I i F {a i } i I (summable) (sum) a i = a i ( a i ) 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. j J i Ij i I 1. i I a i < + {i I; a i } [a, b] [a, b] : a = x < x 1 < < x = b (mesh) = mi{x 1 x, 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 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) 3
1.3. f : [a, b] R S(f) = S(f) b a f(x) dx f (Riemaia itegral). Riema (1857) Darboux (1875) 2 Riema f 1.4. f : [a, b] R b a f(x) dx = lim i=1 f(x j )(x j x j 1 ). (improper itegral) f(x) dx < + R f(x) dx = + R lim R f(x) dx 1.5. (i) (ii) (i) (ii) si x 1 + x 2 dx. si x 1 + x dx. 4
2 2.1 (Bolzao). {a } 1 N N k k {a k } k 1 2.2. R K K K x = (x 1,..., x ) R x = (x 1 ) 2 + + (x ) 2 x, y R d(x, y) = x y R lim x () = x lim x () x =. B r (a) = {x X; d(x, a) < r}, B r (a) = {x X; d(x, a) r} a R r > (ope ball) (closed ball) U R (ope set) a U, r >, B r (a) U F R (closure) F = {a R ; r >, B r (a) F } F = F (closed set) B R (bouded) a R, r >, B B r (a) R X f : X R (cotiuous) x X, ɛ >, δ >, y B δ (x) = f(x) f(y) ɛ f, g : X R Φ : R 2 R Φ(f, g) : x Φ(f(x), g(x)) f + g, fg, f g, f g 2. (a, b) a b, a b R X f (support) [f] [f ] f(x) = (x X \ [f]) [f + g] [f g] [f g] [f] [g], [fg] [f] [g]. 5
3. (i) (ii) R f, g [fg] [f] [g] 4. f f(x) = (x F ) F X R X X C c (X) C c (X) f, g C c (X) = f g, f g, fg C c (X). C c C Cotiuous c compact 2.3. X R f : X R δ > f (the degree of uiform cotiuity) C f (δ) = sup{ f(x) f(y) ; d(x, y) δ} 5. f : R R M = sup{ f (x) ; x R} C f (δ) Mδ. 2.4 (H.Heie). f : X R (uiform cotiuity) Proof. δ = 1/ lim C f (δ) =. δ ɛ >, δ >, 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 ) 6. 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 ) 6
7. * [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 R f(x) dx. (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 8. * [a, b] R Φ : [a, b] R f : [a, b] R lim j=1 f(x j )(Φ(x j ) Φ(x j 1 )) = b a f(t)dφ(t) Stieltjes 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} [, + ] f f < + f f (coverge uiformly) lim f f = f (x) f(x) f f. lim ( a 1 p + + a p ) 1/p = a 1 a p + 7
2.5. f : [a, b] R f(x) dx (b 1 a 1 ) (b a ) f. [a,b] 2.6. f f (uiformly) lim [a,b] f (x) dx = [a,b] f(x) dx. 2.7. *X R f 2.8. 2.9 (Dii). K {f } 1 x K, f (x) lim f = Proof. f r >, N 1, N, f > r. 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 1 2.1. f : R m R f lim R f (x) dx =. m 3 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 } 3.1. 8
(i) R C c (R ). (ii) * X f [f ] L. (iii) * S = {x = (x, x 1,..., x ) R +1 ; (x ) 2 + (x 1 ) 2 + + (x ) 2 = 1} C(S ). 9. f = f f L. L = L + L + 1. X L (i) L (ii) f L f L. (iii) f L f L. 3.2. 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 = I(f). (iii) [Cotiuity] f = I(f ). L I (L, I) (itegratio system) 3.3. Dii (i) ρ : R [, + ) f C c (R ) I(f) = f(x)ρ(x) dx R C c (R ) ρ 1 (ii) * L = {f : X R; [f ] } I(f) = x X f(x) (iii) * L = C(S ) I(f) = < x 1 f ( ) x dx x 11. * Φ : R R L = C c (R) Stieltjes I(f) = f(t)dφ(t) 9
12. f g = I(f) I(g). 13. f f = I(f ) I(f). 14. * L + f {h } 1 f =1 h = I(f) I(h ) =1 h L 3.4. X L L = {f : X (, + ]; a sequecef L, f f}, L = {f : X [, + ); a sequecef L, f f} L + = {f L ; f }. 3.5. (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) = ± f(x) = (ii) L L ± 15. 3.6. L = C c (R ) A R (i) 1 A L A (ii) 1 A L A 16. f(x) = x/(x 2 + 1) C c (R) C c (R) 3.7. L f, g lim f lim g lim f, lim g L lim I(f ) lim I(g ) 1
Proof. f m lim g f m = lim f m g. (f m f m g ) m 3.8. 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 17. (well-defied) 3.9. L = C c (R ) I I (1 (a,b) ) = (b 1 a 1 )... (b a ) = I (1 [a,b] ). (a, b) = (a 1, b 1 ) (a, b ), [a, b] = [a 1, b 1 ] [a, b ] 18. * Φ : R R Stieltjes I : C c (R) R I (1 (a,b) ) = Φ(b ) Φ(a + ), I (1 [a,b] ) = Φ(b + ) Φ(a ). 3.1. (i) I ( f) = I (f) for f L ( L = L (ii) I, I I f L L (f) = I(f) = I (f). I () = I () = (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 3.11 ( ). (i) L f f L I (f ) I (f) (ii) L f f L I (f ) I (f) Proof. (i) f L {f,m } m 1 f,m f {f,m } 1 g,m = f 1,m f 2,m f,m. 11
g 1,m = f 1,m g,m f,m m g,m m f,m g,m f 1 f 2 f = f g,m f {g, } 1 g, L g, f,m g,m g m,m f m, m m f lim m g m,m f f = lim m g m,m L I(f,m ) I(g m,m ) I (f m ), m m I (f ) I (f) lim m I (f m ) lim I (f ) = I (f) 3.12. f L + f L + 1 I f = I (f ). 1 1 Proof. f f L + {f L + } 1 f = =1 f I (f) = I(f ) =1 {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 12
3.13. L = C c (R) Q = {q } 1 ɛ > 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 4 4.1. 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) = +. 4.2. Q R 1 Q Dirichlet I(1 Q ) = 19. (i) 1 Q (x) = lim m lim (cos(πm!x)) 2 (ii) Darboux S(1 Q [a,b] ) =, S(1 Q [a,b] ) = b a 4.3. (i) I(f) = I( f) for ay f. (ii) I(λf) = λi(f) for λ < +. I() = (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() = (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). 13
f L I(f) = I (f) I(f) = I(f). 2. (i) (iv) 4.4. f : X R (Lebesgue itegrable) I(f) = I(f) R L 1 f L 1 I(f) = I(f) R I(f) I 1 3.1 (ii) 4.3 (vi) - L 1 L 1 (R ) 21. [a, b] f : [a, b] R I(f) = b a f(t) dt. ɛ >, S(f) ɛ I(f) I(f) S(f) + ɛ. 22. f : X R g L 1 f(f + g) = I(f) + I(g) 4.5. 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 ) f I (f ) I(f), I (f ) I(f). Proof. I (f ) I(f) I(f) I (f ) 4.6. (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 > 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 ) 14
f L 1 I( f) = I(f). L 1 I L 1 f L 1 = f L 1 f f f f f f f I (f ) I (f ) = I (f f ) I (f f ) f I(f) = I(f ) I (f ) I(f) f L L f f f f L 4.3 (vi) I(f) = I(f) [I(f ), I(f )] 4.7. f : R R f L 1 R lim f(t) dt < + R + R I(f) = R lim f(t) dt R + R f f f f C c (R) + f (t) = f(t) ( t R ) R f (t) dt f(t) dt R 1 si t t R si t dt lim dt = π R R t 23. α > ( ) 1 si t α dt 24. * ɛ >, f L 1, g L, I( f g ) ɛ. 4.8. 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 4.9. 15
(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 4.1. 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 4.11. f L 1 (R ) y R f(x + y) x R f(x + y) dx = f(x) dx. R R 25. f L 1 (R ) λ > f(λx) dx = λ R f(x) dx R 5 5.1 (subadditivity of upper itegrals). f : X [, + ] f = =1 f (f ) I(f) I(f ). =1 Proof. I(f ) = + I(f ) < + ( 1) ɛ > g L + f g, I(g ) = I (g ) I(f ) + ɛ 2 f g 3.12 g L + I ( g ) = I (g ) ( ) I(f) I g = I (g ) I(f ) + ɛ 2 = I(f ) + ɛ 16
5.2 (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 = =1 (f f 1 ) I(f f ) I(f f 1 ) = I(f f 1 ) = (I(f ) I(f 1 )) = lim I(f ) I(f ). =1 =1 =1 I(f) I(f ) + I(f f ) = I(f ) + I(f f ) lim I(f ) f f f L 1 I(f ) = I(f ) I(f) lim I(f ) I(f) I(f) lim I(f ) 5.3. I L 1 f L 1 f I(f ). (L, I) (L 1, I) 5.4 (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 17
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 5.5. R [a, b] f(x, t) (i) t [a, b] f(x, t) x R (ii) x f(x, t) t (iii) g(x) f(x, t) g(x) (x R, t [a, b]) lim f(x, t) dx = t t R R f(x, t ) dx. 5.6. R (a, b) f(x, t) (i) t (a, b) f(x, t) x R (ii) x R f(x, t) t t (iii) g(x) f t (x, t) g(x) (x R, a < t < b) f t (x, t) x d f(x, t) dx = dt f (x, t) dx t Proof. f(x, t + h) f(x, t) h g(x) 5.7. (i) t e x2 +tx dx = = t! x e x2 dx. (ii) t > 26. t > d dt =1 e tx2 dx = ( 1) x 2 e tx2 dx. 1 t = 1 x t 1 Γ(t) e x 1 dx 18
27. f(x) ( x 1) lim f ( x ) e x dx 28. f L 1 (R ) f(x + y) f(x) dx y R 29. f L 1 (R ) R g fg 3. f L 1 (R) a > 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) =, x ± f a (x) dx f(x) dx 6 (L, I) (L 1, I 1 ) (L, I) (L 1, I 1 ) ((L 1 ) 1, (I 1 ) 1 ) (L 1 ) 1 = L 1, (I 1 ) 1 = I 1 (L 1 ) 1 L 1 6.1. X (L, I) A X I- (I-measurable) 1 A L 1 I- L(I) L L(I) = {A X; 1 A L 1 }.. I- 6.2. (i) L. (ii) {A } 1 L A A 1 (iii) A, B L B \ A Proof. L 1 L (iii) A B 1 L 1 + ϕ 1 A, L 1 + ψ 1 B 19
ψ ψ ϕ m ψ m ψ ψ 1 A ψ 1 ψ ψ 1 A = ψ (1 1 A ) ψ (1 1 A ) 1 B (1 1 A ) = 1 B\A L 1 (ii) 3.11 (iii) A = A 1 \ A 1 \ A = A 1 \ 1 \ A ) 1 1(A 1 6.3. (L, I) σ- (σ-fiite) 1 X L 1 6.4. (L, I) σ- L σ- (σ-boolea algebra) (i), X L. (ii) {A } 1 L = 1 A, (iii) A L = X \ A L. A L. 1 6.5. σ- A I- A I = I 1(1 A) 6.6. (i) A I [, + ]. (ii) I =. (iii) A A I = A I. 1 1 1 Proof. (i) (ii) (iii) I 1 σ- 3.12 6.7. I- - 6.8. σ- B 2 X [, + ] µ (measure) (i) µ( ) =, (ii) {A } 1 B A m A = (m ) µ A = µ(a ) 1 =1 2
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). 2 X X (power set) X 6.9. ρ : R [, + ) ρ(x) dx R 1 31. {A } 1 I A A I 1 1 σ- 6.1. f : X R (i) a R, f a L 1, f a L1. (ii) a R, f a L 1, f a L1. (iii) a R, [f > a] L(I). Proof. f f L 1 a >, [f > a] L [ ] f L 1 f f (f L 1 ) [f > a] = 1 [f > a] ϕ 1 X (ϕ L 1 +) ϕ l ((f aϕ m aϕ m )) ϕ l m ϕ l ((f a a)) L 1 + l 1 ((f a a)) L 1 1 ((f a a)) 1 [f>a] ( ) 3.11 [f > a] L [ ] f = 1 k<2 k 2 1 [k2 <f (k+1)2 ] + 1 [f>] [a < f b] = [f > a] \ [f > b] L f L 1 f f L 1 21
6.11. (ii) (L 1 ) + (i) (L, I) σ- I σ- = Proof. (i) X,m = [ϕ > 1/m] 1 m 1 X,m ϕ X,m I mi(ϕ ) < + X = m, 1 X,m (ii) a > [f α > a] = [f > a 1/α ] f g fg 1 A 1 B = 1 A B f g L 1 A f 1 A f f A I(1 A f) = f(x) µ(dx) 32. L(I) A f(x) µ(dx) R A A f L 1 + f L 1 I(f) = lim I(f ) = lim k 2 [k2 < f (k + 1)2 ] I + [f > ] I 1 k<2 I µ( ) = I (L, µ) µ L 6.12. f : X C Rf If f I(f) = I(Rf) + ii(if) L 1 C 33. 6.13. 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 ) 6.14 ( ). {f } g L 1 f g ( 1) f = lim f I(f) = lim I(f ) 6.15. e x2 +itx dx = πe t2 /4 22
34. f : R C a > x R a f a (x) = f(x t)e at2 dt π C 1 T T θ T θ 2π L L 1 2π 2π f(θ) dθ µ µ t 2 T H = {e 2πit ; Z} t Z e 2πit Z H T/H R R T = Z e2πit R R µ 1 = µ(t) = µ(e 2πit R) = µ(r) Z Z µ(r) = µ(r) > 7 (L, I) f ɛ >, f L, f L, f f f, I (f f ) ɛ. ɛ = 1/ ( = 1, 2,... ) f L f L f f f, I (f f ) f = lim f f = lim f f f L 1, f L1 I1 (f f) = 7.1. f : X R f L 1 I( f ) = (ull fuctio) A X (ull set) 1 A A A I = N(I) N 7.2. (i) N(I). 23
(ii) N N(I) (iii) {N } 1 =1 N 35. 7.3. (i) Dirichlet (ii) R (iii) Cator (iv) R 36. R 7.4. f : X R (i) f (ii) [f ] Proof. 1 [f ] = lim 1 ( f ), f = lim f (1 [f ]). I(1 f ) I( f ) = I( f ), I( f 1 [f ] ) I(1 [f ] ) = I(1 [f ] ) f(x) dx, (,1) [,1] f(x) dx 7.5. f : X R (X X) f L 1 N X \ X f (x) = f(x) ( x X \ N) f (almost everywhere) L 1 f a.e. L 1 I(f ) I(f) f, g : X R [f g] f a.e. = g 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). Lebesgue almost everywhere (a.e.) presque partout (p.p.) almost all (a.a.) almost surely (a.s.) 24
37. 38. f a.e. L 1 f a.e. L 1 I(f) I( f ) 39. 7.6. f L 1 a.e. I1 (f) < + f L 1 Proof. f f (f L 1 ) f f f 1 f N = [f = + ] a a1 N f ai(1 N ) I(f) = I (f) < + I(1 N ) = N N(I) f 1 N L 1 f 1 X\N f1 X\N f f 1 N L 1 f1 X\N L 1 f a.e. = f1 X\N 7.7. {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 ) f j L 1 f L 1 1 I f = I( f ) = I( f ) < + 1 1 1 f (x) = + 1 x N N = N j x N f(x) = 1 f (x) = 1 f (x) 25
7.8. {q ; 1} f(x) = 1 e 3 (x q ) 2 x R f(x) dx = 1 7.9. 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+ I(f,m f,m 1 ) = I(f f,m 1 ) I(f f,m ) 1 2 m+ 1 + = 2 2 m+ m, 1 (f 1,m f 1,m 1 ) lim f = f + (f f 1 ) 1 = f, + (f,m f,m 1 ) +, f 1, ) m 1 1(f + (f,m f,m 1 ) (f 1,m f 1,m 1 ) m, 1 m, 1 I( f, f 1, ) I( f, f ) + I( f f 1 ) + I( f 1 f 1, ) 6 2 f, = f, ( f, ), f, f 1, = (f, f 1, ) (f 1, f, ) f(x) = lim f (x) (a.e. x X) (ii) f ± L 1 L f + = f, + (f,m f,m 1 ) + (f, f 1, ) + (f,m f,m 1 ) m 1 1 m, 1 f = ( f, ) + (f 1, f, ) + (f 1,m f 1,m 1 ) 1 m, 1 26
(i) f L 1 lim m lim sup I( f m f ) =. { k } k 1 I( f k f k+1 ) 1/2 k f = lim k f k = f f f k f f + (f k f k+1 ) k= f k f k+1 L 1 k= = lim lim k 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] [ ] [ ] 7.1. (L 1 ) 1 = L 1 Proof. f (L 1 ) 1 f f (f L 1 ) lim I (f ) < + f a.e. L 1 f a.e. L 1 f f L 1 8 σ- π : Ω X Ω σ- F X (L, µ) x X (L x, µ x ) f F (i) x X, f L x (ii) f(ω) µ x (dω) x L 27
(F, (L, µ), {(L x, µ x )}) (fibered itegratio system) F I(f) = X µ(dx) µ x (dω) f(ω) 8.1. π : R m+ R (repeated itegratio system) 8.2. (π : Ω X, F, (L, µ), {(L x, µ x )}) (i) f F x X f (L x ) f(ω) µ x (dω) x L ( ) I (f) = f(ω) µ x (dω) µ(dx). X (ii) f F 1 N X x X \ N f L 1 x L 1 X \ N x I(f) = X ( f(ω) µ x (dω) f(ω) µ x (dω) ) µ(dx). Proof. (i) f F f f (f F ) x X, f f f L x f (L x ) f(ω) µ x (dω) = lim f (ω) µ x (dω). 1, x f (ω) µ x (dω) L x f(ω) µ x (dω) L I (f) = lim I(f ) = lim µ(dx) X = µ(dx) X µ x (dω)f (ω) = µ x (dω) lim f (ω) = X X µ(dx) lim µ(dx) µ x (dω)f (ω) µ x (dω) f(ω) 28
(ii) f F 1 {f } F {f } F f f f, I (f f ) 1 (i) h (x) = L + I (f f) = X µ x (dω)(f (ω) f (ω)) µ(dx)h (x) 7.6 h a.e. L 1 N X µ x (dω)(f(ω) f(ω)) = for x N. lim x N f (L x ) 1 f(ω) µ x (dω) = lim f(ω) µ x (dω) = lim f(ω) µ x (dω) I (f) I(f) 7.6 x f(ω) µ x (dω) L 1 f(ω) µ x (dω) x L 1 (X, µ) I(f) = lim I (f) = lim µ(dx) µ x (dω) f(ω) X = µ(dx) lim µ x (dω) f(ω) = µ(dx) X X µ x (dω) f(ω). 8.3. 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 >, δ(x) otherwise, I(f) = dt R f(t, x) µ t(dx). 8.4 (). 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 dx. 29
x y ( ) ( ) f(x, y) dy dx. = f(x, y) dx R m R R m R dy 8.5. C = e t e tx2 dtdx = e t e tx2 dxdt = C e y2 dy = 1 2 1 x 2 + 1 dx = π 2, e t 1 t dt = 2C 2. e t 1 t dt. 4. t > 41. * α R, β > R e x β t x α dx = 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 /2 42. 1 1 1 dxdy < + x y α α 43. f : R C a > a f a (x) = f(x t)e at2 dt π lim x ± f a (x) = lim f(x) f a (x) dx =. a + 3