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Lebesgue 1 2 4 4 1 2 5 6 λ a<b λ([a, b]) = b a 7 Fubini 1 A.1
ii 110 * 2015 6
i v vi 1 1 1.1 1 1.2 2 1.3 3 1.4 4 1.5 5 2 6 2.1 6 2.2 σ 9 2.3 12 2.4 15 2.5 23 2.6 29 2.7 34 3 0 35 3.1 35 3.2 40 3.3 L 1 44 4 1 49 4.1 1 49 4.2 1 55 4.3 1 58 4.4 1 62 5 68 5.1 68
iv 5.2 * 76 5.3 82 5.4 * 86 5.5 1 94 6 100 6.1 100 6.2 2 111 6.3 119 6.4 * 128 7 135 7.1 135 7.2 146 7.3 155 7.4 * 161 8 171 8.1 171 8.2 175 8.3 179 9 190 9.1 190 9.2 195 9.3 199 9.4 204 9.5 209 9.6 215 218 A.1 218 A.2 222 227 241 242
N := Z := Q := R := A.1 R 0 := {x R: x 0}, R >0 := {x R: x>0} R := R {, + }, R 0 := {x R: x 0} f,g: X R X g(x) f(x) g f X 0 f(x) 0 f f : X Y (image) A X f(a) :={f(x);x A} (range) range f := f(x) (inverse image) B Y f 1 (B) :={x X : f(x) B} f : X R a R {f a} := {x X : f(x) a}, {f <a} := {x X : f(x) <a} {f = a} := {x X : f(x) =a}, {f a} := {x X : f(x) a} {f a}{f >a} {a f<b} A X (complement) A c { 1 (x A) 1 A (x) := 0 (x A c ) A (indicator function)
vii (power set) P(X) X (difference) AB A \ B := {x A: x B} #A: A x + o O p R f(x) =o(x p ) lim x + f(x)x p =0 f(x) =O(x p ) R R >0 sup x>r f(x) x p < + ( ) s s R n Z 0 := n n =0 ( s 0) =1 n 1 k=0 (s k) n! 0 (x =0) sgn : R R sgn(x) = x (x 0) x Bμ σ f 1 μ a.e. f L 1 μ I := {(a, b];a, b R,a<b} { } dv v γ(m; ) m M(γ)N (γ) γ δ a a λλ (2) 1 2
viii σ(a) A σ B(R)B(R 2 ) 1 2 B 1 B 2 μ 1 μ 2 σ C0 r(rd ) C r R d R γ 5 f c f s f var + (ξ; )var (ξ; )var(ξ; ) ξ
1 1.1 Riemann Dirichlet { 1 (x Q) f(x) = 0 (x Q) cos { 1 (x Q) lim lim (cos πm! m n x)2n = 0 (x Q) (1.1) x Q Mx Z M N m M (cos πm! x) 2 =1 x Q m m! x Z lim n (cos πm! x) 2n =0 (1.1) (1.1) 1 0 (cos πm! x) 2n dx = n k=1 2k 1 2k (2k 1)/(2k) k/ k +1 n k=1 (2k 1)/(2k) 1/ n +1 0
2 1 0 Q 0 1 1.2 0 1 Lebesgue 2 f :[0, 1] [0, 1] R 1.1 (b) f [0, 1] [0, 1] 1.2 1/2 n [0, 1] [0, 1] { (x, y) [0, 1] [0, 1]: } k k +1 f(x, y) < 2n 2 n (k Z) (1.2) 1.1 1.2
1.3 3 Lebesgue (1.2) k=1 ( { } ) k k 2 +1 n f<k 2n 2 n (1.3) n (1.3) 1.3 Lebesgue (1.3) 1.3 B μ: B R μ B B [0, 1] [0, 1] B[0, 1] [0, 1] B. A B([0, 1] [0, 1]) \ A B A n B(n =1, 2,...) n=1 A n B B σ μ A Bμ(A) 0 μ( ) =0 A n B(n =1, 2,...) μ ( n=1 A n)= n=1 μ(a n)
4 1 μ 2 σ B μ() = μ B μ 2 f :[0, 1] [0, 1] R {k/2 n f<(k +1)/2 n } B f (1.3) n f f 2 [0,1] [0,1] f(x, y) dxdy = lim n k=1 ({ }) k k 2 μ +1 f<k n 2n 2 n 2 1.4 2.6 2.14 f n f g n Nx R f n (x) g(x) + + f n (x) dx = f(x) dx lim n 1.1 Fourier 8.3 1.4 1.1
1.5 5 1.4 1.1 f : R R lim n + nf(x) dx = πf(0) n 2 x 2 +1 + f(x/n)/(x2 +1)dx M y R f(y) M g(x) :=M/(x 2 +1) g n Nx R f(x/n)/(x 2 +1) g(x) lim n f(x/n)/(x 2 +1) = f(0)/(x 2 +1) + f(0)/(x2 +1)dx (= πf(0)) r lim r + r nf(x)/(n2 x 2 +1)dx + nf(x)/(n2 x 2 +1)dx (1.3) R r lim r + r dx 1.1 Lebesgue 1.5
2 A.1 2.1 σ 2.1 B R d 3 B R d σ (σ field) B A BA c BA c R d A n N A n B n=1 A n B 2.1 (union) (countably infinite) A n (mutually disjoint) n m A n A m = 2.2 R d B μ: B R 3 (B,μ) B R d (measure) σ
2.1 7 B R d σ A Bμ(A) 0μ(A) =+ μ( ) =0 A n Bμ ( n=1 A n)= n=1 μ(a n) 2.2 + σ (σ additivity) μ σ (σ additive) (B,μ) R d 1 (1.2) σ B 2.1 2.3 f : R d R B (measurable) a R {f <a} := {x R d : f(x) <a} B f B (measurable function) σ B B (measurable set) range f := f(r d ) f 2.1 {f = a} {x R d : f(x) =a} 2.1 2.3
8 2 B {f = a}, {f = b},... B 2.3 2.4 f : R d R B (simple function) B + range f f 2.2 0 g f B g y range g yμ({g = y}) f f 2.2 0 g f 2.5 f : R d R + B + μ f (integral) { } fμ:= sup yμ({g = y}); g B 0 g f R d y range g f(x) =max{f(x), 0} max{ f(x), 0} f : R d R B max{f,0} (Darboux) f f B μ σ
2.2 σ 9 max{ f,0} B 2.14 2.6 B f : R d R μ (integrable) max{f,0} μ<+ max{ f,0} μ<+ R d R d f (B,μ) (integrable function) μ f fμ:= max{f,0} μ max{ f,0} μ R d R d R d 2.14 μ R d f μ<+ 2.2 ff σ 2.3 {f <a} {x R d : f(x) <a} {f a} 2.1 f : R d R 4 a R {f <a} B 2.3 a R {f a} B a R {f a} B a R {f >a} B 2.1 {f <a} {f a} {f <a} B {f a} B {f >a} = n=1 { f a + 1 } n (2.1) (2.1) B
10 2 2.1 B {f <a} = n=1 { f a 1 } n 2.1 (2.1) c R c c R d 2.2 c R c B 2.2 (i) R d B (ii) A, B BA BA BA \ B B (iii) n N A n B n=1 A n B 2.1 R d = c B(ii) A B B A 1 := A, A 2 := B, A n := (n 3) n N A n B 2.1 A B = n=1 A n B(ii) (iii) A B =(A c B c ) c, A\ B =(A c B) c, ( A n = n=1 n=1 A c n ) c 2.1 A, B B(A c B c ) c B (A c B) c B 2.1 n N A n B( n=1 Ac n )c B 2.1 f : R d R B a <b {a f<b} B y R {f = y} B
2.2 σ 11 {a f < b} = {f < b}\{f < a} 2.2 (ii) B y R {f = y} B{f = + } = n=1 {f n} 2.2 (iii) {f =+ } B 2.3 2.1 {f = y} B 2.4 f : R d R {f = y} y range f R d 2.3 f : R d R range f f B y range f {f = y} B B {f = y} B 2.1 {f <a} = y range f,y<a {f = y} range f 2.4 A, B B (i) (finite additivity)a B = μ(a B) =μ(a)+μ(b) (ii) A B μ(a)+μ(b \A) =μ(b) μ(a) μ(b) (iii) A B μ(a) < + μ(b \ A) =μ(b) μ(a) (iv) (subadditivity)μ(a B) μ(a)+μ(b) (i) 2.2 (ii) σ μ( ) =0 (ii)(iii) A (B \ A) = μ(a)+ μ(b \ A) =μ(b) μ(b \ A) 0 μ(a) μ(b) μ(a)+μ(b \ A) =μ(b) μ(a) < + μ(a) 2.5 2.4 (iv) σ
4 1 1 λ a<b λ([a, b]) = b a 1 6.2 [a, b] b a 2 4.1 1 1 (a, b] :={x R: a<x b} a, b Ra <b 4.1 R (B,λ) (a, b] (a, b] B λ((a, b]) = b a 6.2 5.7 B R σ a<b (a, b] B λ B a<b λ((a, b]) = b a 2 1 4.1 a, b, c 1,c 2 R a<b 2.2
50 4 1 (a,b] c 1 (b a) (b 0) (c 1 1 (,0] + c 2 1 (0,+ ) ) λ = c 1 a + c 2 b (a <0 <b) c 2 (b a) (0 a) B 4.1 (i) a<b (a, b)[a, b][a, b) B (ii) I f : I R B (iii) A R B (iv) A R B A R B (i)(iii) (ii) I =(0, 1) x R [x] :=max{n Z: n<x} n N f n :(0, 1) Rx f([nx]/n) f(0) = 0 N := {k Z: 0 k < n} range f n = {f(k/n);k N} B {f n = y} = k N : f(k/n)=y (k/n, (k +1)/n] (0, 1) f n B f f n f f B (iv) BA x R { x z ; z A} f : R Rx inf{ x z ; z A} x, y R ε R >0 x z f(x)+ε z A f(y) y z y x + x z y x + f(x)+ε xy f(x) x y + f(y)+ε x, y R f(x) f(y) x y (ii) B {x R: f(x) =0} B A = {x R: f(x) =0} 4.1 (2) B B 4.1 (1) 4.1 (i)(iii)
4.1 1 51 (2) R A A = {x R: inf{ x z ; z A} =0} (3) I f : I R B 4.2 (i) A R λ(a) =0 (ii) a, b R a<bλ((a, b))λ([a, b])λ([a, b)) b a (i) 4.1 (iii) A B x A n N 0 λ({x}) λ((x 1/n, x]) = 1/n λ({x}) =0σ λ(a) = x A λ({x}) =0 4.2 1 Q (i) 4.2 (i) λ(q) =0 (0,1) 1 Q λ =0 (ii) m N A := {x R: m! x Z} A Q (0,1) 1 A λ =0 x R lim n (cos πm! x) 2n = 1 A (x) 0 (cos πm! x) 2n 1 lim n (0,1) (cos πm! x)2n λ(dx) =0 (iii) x R lim m lim n (cos πm! x) 2n =1 Q (x) lim m (0,1) lim n (cos πm! x) 2n λ(dx) =0 4.3 J B f : J R I 1,...,I n J J \ n k=1 I k (i) f J λ I k λ (ii) f J λ J fλ = n k=1 I k fλ 4.2 (i) λ (J \ n k=1 I k)=0 2.2 2.5 (ii) 4.3 I F : I R F f : I R F : I R F = f
52 4 1 F f (primitive function) 4.2 a, b Ra <b f :(a, b) R λ (i) (a, b) Rx (a,x) fλf (ii) f F lim x a F (x)lim x b F (x) (a,b) fλ= lim x b F (x) lim x a F (x) (i) c (a, b) ε R >0 δ R >0 y c <δ f(y) f(c) <εc x<b λ((c, x)) = x c x = c (c, x) = 4.3 (ii) (a,x) ( ) fλ fλ f(c)(x c) = f f(c) λ (a,c) (c,x) 2.4 (i) c x<min{c + δ, b} fλ fλ f(c)(x c) f f(c) λ ε x c (a,x) (a,c) max{c δ, a} <x c x (a,x) fλc f(c) (ii) K R (a, b) F (x) = (a,x) fλ+ K ab a = b =+ 2.4 (i) F (x) K = (a,x) (c,x) fλ (a,x) f λ lim sup F (x) K lim sup f λ f λ (n N) x a x a (a,x) (a,a+1/n) (a, a +1/n) n=1 (a, a +1/n) = 2.6 (ii) (a,a+1/n) f λ 0 F 0 F
lim x b F (x) =K + (a,b) fλ 4.1 1 53 x a F (x) K F (x) K fλ = fλ fλ = fλ f λ (a,b) (a,b) (a,x) (x,b) (x,b) λ 4.1 a, b Ra <b f :(a, b) R a = b =+ F : R R x <0 F (x) := (x,0) fλf (0) := 0 x >0 F (x) := (0,x) fλc R 4.3 (ii) x > c 1 F (x) = fλ fλ ( c 1,x) ( c 1,0) 1 4.2 (i) F (c) =f(c) 4.2 a, b Ra <b f :(a, b) R f F (i) (a,b) fλ=sup x (a,b) F (x) inf x (a,b) F (x) + =+ (ii) F f λ a = b =+ a n a b n b f F 4.2 (ii) n N ( n,n) fλ= F (n) F ( n) 2.6 (i) 4.3 (max{a, 0}) 2 a, b Ra < b (a,b) 2max{x, 0} λ(dx) = (max{b, 0})2 4.2 F lim x a F (x) lim x b F (x)
54 4 1 4.2 lim x b F (x) lim x a F (x) lim x b F (x) lim x a F (x) F (x) x=b x=a F b a 4.4 a, b Ra <b f :(a, b) R λ F (a,b) fλ= F b 4.2 a 4.5 4.2 (0,+ ) e x λ(dx) = e x x=+ x=0 =1 4.6 (i) (ii) (0,1) s>0 ( x s 1 x s +1 1 x +1 ( x s 1 x s +1 1 x +1 (1,+ ) (iii) x ) λ(dx) = 1 s s log 2 ) λ(dx) = s 1 log 2 s xs 1 x s +1 1 (0, + ) λ x +1 (i) x (1/s)log(x s +1) log(x +1) 0 <s<1 (0, 1) Rx x s 1 /(x s +1) 1/(x +1) 4.2 (ii) 0 <s<1 (1, + ) Rx 1/(x +1) x s 1 /(x s +1) 4.2 lim x (x s +1) 1/s /(x +1)=1 4.2 4.2 A.3 2.5 4.7 R Rx e x2 λ x e 2 x +1 4.2 s>0 (0,1) (1 x2 ) s 1 λ(dx) < +
a.e. 37 a.e. 47 C 69 C 70 L 1 85, 134 L 1 44 L 1 44 L p 46 p 46 σ 7 σ 6, 77 σ 103 σ 7, 73, 192 σ 100, 194 2 117 172, 184 143 139 76 65 163 103 C 71 21, 38, 76 72 9, 19, 204 9 22, 54, 156 33 7, 103 102 7, 76, 80, 103 225 76 69, 191 84 79 57, 150 1/2 147 147, 150 139, 150 150 147 66 147 42 143 130 209 164 52 52 20, 196 29, 41 176 166 29 215 176 46, 172 197 161 65, 149 137 154 197 σ 102 121 190 8, 9, 17, 19, 36, 204 36 σ 32 91 19 28 16, 19, 22
88, 202 32 203 20, 192, 196 6 129 81 C r 132 84 2 179 6 186 125 8 24, 40 11, 71 111 161 σ 115 115 115 10 84 182 183 1 167 120 122 179 203 202 203 136 224 176 174, 185 70 198 52 49 43 217 20, 24, 41 195 198 191 197 195, 206, 213 136 124, 136, 157 55, 211 190 188 177 171 174 171 177 243 97 187 178 175 69 95 59, 94, 108, 118 94, 118 128 161 57 46 149 143 1 58, 94 145 137 190 131 168 178 81 37 106 21 106, 114 106, 114 106, 114 17 89, 195, 213
244 46 180, 209 11, 68 68 71 23 17 166 153 215 106, 114 195 202 161 161 88 154 98 124, 136, 157 75, 83 83 83 83, 114 75 83 83, 211, 214 106 83 137 83, 114 114 111 108 127 34 33, 41, 206 33 203 37, 79 207 11, 71 166
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