B [ 0.1 ] x > 0 x 6= 1 f(x) µ 1 1 xn 1 + sin sin x 1 x 1 f(x) := lim. n x n (1) lim inf f(x) (2) lim sup f(x) x 1 0 x 1 0 (

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1 [.1 ] x > x 6= 1 f(x) µ 1 1 xn 1 + sin sin x 1 x 1 f(x) := lim. 1 + x n (1) lim inf f(x) (2) lim sup f(x) x 1 x 1 (3) lim inf x 1+ f(x) (4) lim sup f(x) x 1+ [.2 ] [, 1] Ω æ x (1) (2) nx(1 x) n 1 + n 2 x 2 [.3 ] f(x) = sin(x 2 ) R [.4 ] f(x) := x x e x x = 1 1P e n x x e x dx n P [.5 ] (1) 1 a n lim a n = (2) f(x) f(x) dx lim sup x 1 1 f(x) = 1 [.6 ] a > b 2 < ac Q(x, y) := ax 2 + 2bxy + cy 2 2 dxdy I := p D := {(x, y) R 2 ; Q(x, y) < 1}). 1 Q(x, y) D [.7 ] (x y) 2 e (x2 +y2) dxdy R 2 [.8 ] (X, d) X A, B d(a, B) := inf d(a, b) a A, b B A B A B =? d(a, B) > A

2 1. Lebesgue [ 1.1 ] f(x) f (x) x = 8 < x 2 sin 1 (x 6= ) f(x) := x : (x = ) [ 1.2 ] Riemann [ 1.3 ] f n (x) := x 2 x 2 + (1 nx) 2 log sin x dx. (n = 1, 2,... ) [, 1] [ 1.4 ] f n (x) lim f n (x) dx = = lim f n (x) dx Hint: ε > [, ε] [ε, 1] 1X [ 1.5 ] t t dt = n n Hint: t t = e t log t = 1 P n= ( 1) n n! (t log t) n [ 1.6 ] A n (n = 1, 2,..., ) X A := 1 S χ An lim inf 1T k=n χ A n (x) = χ A (x) A k A n [ 1.7 ] X, Y f : X Y A Ω X B Ω Y (1) f f 1 (B) Ω B f 1 (B) f (2) f 1 f(a) æ A f [ 1.8 ] [, 1] f kfk kfk := f(x) dx ( ) (1) k k (2) [, 1] ( )

3 [ 2.1 ] (1) E 3 := {(a, b ] ; 1 < a < b < 1} B(R) = σ[ E 3 ] (2) E 8 := {( 1, b ] ; b R} B(R) = σ[ E 8 ] Remark: B(R) = σ[ E 1 ] [ 2.2 ] E := {R h-intervals} E elementary family [ 2.3 ] E Ω P(X) elementary family A := { E } A algebra [ 2.4 ] B 1, B 2 Ω P(X) σ-algebras (1) B 1 B 2 algebra B 1 B 2 σ-algebra (2) B 1 B 2 σ-algebra [ 2.5 ] B σ-algebra (1) B {C n } (2) a = {a n } (a n = or 1) D a := S C n B a n =1 [ 2.6 ] S X X = F S X E S B := {A Ω X ; A S } {?} σ-algebra [ 2.7 ] S E σ[s] = {A Ω X ; A A c S } {?} Hint: σ-algebra [ 2.8 ] X A, B A B := (A B c ) (A c B) A B (1) (A B) C = A (B C) (2) A B Ω (A C) (C B)

4 [ 3.1 ] X B ( := {E Ω X ; E E c } (E : ) µ(e) := µ 1 (E c : ) [ 3.2 ] X? B = P(X) f X 1 E B nx o µ(e) := sup f(x) ; F E (E 6=?), µ(?) = x F A := {x X ; < f(x) 5 1} (1), (2) S (1) A E µ(e) = 1 A E = 1 x E ; f(x) > 1 n (2) A E τ : {1, 2,..., n,... } A E P µ(e) = 1 f(τ(n)) [ 3.3 ] [3.2] µ µ σ-finite x X f(x) < 1 A [ 3.4 ] A algebra : A [, 1) µ [ n nx X E j = ( 1) p 1 (E j A). j=1 p=1 i 1 < <i p E i1 E ip P [ 3.5 ] (X, B, µ) E n B (n = 1, 2,... ) 1 µ(e n ) < 1 T lim sup E n := 1 1S E k µ lim sup E n = k=n Borel Cantelli [ 3.6 ] (X, B, µ) µ(x) < 1 (X) < 1 : B [, 1) ( ε > δ > µ(e) < δ E B (E) < ε. Hint: E n B (n = 1, 2,... ) P µ(e n ) < 1

5 [ 3.7 ] (X, B, µ) µ(x) < 1 E F E F cf. [2.8] (1) µ(e F ) = E ª F ª B (2) Ė, F B/ ª E, F d(ė, F ) := µ(e F ) d well-defined B/ ª [ 3.8 ] (1) lim sup E n lim inf E n Ω lim sup (2) P µ(e n E n+1 ) < 1 µ lim sup (E n E n+1 ) E n lim inf E n = (Remark: [3.7] (B/ª, d)

6 >< (E =?) [ 4.1 ] N E µ (E) = 1 (E 6=?, N) >: 2 (E = N) µ [ 4.2 ] [4.1] µ Carathéodory µ? N 8 < ]E (]E < 1) [ 4.3 ] N E µ (E) := 1 + ]E : ]E E 1 (]E = 1) (1) E Ω F µ (E) 5 µ (F ) S 1 (2) E 1 Ω E 2 Ω Ω E n Ω lim µ (E n ) = µ E n [ 4.4 ] [4.3] µ Carathéodory µ? N [ 4.5 ] X = {} N A A := {A Ω X ; A A c }. (1) A algebra (2) A A µ (A) := ]A µ A premeasure σ[a] = P(X) [ 4.6 ] [4.5] α E Ω X ( ]E ( / E) α (E) := α + ](E \ {}) ( E) α P(X) α Ø ØA = µ µ Hopf σ[ A ] µ µ = 1 [ 4.7 ] (X, B, µ) N := {N B ; µ(n) = } B := {B F ; B B, F Ω N for some N N } B σ-algebra [ 4.8 ] [4.7] (1) µ(b F ) := µ(b) B µ (2) µ (3) B Ø Ø B = µ = µ

7 5. R [5.5] 5 23 [ 5.1 ] X µ X E M µ G Ω X µ (E G) = E G E G G M µ Hint: µ (F ) = =) F M µ [ 5.2 ] Lebesgue Stieltjes µ E µ(e) < 1E G δ : Borel B µ(e) = µ(b) [ 5.3 ] F : R R µ F (a, b ] µ F ((a, b ]) = F (b) F (a) Borel (1) (3) (1) µ F ([a, b]) = F (b) F (a ) (2) µ F ([ a, b)) = F (b ) F (a ) (3) µ F ((a, b)) = F (b ) F (a) (1) (3) 1 < a < b < 1 [ 5.4 ] f : R R B(R) R Borel σ-algebra B B(R) f 1 (B) B(R) (1) C := {C B(R) ; f 1 (C) B(R)} σ-algebra (2) C R (1) B(R) C æ B(R) [ 5.5 ] C Ω [, 1] Cantor x, y C x < y x < z < y z / C z [ 5.6 ] R Lebesgue m ε > [, 1] R G ε G ε = [, 1] m(g ε ) 5 ε Hint: ε/2 k [ 5.7 ] m [5.6] R Lebesgue E L < m(e) < 1 < α < 1 I m(i E) > α m(i) (Hint: G æ E m(g) < 1 α m(e) G = P I n [ 5.8 ] N Ω R Lebesgue α N +α := {x + α ; x N} S Hint: Q = {r 1, r 2,... } R \ Q Ω 1 ( N + r j ) N := { x ; x N} j=1

8 [6.1]ª[6.7] (X, B) [ 6.1 ] A Ω X A B A B A := {E Ω A ; E B} σ-algebra A {F A ; F B} [ 6.2 ] f : X R X := f 1 (R) f (1) (3) (1) f 1 ({ 1}) B (2) f 1 ({1}) B (3) f := f Ø Ø X : X R [ 6.3 ] f, g : X R (±1) = fg [ 6.4 ] f, g : X R a R ( a (f(x) = g(x) = ±1) h(x) := f(x) + g(x) h a R a R [ 6.5 ] f n (n = 1, 2,... ) X R L := {x X ; lim f n (x) } T L = 1 1S 1T n x X ; f n (x) f m (x) < 1 o k k=1 l=1 n,m=l L B [ 6.6 ] {x X ; lim f n (x) = 1} {x X ; 1} lim f n (x) = [ 6.7 ] f : X R r Q f 1 ((r, 1]) B f [ 6.8 ] f : R R Borel

9 (X, B, µ) B [ 7.1 ] Fatou [ 7.2 ] f = f du < 1 ε > µ(e) < 1 E B f dµ > f dµ ε E [ 7.3 ] f = (E) := f dµ (E B) E g = g d = gf dµ Hint: g [ 7.4 ] f : X [, 1] E n := {x X ; f(x) = n} P 1 P µ(e n ) 5 f dµ 5 µ(x) + 1 µ(e n ) 1X µ(x) < 1 f dµ < 1 () µ(e n ) < 1 [ 7.5 ] f = f dµ < 1 {x X ; f(x) = 1} X {x X ; f(x) > } σ-finite [ 7.6 ] Fatou lim inf lim sup. [ 7.7 ] f n = (n =, 2,... ) x X f n (x) f(x) lim f n dµ = f dµ < 1 E B lim f n dµ = f dµ E E Hint: χ E f n χ E cf n Fatou [ 7.8 ] lim f n dµ = f dµ = 1

10 (X, B, µ) R Lebesgue m 1P [ 8.1 ] f n dµ < 1 f n (a.e.) [ 8.2 ] C Cantor [, 1] f ( (x C) f(x) := n x 1 3 n f dm = 3 [,1] [ 8.3 ] [, 1] ( f(x) := (x Q) n x 1 n f dm = 1 9 [,1] [ 8.4 ] {f n } f n (x) (8x) lim f n (x) 5 g(x) (a.e.x) g(x) f n dµ = [ 8.5 ] f n (n = 1, 2,... ) f (1) µ(x) < 1 f f n d u f dµ (2) µ(x) = 1 (1) R Lebesgue [ 8.6 ] f n, g n (n = 1, 2,... ) f, g f n f (a.e.) g n g (a.e.) f n (x) 5 g n (x) (8x X) g n dµ g dµ f n dµ f dµ [ 8.7 ] f ε > δ > µ(e) < δ f dµ < ε E [ 8.8 ] f = f n dµ n = 1, 2,... f

11 9. Riemann Fubini [9.7] [ 9.1 ] (X, B, µ) X f g µ-a.e. f [ 9.2 ] lim 1 + nx 2 (1 + x 2 ) n dx k [ 9.3 ] lim x n 1 x k dx = n! k 1 k (Hint: 1 k x k 5 e x ( 5 x 5 k) [ 9.4 ] e tx2 dx = r π t (t > ) t x 2n e x2 dx = (2n)! π 4 n n! x α 1 [ 9.5 ] α > 1 e x 1 dx = Γ(α) P 1 µ 1 n α. Hint : x > 1 e x 1 = 1 P e nx. [ 9.6 ] a > 1 ax sin x e x dx = Arctan 1 a Re a > [ 9.7 ] f(x, y) := e axy sin x E := (, 1) (1, 1) x, y E f(x, y) 5 xe axy [ 9.8 ] a > f (, a) g(x) := ( < x < a) g (, a) a f(x) dx a x a f(t) t dt g(x) dx =

1 Introduction 1 (1) (2) (3) () {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a, b] lim f n (x) f(x) (1) f(x)? (2) () f(x)? b lim a f n (x)dx = b

1 Introduction 2 2.1 2.2 2.3 3 3.1 3.2 σ- 4 4.1 4.2 5 5.1 5.2 5.3 6 7 8. Fubini,,. 1 1 Introduction 1 (1) (2) (3) () {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a, b] lim f n (x) f(x) (1) f(x)?