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B2 3 2005 (10:30 12:00) 201 2005/10/04 10/04 10/11 9, 15 10/18 10/25 11/01 17 20 11/08 11/15 22 11/22 11/29 ( ) 12/06 12/13 L p L p Hölder 12/20 1/10 1/17 ( ) URL: http://www.math.tohoku.ac.jp/ hattori/hattori.htm B2 T.A. (TAWARA Yoshihiro) (D1) hattori@math.tohoku.ac.jp 512

1 (, F,µ) F = σ[o] Borel = R n B n = σ[o n ] n µ n ((a 1,b 1 ) (a n,b n )) = (b k a k ) k=1 R n (Lindelöf) µ n : n Borel (, F,µ) (, F, µ) σ n n (R n, F n,µ n ) µ n σ (Carathéodory) σ σ F f {f n } f fdµ= lim f n dµ f = f + + f, f = Re(f)+ 1Im(f) 0 ± ± 0 0 (af + bg) dµ = a fdµ+ b gdµ σ

2 2.1 Fubini [Durrett Chapt. 5], [Williams 3.14, A.3] [Williams 8] 2 σ (, F,µ), (Y,G,ν) ( Y,F G,µ ν) (µ ν)( F )=µ() ν(f ) 0 =0 Fubini n + m µ n+m f ( ) ( ) fdµ n+m = fdµ m dµ n = fdµ n dµ m. Ê n+m Ê n Ê m Ê m Ê n m + n m + n 3 f 3 a.e. µ µ f f 0 a.e. σ 2 (, F,µ), (Y,G,ν) µ f = χ Fubini σ (, F,µ), (Y,G,ν) F G x [] x = {y Y (x, y) } G, x ν([] x ) F ν([] x ) dµ(x) =(µ ν)() (µ ν)() < µ([] x ) <, a.e.. µ ν x a.e. G 2.2 [ ][Williams, 1.6, A1.2 4] [Durrett, App. (2.1)] µ n σ F H σ σ σ

Γ σ σ 0 π Cylinder sets d (λ ) σ π d Dynkin I π I d d[i] =σ[i] π I d G σ[i] G π d σ d[i] σ[i] d[i] π Step 1: D 1 = {B d[i] B C d[i], C I} I d d[i] Step 2: D 2 = {A d[i] B A d[i], B d[i]} Step 1 I Step 1 d = d[i] π Dynkin Dynkin 2 π I σ[i] Carathéodory D = {F σ[i] µ 1 (F )= µ 2 (F )} I d Dynkin Fubini Dynkin (, F) I π H {f : R bdd} (i) (ii) χ I H, I I (iii) {f n } H f n f f H H σ[i] F = {F χ F H} Dynkin F 2.3 Fubini σ (, F,µ), (Y,G,ν) ( Y,F G,µ ν) (µ ν)( F )=µ() ν(f ) 0 =0

σ Carathéodory σ π Dynkin Γ Γ( ) =0 Γ Dynkin [ 5 5.3] d Dynkin σ Γ=m R n+m n + m Dynkin OK n + m Fubini R n+m σ Fubini (, F,µ), (Y,G,ν) (i) F G x [] x = {y Y (x, y) } G, (ii) x ν([] x ) F (iii) ν([] x ) dµ(x) =(µ ν)() (i) A = { F G [] x G,x } A F G π A d Dynkin F [] x [F ] x [ \ F ] x G G (ii) Dynkin G = { F G x ν([] x ) F } (i) d I π Dynkin (iii) G (ii) 2.4 Fubini (, F,µ) f : R + fdµ = χ f(x) t dt dµ(x) µ({f > t}) dt {f > t} = {x 0 0 f(x) t} µ() =1 µ P P[ f >t} ]=:P[f t ] f t fdp=:[f ] f [ f ]= 0 P[ f t ] dt

3 [ 5.6-5.7][ ][ IV 17 18] (, F) R n 3.1 Φ: F R ± σ σ ± µ() < f : R µ Φ() = fdµ µ(),ν() < 2 Φ=µ ν µ() < 1 F, Φ( ) =µ( 1 ) µ( 1 c) Hahn Φ( ) =0 Φ( lim A n ) = lim Φ(A n ) V (Φ; ) = sup Φ(A) A V (Φ; ) = inf Φ(A) A V (Φ; ) = V (Φ; ) + V (Φ; ) Φ A = V 0 V V () =0 Φ V () =0 Φ [ ][ ] Hahn support Jordan [ ] (Jordan ) Hahn [ ] [ ] ([ 5.2]) A Φ(B) Φ(A) B A [ 8.8] Φ(B) Φ(A) a 1 = inf Φ() < 0 A 1 A; Φ( 1 ) > A a 1 /2 > 0 Φ(A) > 0 Φ(A \ 1 )=Φ(A) Φ( 1 ) > Φ(A) A 1 = A \ 1 a j, j,n j,a j, j =1,,k 1, a k = inf Φ() < 0 A k 1 A k 1 k A k 1 ; Φ( k ) >a k /2 > 0 Φ(A k 1 ) > 0 A k = A k 1 \ k Φ(A k ) > 0

B = A \ k = A k Φ σ Φ(B) =Φ(A) Φ( k ) > Φ(A)+ k=1 k=1 k=1 1 a k > 0 < Φ(A) Φ(B) = Φ( k ) < 1 a k a k 0 2 2 k=1 k=1 k=1 Φ(B) > Φ(A)+ 1 2 k=1 a k > Φ(A) C B C A k 1 Φ(C) a k k Φ(C) 0 B A Φ(B) > 0 Hahn Φ V (P )=V (P c )=0 P Φ(A k ) V () =:a {A k } P = A k Φ(P )=a C P c a Φ(C P )=Φ(C)+a Φ(C) 0 P c Hahn P Φ + (A) =Φ(A P ) Φ (A) = Φ(A P c ) ±Φ ± Hahn Φ=Φ + Φ Jordan V =Φ +, V = Φ Φ=V + V V = V V F P P c Φ(F )=Φ(F P )+Φ(F P c ) Φ( P ) F sup V () Φ + () V () Φ( P ) V () = sup n Φ( j ) sup V () =Φ + ()+Φ () =Φ( P )+ Φ( P c ) OK V n n n = j V () = V ( j ) Φ( j ) OK j=1 j=1 (, F,µ) f : R Φ: Φ ± ( ) = f± dµ j=1 j=1 fdµ 3.2 (, F) (signed measure) µ ν ν µ ν µ µ() =0 ν() =0 µ ν ν µ A µ(a) =0,ν(A c )=0 δ 0 (A) =χ 0 A 0 ν µ ( ɛ >0) δ >0; (µ() <δ ν() <ɛ)

ɛ >0, j ; µ( j ) < 2 j, ν( j ) >ɛ = j ν µ ν µ ( ɛ >0) F; µ() <ɛ, ν( c ) <ɛ k=0 j=k µ( j ) < 2 j, ν(j c) < 2 j, j N, = Fatou k=0 j=k j µ, ν Lebesgue Radon Nikodym µ σ Φ Φ ac µ V Φac µ Φ s µ Φ=Φ ac +Φ s µ a.e f : R Φ s () = fdµ, F ν µ µ a.e f : R + ν() = fdµ, F Radon Nikodym f ν µ Radon Nikodym f = dν dµ Φ s [ ] Lebesgue Radon Nikodym µ σ Hahn Φ ν (*) φ : R + Ψ (*) φdµ< F φ : φdµ F φ() ν(), F, 0 χ Ψ Ψ α = sup F φ () 0 α φ Ψ ν() < {φ n } Ψ lim F φ n () =α f(x) = sup φ n (x) n 1 f f Ψ F f () =α n f n = max{φ 1,,φ n } F = {f n = φ i } i {f n = n n φ i } = i i j =, i j, f n dµ = F φi ( i ) ν() i=1 i=1 lim f n = sup φ n = f F f () = lim f n dµ n 1 F f () ν(), F, f Ψ α F f () = lim f n dµ lim φ n dµ = α F f () =α F f Φ = ν F f 0 µ, ν ν 0 µ n, n F; µ( n ) > 0; ( n ) ν() n 1 µ() Φ n = ν n 1 µ Hahn n ; n ν() n 1 µ(), n c ν() n 1 µ() µ( n ) > 0 n 0 = n µ( 0 )=0 0 c ( c n ) 0 ν() n 1 µ() n 1 µ() n=1 n ν() =0 ν µ i=1

Φ = ν F f n, n F; µ( n ) > 0, ( n )Φ () n 1 µ() g = n 1 χ n f + g F f+g () =F f ()+n 1 µ( n ) F f ()+Φ ( n ) F f ()+Φ () =ν(), f + g Ψ F f+g () >F f () =α α Φ ν 0 3.3 1 [ 7.4][ IV 21 22] (R, F 1,µ 1 ) µ 0 D µ(d c )=0 x µ({x}) =0 (R, F) µ µ = µ c + µ d µ c µ d R n D = {x µ d ({x}) 0} µ d ( ) =µ( D) µ c = µ µ d R 1 F : R R ν([a, b]) = F (b) F (a) π σ Caratheodory B 1 F ν F 1 ν ν g(x) dν g(x) df (x) F F = F + F f + =0on, F =0on c n F M [a, b] F (x i ) F (x i 1 ) <M F = F 1 F 2 V = F 1 + F 2 V = V F F g dv F < g g(x) df (x) := g(x) df 1 (x) g(x) df 2 (x) Lebesgue Stieltjes g Riemann Stieltjes F : [a, b] R ( ɛ >0) δ >0; [a, b] n n {(a i,b i ] i =1,,n} (b i a i ) <δ F (b i ) F (a i ) <ɛ i=1 i=1 i=1

F F ν F ν F 1 µ 1 (N) =0 ν(n) =0 ν µ 1 F ν F F F i V F x f : [a, b] R F (x) F (a) = f(y) dy Radon Nikodym Φ s Φ s ((a, x]) F : [a, b] R Radon Nikodym f x f(x) = lim (F (x + h) F (x)) f h Radon Nikodym OK Φ µ 1 Φ( ) = φ(x) dx g Φ V Φ g(x)φ(x) µ 1 fdφ= f(x) φ(x) dx a h 0 1 R F F (x) F (a) = fdf = f(x) F (x) dx F g = f F 1 Riemann F (b) F (a) g(y) dy = [a, b] F φ F (b)φ(b) F (a)φ(a) b Riemann a b a x a F (y) dy g(f (x)) F (x) dx b a φ(x) df (x) = F (x)dφ(x) Riemann

4 L p (, F,µ) fdµ 4.1 L p f n f n lim f n ( f p = f p dµ) 1/p L p L p Banach c x = {y A x y}, x Λ Λ A/ = {c x x Λ} x y x y f, g f = g, a.e., µ({x f(x) g(x)}) =0 f g Y = Y/ L p p 1 L p = L p () f : R a.e. f = g, a.e., (1 + x ) q x q χ x 1 { } 1/p f L p f p = f p dµ f p dµ < f p dµ = f n L p f L p p lim f n f p =0 g p dµ

f n f lim f n (x) =f(x), x-a.e. Hölder Schwarz f, g L 2 fg dµ f 2 dµ g 2 dµ Hölder p>1, 1 p + 1 q =1, f Lp, g L q, fg µ(dx) f p g q. Schwarz Hölder Hilbert f(x) =x p /p +1/q x 0(x 0) x = ab q/p b q ab ap p + bq q (a 0, b 0) 0 < f p g q < a = f(x) / f p, b = g(x) / g q Minkowski p 1, f,g L p f + g p f p + g p. p p =1 p>1 f + g p p = f + g p 1 fdµ+ f + g p 1 gdµ Hölder Banach ( ) f =0 f =0) ( af = a f ) (, ) 2 ρ (ρ(f,g) 0) ρ(f,g) =0 f = g) ρ(x, y) ρ d(x, y) = 1+ρ(x, y) (, ) ρ(f,g) = f g Banach (, ) R n, C n R, C ( n ) 1/p Banach x p = x i p l p p 1 Banach p p C 0 ([0, 1]) sup sup u(x) = lim u n(x) ɛ>0 y N(y) m N(y) u m (y) u(y) <ɛ m(y) N(y) u n u sup u n (y) u m(y) (y) + ɛ. y [0,1] i=1

sup n, m(y) ɛ u n u in norm u C 0 3ɛ ɛ x, y n u(y) u(x) 2ɛ + u n (x) u n (y) u n C 0 u C 0 Lebesgue Riemann L p Riemann L p Banach {f n } L p Cauchy lim f n f m n,m p =0. fn f p n(k) < 2 k, n>n(k), n(k) fn(k+1) f p n(k) < 2 k, f k = f n(k) g n = f n 1 1 + f j+1 f j L p g(x) = lim g n(x) j=1 g n p f p 1 +1 lim g n = lim g n p p f p 1 +1 g L p f 1 + ( f j+1 f j ) = lim f n = f g(x) < a.e.-x f g, a.e. j=1 f L p f(x) f n (x) g(x) lim lim fn f p n(k) + lim fn(k) f p n,k k f f p n =0 lim f n f p f n L p, n N, lim f n f p =0 lim f n(k)(x) =f(x), a.e.-x, L p k n(k) f; lim f n(k) = f, a.e., k lim f p f n(k) =0. k f f p f f n(k) p + fn(k) f p 0(k ). 4.2 1. µ() < 1 <p<p L p L p f = f p g =1 f p Hölder 2. 1 p <, f L p (R n ), g L 1 (R n ) (f g)(t) := Ê n f(x y)g(y)dy f g L p (R n ) 3.(a) = {χ A µ(a) < } L p L p χ n φ L p {0, 1} (b) f L 1 F f : R φ = {χ A µ(a) < } L p F f (φ) = f(x)φ(x) dµ(x) F f χ A χ A p = µ(a A) 1/p χ A χ A (L p ) µ(a A) 0 µ(a) 0 A f dµ(x) 0 f n f n A f dµ A f n dµ sup f n < µ(a) F f (χ A ) F f (χ A ) f(x) dµ(x) 0, χ A χ A p 0. A A

4. Banach (, d) T : ; a (0, 1); d(tf,tg) ad(f,g), f,g (a) f 0 f n = Tf n 1, n =1, 2,, {f n } (b) f = lim f n Tf = f (c) Tf = f f f 5. (C([0, 1]); 1 ) 4.3 L p = Banach [ 23] M 24 L f a f(x) a, a.e.-x, a ess. sup x f(x) f L a.e. f = ess. sup x f(x) (L, ) Banach Hilbert ((f,g) =(g, f)) ((af +bg, h) =a(f,h)+ b(g, h)) ((f,f) 0) ((f,f) =0 f =0) 2 (, ) (f,g) = fgdµ, f, g L 2 g g L 2 f = (f,f) Hilbert Banach L p L 2 Hilbert Hilbert Banach

5 p 1, x, a, b > 0 (a + b) p (1 + x) p 1 a p + x ( 1+ 1 x) p 1 b p x =1

6 n 6.1 σ σ σ σ σ f 1 ((a, )) F 1 0 0 σ 6.2 6.2.1 0 sup [0, 1] L 1 L 1 sup (decay) 1/x decay 6.2.2 n Wiener