untitled

Transcription

1 1 kaiseki1.lec(tex) ; ;16 26;0329; 0410;0506;22; ;08;20;0707;09;11-22;24-28;30;0807;12-24;27;28; (σ,F = µ);0212( ); 0429(σ- A n ); 1221( ); ;30(L p ); ( subsubsection phi ); 24( ); 0926( Omega Ω ); ((5), 11) ( 5.2) cf. ( 16.3) ( 100) σ = Hopf ( 4.2) ( 4.3) Fubini ( 8) Radon Nikodým ( 12) Lebesgue Riemann ( ) Radon Nikodým ( 12) = cf. ( 16.3) Stieltjes ( 14) = L p

2 c b c (i) = +, f + g = f + g, a a b (ii) Riemann (iii) (iv) g = g σ Wiener Lebesgue σ Wiener 2 Cantor 3 Riemann ( ) f n (x) dx = f n (x)dx, A A ( ) ( ) f(x, y)dy dx = f(x, y)dy dx = A B B A A B f(x, y)dxdy, 1 [Lebesgue, 40 (p.56)] 2 cf. Lebesgue [ ] 3 [, p.106] Borel

3 0 3 robustness [, p.2] Banach Radon Nikodým Stieltjes Ω 4 φ i=1,. i Λ A i = A 3, i 3, A i = A 1 A 2 A 3 i=1 A + B, A i, cf. A B i=1. de Morgan 1,2 [, p.7] de Morgan 3 Λ (index set) A λ, λ Λ. B λ,1 = A λ, B λ,0 = A λ c, Ω= (A λ A c λ )= B λ,σ(λ). λ Λ σ Σ λ Λ Σ={σ : Λ {0, 1}} =2 Λ lim sup A n = ν=n A ν lim inf A n = ν=n A ν N, Z +, Z, Q, R. [, p.9] [, p.11] 5 4 [ ] X 5 [, p.3]

4 R (a, b] (a, ), (,b], (, ) =R, φ I 1 = {(a, b] a i b i } I 1 b = (a, b] =(a, ) ± (, ] ± R {+ } [, p.12] Z = X Y = {(x, y) x X, y Y }: [, p.7] R N, R N R I N = {R N } = {(a 1,b 1 ] (a 2,b 2 ] (a N,b N ] a i b i }. 1 I N 1 I N 1 I N 1 (a, b] I N.. R N =(, ] N,(a, a] I N 1 = φ A o = {x Ω ɛ>0; Ball(x, ɛ) A} Ā = {x Ω {x n } A; lim x n = x} 0.3. (family) 6 cf. ZF {u A(u)} class von Neumann class Ω 02. Ω 2 Ω Ω, F 2 Ω, Φ: F R. 9 N : J N = {R N n } = { I i n N, I i I N }. f : R n R i=1 J N Φ() = f(x) dx; dx = dx 1 dx 2 dx N.. R {+ } 0.4. exp, n!, Stirling [, pp ]: lim N N! N N e N 2πN =1. 6 [, 162B ] ( ) 7 123C BG 8 [, 28 (p.209)] 9 [, p.13]

5 ( +1) σ Ω Ω F 2 Ω σ A F A c F, A n F, n N, A n F, A B F de Morgan, φ F, Ω F,. σ σ F 2 Ω : σ µ : F R {+ } 0 µ(a), A F, σ µ( A i )= µ(a i ), A i F, i N, i=1 i=1 (Ω, F,µ) F (µ-) n n µ(φ) =0, µ( A i )= µ(a i ) A B µ(a) µ(b), i=1 A B, µ(b) < µ(a B) =µ(a) µ(b) µ( A i ) µ(a i ), 2 {A n } F (i) A 1 A 2 µ( A n ) = lim µ(a n), (ii) µ(a 1 ) <, A 1 A 2 µ( (iii) µ( k=n A k ) lim inf µ(a n), i=1 A n ) = lim µ(a n), i=1 i=1

6 1 6 (iv) µ( A n ) < µ( (v) µ( A n ) <, k=n k=n A k = A k ) lim sup µ(a n ), k=n A k = lim A n µ( lim A n) = lim µ(a n).. [, 6.2 (pp.31 32)].. 2 µ(a 1 ) < Ω =N, F =2 Ω, µ({n}) =1, A n = {n, n +1, }. 3 A n F, n N, µ(a n ) < µ( A k )=0, µ( A k )=µ(ω).. 2 de Morgan 10 k=n k=n µ(ω) = 1 µ P (Ω, F,µ) Ω=N, F =2 Ω, µ({n}) =1,n Ω. µ(a) = A <p<1, N N fix (Ω N, F N,P N,p ) Ω N = {0, 1,,N}, F N =2 ΩN, P N,p ({n}) = N C n p n (1 p) N n, n Ω N Poisson λ >0 fix. (Ω, F,P λ ) Ω=Z +, F =2 Ω λ λn, P λ ({n}) =e n!, n Ω. Poisson p = p N = λ/n, N, P N,pN ({n}) P λ ({n}) Stirling 10 [, 6 3 (pp.35, 273)]

7 (Ω, B,P) Ω=N, B = {φ, 2N, 2N 1, N}, P( ) =P ( ) = 1 2. P well-defined (P ({n}) =α) Ω (Ω, F,µ) F =2 Ω, µ({n}) =1, n N. P µ m/n 11 m, n p p 2 P m/n p P m, n p p P(1 2 )= ( ) 1 1 n 2 = ζ(2) 1 =6/π 2. A = {(A 2,A 3,A 5, ) A p {( emptyset, pn pn, (pn pn) c, N N}, p P} (Ω, F,P) Ω=N N, F pn pn, P(pN pn) =p 2, p P, ( (A p ) A) P p P A p = P (A p ), p P 12 (Ω, F,P) n N p P g : N N A g(n, m) (φ, φ) N N p n, p m, g p (n, m) =pn pn, g p (n, m) =(pn pn) c, g : N N A g =( g 2, g 3, g 5, ) F = { g 1 (s i )( Ω) {s 1,s 2, } A, i=1 s i s j,i j}, s =(A p ) A P (g 1 (s)) = χ(a p φ; p P) p P; A p=pn pn = χ(a p φ; p P) ζ(2) 1 p 2 p P: A p=pn pn (1 p 2 ) p P; A p=(pn pn) c (p 2 1) 1, σ P F σ F 11 [Cacoullos, p.33, Q.155, p.176] 12 A p = φ p 0, A p = N p

8 1 8 F {1} = g 1 (0, 0, 0, ) g 1 (1, 0, 0, )={2 m m N}, g 1 (1, 1, 0, )={2 m2 3 m3 m 2,m 3 N}, g 1 (a 1,a 2, ) φ k N; a n =0,n k P Ω=N, F pn, P(pN) =1/p, p P, (Ω, F,P) ( ) A p {pn, (pn) c } ( F) p P, P A pi = P (A pi ), i=1 i=1 {p i } P. N = A p B i, i N, {A p} p P B i F, 1 =P (N) P (B i ). P (A p ) {1/p, 1 1/p} P (A p ) 1 1/p. i=1 P (B i ) p P(1 1/p) = (Ω, F,µ) µ(ω) < A = {x {x} F, µ({x}) > 0} A = {x µ({x}) > 1 n } 13 µ({x}) =0 14 Lebesgue Lebesgue 5 (Lebesgue ) R N I =(a 1,b 1 ] (a 2,b 2 ] (a N,b N ] I N N µ N (I) = (b ν a ν ) µ I N σ F N 2 RN ν=1 I N σ Lebesgue 13 [, p.35, 2]

9 Wiener N N t >0, x R N, p(t, x) =(2πt) N/2 exp( x 2 /(2t)) W R + R N w : R + R N 6W I = {{w W w(0) = 0, w(t 1 ) I 1,,w(t n ) I n } n Z +, 0 <t 1 <t 2 < <t n R +, I i I N, i =1, 2,,n} I i Borel sets I cylinder set I = {w W w(0) = 0, w(t 1 ) I 1,,w(t n ) I n } P (I) = dx 1 dx 2 dx n p(t 1,x 1 ) p(t 2 t 1,x 2 x 1 ) p(t 3 t 2,x 3 x 2 ) p(t n t n 1,x n x n 1 ) I 1 I 2 I n n =1, I 1 = R P (W) =1 P I σ I σ Wiener Lebesgue 2, σ ( 14) (1) [, 2.1 (pp.51 56)] [, pp.5 8] Wiener Brown [, p.35, 1,3, p.273] 2. Lebesgue Ω J 2 Ω A J A c J, A, B J A B J, A B J de Morgan, A B J φ J, Ω J, σ

10 Ω J 2 Ω m : J R {+ } J Jordan 15 0 m(a), A J, m(a + B) =m(a)+m(b), A, B J, n n m(φ) =0, m( A i )= m(a i ) A B i=1 i=1 n m(a) m(b), m(b) < m(a B) =m(a) m(b) m( A i ) i=1 i=1 n m(a i ) σ (Ω, J,m) J (σ) J {A i } J J m( A i )= m(a i ) i=1 i=1. σ [, p.55] σ 7 m J σ (1) A n J, n N, A 1 A 2, m(a 1 ) <, A n = φ = lim m(a n)=0, (2) A n J, n N, A 1 A 2, A =. [, pp.53 54] A n J, m(a) = = lim m(a n)=.. (i) m(ω) < (e.g. ) (1) m() < (ii) m σ (2) (25) ( ) Γ: 2 Ω R {+ } 16 0 Γ(A), Γ(φ) =0, Γ(A) Γ(B), A B ( ) Γ A i Γ(A i ), i=1 i=1 15 [, p.17] 16 [, p.23] [, pp.25 26] [, pp.59 60] R R 2 Haussdorff d

11 (Ω, J,m): J m Γ (3) Γ(A) = inf{ m( n ) { n } J, A n } well-defined on 2 Ω A Ω 8([, p.23] ) (i) Γ (ii) m J σ Γ m Γ J = m 17.. [, p.24] Γ σ 1 (Carathéodory, [, p.26]) Ω Γ (4) ( A Ω) Γ( A)+Γ( c A) =Γ(A). F. Γ Γ( A)+Γ( c A) Γ(A) Ω F (3) 9 F σ µ def =Γ F F F µ-. F µ µ σ [, 5.3 (pp.27 28)] Γ J m (3) Γ F J F.. [, p.27]. (i) m σ (ii) Γ() m(). 18

12 J σ m Γ m (3) F 11 Carathéodory (4) A Lebesgue 2 ([Lebesgue, 3,4 (p.9)]) Ω Γ L- (5) I J; I, m(i) <, Γ() =m(i) Γ(I ).. Lebesgue I 19 Ω σ m(i) < L- F f L (4) (5) F f def = F { Γ() < } F f L. 11 ([, p.425], [, p.95]) Γ() < F f L = F f, Lebesgue Carathéodory F L = { 2 Ω Γ( J)+Γ( c J) =m(j), J J} F F L 12 F = F L.. 20 F L ɛ>0, A 2 Ω, m(j n ) Γ(A)+ɛ, J n J (n N) J n A, Γ(A) Γ( A)+Γ( c A) Γ( n J n )+Γ( c n J n ) = n (Γ( J n )+Γ( c J n )) = n m(j n ) Γ(A)+ɛ. F L J n J ɛ 0 F F f L 12 F L, J J Γ( J)+Γ( c J) m(j) m(j) = m(j) < Γ Γ J = m ( 8) J F( 10) F f L I J; m(i) <, I, Γ()+Γ( c I) =m(i). J J F I Γ(I J )+Γ(I J c ) =Γ(J )+Γ(J c ) =Γ() Γ(I c )=Γ(I J c )+Γ(I J c c ). 19 [Lebesgue, 3,4 (p.9)]. 20 [ ] [ ] 12 [ ] [ ] suspend Lebesgue

13 1 13 Γ(I J )+Γ(I J c )+Γ(I J c )+Γ(I J c c )=m(i) Γ(I J c )+Γ(I J c c ) Γ(I J c )=m(i J c )=m(i) m(i J) m(i) < (6) Γ(I J )+Γ(I J c ) m(i J). I Γ(I c J ) =0. m(i c J) =Γ(I c J) Γ(I c J c )= Γ(I c J c )+Γ(I c J ) (6) m(j) Γ(J )+Γ(J c ) 2.2. Lebesgue 5( 1.2.8) Lebesgue N (Ω, F) =(R N n, J N ); J N = { I i n N, I i I N, i =1, 2,,n}; I N = {(a 1,b 1 ] (a 2,b 2 ] (a N,b N ] a i b i }. i= J N. 22 Ω=(, ] N J N, A + B J N n n A B I I N I c J N ( I i ) c = I c i A c A B = A +(B A c ) I i = {(a i1,b i1 ] (a i2,b i2 ] (a in,b in ]} I N, i =1, 2 (a 1ν,b 1ν ] (a 2ν,b 2ν ]=φ ν I 1 I 2 = φ I N, (a 1ν,b 1ν ] (a 2ν,b 2ν ] = (min{a 1ν,a 2ν }, max{b 1ν,b 2ν }] I 1 I 2 I N, I N J N I N J N I = {(a 1,b 1 ] (a 2,b 2 ] (a N,b N ]} I N I c = (,a 1 ] R N 1 +(b 1, ] R N 1 +(a 1,b 1 ] (,a 2 ] R N 2 +(a 1,b 1 ] (b 2, ] R N 2 + +(a 1,b 1 ] (a k 1,b k 1 ] (,a k ] R N k +(a 1,b 1 ] (a k 1,b k 1 ] (b k, ] R N k + +(a 1,b 1 ] (a N 1,b N 1 ] (,a N ]+(a 1,b 1 ] (a N 1,b N 1 ] (b N, ] J N. i=1 i=1 J N = n I i {I i =(a i1,b i1 ] (a i2,b i2 ] (a in,b in ] I N,i=1, 2,,n} i=1 (7) n N m N () = (b iν a iν ), i=1 ν=1 m N () =, m N (φ) =0 m N N 14 m N J N

14 1 14 (i) well-defined (ii) σ 23. (i) [, p.14]. σ, n J N, n N, p = n m N m N () m N ( n ), p N. p (8) m N () m N ( n ). n = k n i=1 I ni k n N I ni I N = ɛ>0 I ni J ni o (9) (10) = i=1 k n i=1 k n I ni m N (J ni ) k n i=1 k n i=1 J ni o, m N (I ni )+ɛ, k n i=1 I ni. I ni {J ni } m N (J ni ) m N (I ni )+ ɛ 2 n k n k J N = Ĩ i {Ĩi} I N α<m N () F J N (11) F, m N (F ) >α, i=1 i Ĩi m N (K i ) >m N (Ĩi) (m N () α)/n, Ki I i, K i I N {Ĩi} k {K i } F = K i F (9), (11) F k n i=1 J ni o, i=1 {J o ni } F Borel Lebesgue [, p.258] n 0 F F n 0 k n i=1 J ni o n 0 k n i=1 J ni. m N (10), (11) α<m N (F ) < n 0 k n i=1 α m N (), ɛ 0 (12) m N () m N ( n ). 23 [, pp.19 22] m N (J ni ) k n i=1 m N (J ni ) k n i=1 m N (I ni )+ɛ = m N ( n )+ɛ.

15 1 15 (8) (12) σ. σ (8) N I =(a 1,b 1 ] (a 2,b 2 ] (a N,b N ] I N m N (I) = (b ν a ν ) J N m N ( (7)) 14 σ (3) 9 (Ω, F N,µ N ) 10 8 J N F N µ N JN = m N (Ω, F N,µ N ) (Ω, J N,m N ) 3 (R N, F N,µ N ) (R N ) Lebesgue (3) (Ω, J N,m N ) µ N =Γ Lebesgue. R n Lebesgue Lebesgue 11.2 ν= Lebesgue (R N, F N,µ N ): Lebesgue R N (Lebesgue) R N Lebesgue ( 15) O N 2 RN : R N O N = {G R N ( x G) a <x<b; (a, b) G} = { }. φ, R N 15 Lebesgue O N F N. G O N x G x I o x I x G I x I o x I x x G 1/2 I x I x I o x G = I o x Lindelöf [, p.260] G I o n, n N, I n G. I n G I n G. G = I n σ F N x G ([, 7.3]) µ N Lebesgue A RN µ N (A) = inf{µ N (G) A G, G O N }.. [, p.36]

16 A ɛ>0 F A G, µ(g F ) <ɛ, G F. [, (p.38)] 4 G δ F δ 18 A F A G, µ(g F )=0, G δ G F δ F. n N F n A G n, µ(g n F n ) < 1 n, G n F n F = F n, G = G n, R N O O ( 4.1) σ Borel Borel Borel R N Borel B N 15 R N Lebesgue Borel Borel Borel ( 4.2) Borel Lebesgue Borel ( 4.3) Borel I =[0, 1] (cf. Cantor set 0) 24 M =1 2 n ɛ n > 0 ɛ n > 0 ɛ n =2 2n 1 M =0.5 I I ɛ 1 J 1 (I 1 = I J 1 ) I 1 ɛ 2 J 21, J 22 I 1 (I 2 = I 1 J 21 J 22 ) = I M>0 Riemann Lebesgue d dx 0 ( ) x f(y)dy = f(x). 25 n I n I n < 1 =[0, 1] \ I n I n [0, 1] 1 I n > (R N, F N,µ N ) Lebesgue 19 A µ N (A) = nabe35 26 Jan 96 22:37:07, (2), 28 Jan 96 00:05:05 [Lebesgue, 29 (p.37)] 25 [Lebesgue, 34 (p.46)].

17 a>0 a/2 σ 0 0 Cantor [, p.41] Cantor [, p.43] Cantor Cantor R [, p.42 43] Lebesgue R 2 Lebesgue R Lebesgue 3 (X, F X,µ X ), (Y,F Y,µ Y ) Z = X Y = {(x, y) x X, y Y } 07 5 Z K = F = {(x, y) x, y F }, F X, F F Y, 6 σ µ { 0, µ X () =0or µ Y (F )=0 (13) µ(k) = µ X () µ Y (F ), otherwise µ X µ Y. (13) σ µ Y (F )= µ X () =0 µ(k) =0 20 J Z def = { n i F i i F X, F i F Y, i =1, 2,,n, n N}, Z F i=1 m( F ) (13) µ = m J Z A = n i F i m(a) = i=1 n m( i F i ) (Z, J Z,m) σ (3) 9 (Z, F,µ). F J Z σ ( 33) µ i=1. J Z m J Z σ 10 8 J Z F µ JZ = m (Z, F,µ) J Z m J Z σ n 21 Z = X Y, J X 2 X J Y 2 Y J Z = { i F i i J X, F i J Y, i =1, 2,,n, n N} i=1

18 Z = X Y J Z J X J Y A + B J N J X, F J Y, ( F ) c =( c F c )+( F c )+( c F ) J Z. n n A B ( ( i F i )) c = ( i F i ) c A c A B = A +(B A c ) n n n n A = Ai F Ai, B = Bi F Bi, A B = ( Ai F Ai ) ( Bj F Bj ) i=1 i=1 i=1 i=1 i=1 j=1 A, B J X, F A, F B J Y K =( A F A ) ( B F B ) J Z K = K def =( A B ) (F A F B ) J Z. (x, y) K = x A B, y F A F B, K K, (x, y) K = (x, y) A F A,(x, y) B F B, K K. 22 (Z, J Z,m) 20 m J Z (i) well-defined (ii) σ 23 J Z, F X, F Y, 20 A J Z n N, j F X, F j F Y, j =1, 2,,n, n (14) A = j F j ; j k j k = φ, j=1. [, p.57]. 24 J Z, F X, F Y, 20 A n J Z, n N, A 1 A 2 A 1 A 27 2 k n N, n N, nj F X, F nj F Y, j =1, 2,,k n, n N, n N (15) A n = k n j=1 nj F nj ; j i nj ni = φ, (1 j k n+1 )1 i k n ; n+1,j ni, A 1 A 2 ( (A 1 A 2 ) (16) ( n, i, j) n+1,j ni F n+1,j F ni ( F n+1,j F ni ),. (i) (ii) n+1,j ni n A 1 A 2 A 1 A 2. [, p.57]. A 1 A 2 (16) (15) [, pp.16 17] cf [, 2 (p.57)]

19 (i) R N [, p.14] σ 7 (1) (2) (2) A n J Z, n N, (17) A 1 A 2, A = A n J Z, m(a) =, A J Z A m(a) = A F m( F )=. (13) µ = m µ X () µ Y (F ) (18) A F, F X, F F Y, µ X () =, µ Y (F ) > 0, 24 (15) k n, nj, F nj, A n = (15) n F = (19) ni F ni F ni A n. i: F F ni i: F F ni (20) F X σ µ X σ n n+1, n N, n, k n j=1 nj F nj n = i: F F ni ni (18) 7(2) lim µ X( n) = (18) µ Y (F ) > 0 (19) m(a n ) m( n F )=µ X( n ) µ Y (F ), n, (2) (20) x n n (15) i, j x n+1,j ni. n (16) F F ni F n+1,j. x n+1 n n+1 x n {x} F A n, 1 i; x ni, {x} F ni F ni. x n n (1) A n J Z, n N, (21) A 1 A 2, A n = φ, m(a 1 ) <, 24 (15) k n, nj, F nj, A n = k n j=1 nj F nj µ X ( nj )=0 µ Y (F nj )=0 j A n (13) µ = m m(a n )=m(a n) lim m(a n)=0 µ X ( 1j ) > 0, µ Y (F 1j ) > 0, j =1, 2,,k 1, (13) (22) M def = k 1 j=1 (µ X ( 1j )+µ Y (F 1j )) < cf. 14 (2) [, p.58] σ Borel σ- Borel Stieltjes 14

20 1 20 (21) k n j=1 c>0 n = (µ X ( nj )+µ Y (F nj )) M n N i: µ Y (F ni) c M(c + µ X ( n )), n N, (23) lim sup ni, n = i: µ Y (F ni)<c m(a n ) M(c + lim sup µ X ( n )), c>0. F X σ µ X σ (24) n n+1, n N, n = φ, ni, m(a n ) cµ X ( n )+Mµ X( n ) (22) 7(1) lim µ X( n )=0 (23) c 0 (1) (24) x n+1 n (15) i, j x n+1,j ni. n+1 (16) c µ Y (F n+1,j ) µ Y (F n,i ). x n n+1 n x n n 1 i n k n x nin, µ Y (F nin ) c>0. 24(15)(16) F n+1,in+1 F n,in. F X σ µ X σ (22) 7(1) F nin φ y F nin (x, y) nin F nin A n. (x, y) A n, n N (21) n = φ (Ω, J,m) 9 (Ω, F,µ) R p+q Lebesgue ( 2.2) R p Lebesgue R q Lebesgue ( 3) I p+q F p, F q, Ω A A σ σ[a] A σ C A σ[a] C σ[a] A σ Ω A σ σ[a] = F F: A σ 29 [, 6.3 (p.32)]

21 ( ) (Ω, O) σ[o] Borel 30 Borel Borel R N ( 2.3.1) 26 B N def = σ[o N ]=σ[j N ]=σ[i N ].. = n i=1 ν=1 N (a iν,b iν ] J N O M = n i=1 ν=1 b iν + 1 M = N (a iν,b iν + 1 M ), M N, b iν = M=1 O M = σ[o N ]. J N σ[o N ] σ[j N ]. σ[j N ] J N I N. σ[j N ] σ[i N ]. G O N x G x I o x I x G I x I o x I x x G 1/2 I x I x I o x G = I o x Lindelöf [, p.260] G I o n, n N, I n G. I n G σ[o N ] σ[i N ]. I n G. G = x G I n G σ[i N ]. O N σ[i N ] 8 F Ω σ J F µ J m µ J = m m F F 4.2. (Ω, J,m) 9 σ m(x n ) <, X n J, X n, n N, Ω= X n. (i) X n = X n \ (ii) X n = k=1 ( n 1 k=1 X k ) Ω= X n, m(x n ) <, n X k X 1 X 2,Ω= X n, m(x n ) <, (ii) 8 (iii) σ Hopf ( 28) ( 4.3) Fubini ( 8) Radon Nikodým ( 12) 27 m σ σ 7( 2.1.3) (2) (25) σ- Borel Stieltjes 14

22 1 22 (25) ( A J; m(a) = ) lim k m(a X k)=.. [, pp.54 55] 28 (Hopf ) (i) J m σ[j ] m J σ (ii) m σ. (3), 9 µ 10 σ[j ] F, 8 µ J = m. [, pp.52 53]. σ [, p.53] σ[j ] (3) 9 µ 10 F (Ω, F,µ) F µ σ (3) A Ω (26) µ (A) = (inf{ µ( n ) { n } F, A n } =) inf{µ(b) B F, A B} 8 µ µ B 9 B σ (Ω, B,µ ) 8 10 F B µ F = µ, µ µ 29 (Ω, F,µ) J σ m (3) 9 µ =Γ (Ω, F,µ)=(Ω, B,µ ).. 9 σ (3) A B Γ(A) Γ(B). B Γ(A) µ (A). ɛ>0 A n, { n } J,Γ(A) m( n ) ɛ, { n } 8 m( n )=Γ( n ), 10 n F m( n )=Γ( n ). A n F µ (A) Γ( n )= m( n ) Γ(A)+ɛ. ɛ 0 µ (A) Γ(A)

23 (Ω, F,µ) 9 10 N F µ(n) =0. R N def 30 F = { Ω F 1,F 2 F; F 1 F 2, µ(f 2 F 1 )=0} µ F σ F c 2 c F c 1, µ(f c 1 F c 2 ) = µ(f 2 F 1 ) = 0 F 1n n F 2n, µ(f 2n F 1n )=0,F 1n, F 2n F, n N, = n, F i = F in F, i =1, 2, F 1 F 2, µ(f 2 F 1 ) µ(f 2n F 1n )=0 F σ F µ σ B F 1,F 2 F, F 1 F 2, µ(f 2 F 1 )=0, F 1 B F B. 11 (Ω, F,µ) F 0 31 F 30 F 1,F 2 F µ() =µ(f 2 )(=µ(f 1 )) µ F. F 1, F 2 F F 2 F 2 F 2 F 2 F 1, F 2 F 2 F 2 F 2 F 1, µ(f 2) µ(f 2 ) max{µ(f 2 F 1), µ(f 2 F 1 )} =0, µ(f 2)=µ(F 2 ) µ well-defined. A N, N F, µ(n) =0, N F; (φ ) A N N, µ(n )=0. A F, µ() 0 = n F 1n,F 2n F; F 1n n F 2n,µ(F 2n F 1n )=0,µ(F 1n )=µ(f 2n )= µ( n ), n N, µ( F 2n F 1n ) µ( (F 2n F 1n )) µ(f 2n F 1n )=0, F 1n µ() =µ( F 1n )= µ(f 1n )= µ( n ). σ F 2n µ F = µ 31 µ 31 (Ω, F,µ) F (Ω, F, µ) (Ω, F,µ) (Ω, F,µ) (26) (Ω, B,µ ) (i) B F (ii) (Ω, B,µ ) 32 [, (8.1) (p.44)] µ(f 2 F 1) = 0 µ(f 1 )=µ(f 2 )

24 1 24 (iii) µ σ ( B) F 1,F 2 F; F 1 F 2, µ(f 2 F 1 )=0.. A N, µ(n) =0,N F, µ (A) µ(n) =0 µ (A) =0. C Ω µ (A C)+µ (A c C) µ (A)+µ (C) =µ (C) A B. B µ (26) 30 (i) A N, µ (N) =0,N B, A B. (ii) F 2 def = (iii) µ () < µ ( n N) F 0n F; F 0n, µ(f 0n ) µ ()+ 1 n. F 0n F 2 F µ (F 2 ) =µ (F 2 ) µ () 1 n, n N. µ (F 2 ) =0. F 2 N F; F 2 N, µ(n) =0. F 1 = F 2 N F F 1, µ(f 2 F 1 ) µ(n) =0, (iii) µ () < X k =Ω,µ(X k ) <, k N, {X k } F( B) k=1 k = X k ( B) µ ( k ) <µ (X k )=µ(x k ) < (µ F = µ) F 1k F 2k, µ(f 2k F 1k )=0, F 1k,F 2k F F i = F ik F, i =1, 2, (iii) k=1 33 µ σ (Ω, F,µ) (26) (Ω, B,µ ) (Ω, F, µ). σ (J,m) (3) 9 (F,µ) 28 σ[j ] 33 F σ[j ], B F, µ µ m. 32(i)(iii) F = B. B B 32(ii) F 1, F 2 F= B µ () = µ (F 1 )+µ (F 1 ) =µ(f 1 )= µ(). µ F = µ Lebesgue 34 R N (R N, F,µ) I N F I =(a 1,b 1 ] N (a 2,b 2 ] (a N,b N ] I N µ(i) = (b ν a ν ) F N = σ[j N ] Lebesgue (R N F N,µ N ). Hopf F =2 Ω 2 Ω R N Lebesgue ( ) ν=1. µ N σ[j N ] m N (F N,µ N ) (σ[j N ],µ N )

25 1 25 R p+q Lebesgue R p Lebesgue R q Lebesgue 34 [, p.276] - 34 [, p.59, 2]

26 ( +1, +2) σ- (Ω, F), (Ω, F ) f : Ω Ω (27) A F = f 1 (A) ={x Ω f(x) A} F. σ- F/F f f σ- F F σ- σ- 35 f : Ω Ω σ- F 2 Ω F def = {A Ω f 1 (A) F} σ-. f 1 (Ω )=Ω F Ω F A F f 1 (A) F F σ- F f 1 (A) c = {x Ω f(x) A} = f 1 (A c ). A c F. A n F, n N, F σ- F A n } = f 1 ( n A n ). f 1 (A n )={x Ω f(x) n A n F. (27) Ω = R ( {± }) ( 38) (27) 40 Banach (27) Radon Nikodým R ( 16.3 ) Ω = R ( {± }) Ω R N remark (Ω, F) Riemann x Lebesgue y Lebesgue Lebesgue x Lebesgue ( )

27 2 27 Riemann Ω =R N F Lebesgue 2.2 F N (Ω, F) A Ω { 1, x A, χ A (x) = 0, x A, χ A A χ χ Ω 1 χ 0 F =Ω k 14 j j F, = j j, j =1, 2, 3,,k, a j R, j =1, 2, 3,,k, f =. ± (a j R) j=1 j=1 k a j χ j (Ω, F) Ω = R {± } Ω = R, F = B 1, [, 10 ] ± Ω Ω =Ω {a} a Ω Ω J Ω F = σ[j ] A n J, n N; A n =Ω, Ω n σ[j {a}] =F {B {a} B F}.. L R R Ω σ- B F= B c = {a} Ω\ B R L R L L = L 2 Ω L σ- Ω= A n L A L L n Ω \ A L Ω Ω, Ω \ A L. A n L L n A n L L Ω σ- J L F L L L {a} L σ- L R (28) Ω = R {± }, F = {B C B B 1,C 2 {± } }, 36 Borel F = B N = σ[o N ] 37 36, 37, [ ]

28 {± } = {, {+ }, { }, {± }}. B 1 Borel B 1 = σ[o 1 ]( 2.3.1, 4.1) O 1 R 36 (29) F = σ[o 1 {{+ }, { }}] 2 Ω σ-. F = σ[o 1 {{+ }, { }}] 40 R F O 1 F {{+ }, { }} F σ- 35 [, 10] Ω F = σ[{(a, + ] a R}]. (a, + ] ={x R x>a} {+ }.. 26 B 1 I 1 σ- : B 1 = σ[o 1 ]=σ[i 1 ]. {+ } = (n, ], { } = ( n c ( ) ( n, ]),(a, b] =(a, ] \ (b, ], R = ( n, ] \ { }, σ[{(a, + ] a R}] n I 1 {{ }, { }} 36 (28) σ[{(a, + ] a R}] F. F O 1 (a, b) ( <a<b< ). F { }. F n n (a, n) { } =(a, ]. F σ[{(a, + ] a R}]. 38 (Ω, F ) (28) f : Ω Ω F/F ( 5.1.1) (30) f 1 ((a, + ]) F, a R,. (30) F = {A Ω f 1 (A) F} 35 Ω σ- (30) (a, + ] F, a R. 37 F σ[{(a, + ] a R}] =F. (27) f F/F f F/F 37 F (a, + ], a R. (27) (30) Ω = R {± }, F = {B C B B 1, C 2 {± } }, F/F 39 f f, max{f, 0}, min{f,0},. a R ( ) c ( f) 1 ((a, + ]) = f 1 ([, a)) = f 1 ([ a, ]) c = f 1 (( a 1 n, ]). 38 ( f) 1 ((a, + ]) F 38 f { (max{f, 0}) 1 Ω, if a<0, ((a, + ]) = f 1 ((a, + ]), if a 0, [ ] ( 5.1.1) Borel 38 [ ] 36, 37 [, p.73] A n

29 2 29 max{f, 0} min{f, 0} = max{ f, 0}, 40 f max{f, 0}, min{f,0} {f n } max{f, 0} ( min{f,0}). 39 max{f, 0}, min{f,0} f 0, x f 1 (0), (31) f n (x) = (k 1)2 n, x f 1 (((k 1)2 n,k2 n ]), k =1, 2, 3,, 2 n n, n x f 1 ((n, ]), f f 1 (((k 1)2 n,k2 n ]) F f 1 n (a) F f n k. (i) f f = a j χ j (31) j f n = j=1 j=1 k a nj χ j j a nj (ii) (Ω, F,µ) F =Ω f : Ω fdµ= f(x) dµ(x) = f(x) µ(dx) = f = j=1 n j j F, j =1, 2, 3,,n, a j 0, j =1, 2, 3,,n, j=1 n a j χ j fdµ= j; a j 0 a j µ( j ) f µ( j ) j a j =0 j 1 j n a j 0 µ( j )= j fdµ= f, g f + g (32) (f + g) dµ = fdµ+ gdµ. f g = A + B (f A = (33) A fdµ gdµ. F f A, B F n a j χ j A ) j=1 fdµ+ fdµ= fdµ. B

30 2 30 f F F fdµ fdµ.. a j R (i) a j =0 j µ( j )= 0=0 µ() < (ii) a 1 = a 2 =1,µ( 1 )=µ( 2 )= µ() < f 40 (31) {f n } fdµ= lim f n dµ f (32) f f f + = max{f, 0}, f = min{f, 0}, 39 f = f + f. f ± dµ f µ fdµ= f + dµ f dµ f (Ω, F,µ)=(R N, F N,µ N ) (Lebesgue ) f(x) dx. fdµ Lebesgue f fdµ = (34) f + dµ f dµ f + dµ + f dµ = f dµ < Ω (Ω, F) Ω = R {± }, F = {B C B B 1, C 2 {± } }, F/F f : Ω Ω 11

31 f, g a, b f a (a 0), af+ bg, fg f a a<0 0 a = {f n } sup f n, inf f n, lim sup f n, lim inf f n, lim f n, f n n 1 n 1 n,. f 39. [, ] goroff Lusin ( 44) 42 (goroff) F, µ() <, f n : Ω, n N, µ( {x Ω f n (x) =± }) =0 µ( {x Ω f(x) = lim f n(x) < } c )=0 ɛ>0 F, µ( F ) <ɛ F F F {f n } f. [, pp.64 65, 10.9] Ω (Ω, F) =(R N, F N ) R N Lebesgue Lebesgue Lebesgue Ω R N (Ω, F,µ)=(R N, F N,µ N ) Lebesgue F N 2.2 Lebesgue 43 F N f : R R {± } F N /F 39. f 1 ((a, ]) = f 1 ((a, )) O N B N F N.. f (f ) 1 (A) =f 1 (A) 44 (Lusin) F N f : R ± ɛ>0 F µ N ( F ) <ɛ f F 39 F N B N

32 2 32. goroff 42 [, p , pp ] Ω=R N goroff 45 ([, p.71, p.277, 11 2]) Lusin F N f : R ɛ>0 µ N ( F ɛ ) <ɛ F ɛ f F ɛ f F N /F 40. F = F 1/n F F N f F 43 f F f 1 ((a, ]) F F N. F ɛ>0 µ N ( F ) <ɛ F Lebesgue 32 F N f 1 ((a, ]) = (f 1 ((a, ]) F )+(f 1 ((a, ]) ( F ) F N Lebesgue (Ω, F,µ) 41 ( 5.2) (31) 46 ([, pp.75 76, 2]) {f n } g x lim f n(x) g(x) lim f n dµ gdµ.. f n = k n j=1 a nj χ nj, g = k a j χ j, a = min{a j a j 0, j =1, 2, 3,,k}, a >ɛ>0 j=1 ɛ g ɛ = max{g ɛ, 0} = p = j: a j 0 j g ɛ = g ɛχ p j: a j 0 (a j ɛ) χ j g ɛ g ɛ F n = {x Ω f n (x) g ɛ (x)} 41 f n g ɛ F n F. {F n p } (F n p )= p ( 2 ) (35) lim µ(f n p )=µ( p )., F n, p f n, g ɛ µ( p )= (33) lim f n dµ f n dµ (g ɛ) dµ (a ɛ) µ(f n p ). F n p F n p n (35) lim f n dµ = µ( p ) < 40 F = B N [, p.79, 12.2], 49 [, p.81, 1], 50 [, p.81, 2].

33 2 33 lim f n dµ (g ɛ) dµ = gdµ ɛµ(f n p ) F n p F n p gdµ gdµ ɛµ( p ). ( F n) p (33) gdµ= gdµ (32) p 0 gdµ max{a 1,a 2,,a k }(µ( p ) µ(f n p )), ( F n) p n lim f n dµ gdµ O(ɛ) ɛ 47 f {f n } f lim f n dµ. fdµ =. fdµ (31) {f 0n } {f n }, {f 0n } m lim f n(x) f 0m (x), lim f 0n(x) f m (x), x 46 m lim f n dµ f 0m dµ, lim f 0n dµ f m dµ. m fdµ= lim f 0n dµ 48 (i) µ() =0 f fdµ=0. (ii) f µ({x Ω f(x) = }) =µ({x Ω f(x) = }) =0.. (i) f ± (ii) f ± > f + dµ nµ(f =+ ). 49 ( ) (i), A, B = A + B f A, B f fdµ= fdµ+ fdµ. A B (ii) f, g a, b R af + bg (af + bg) dµ = a fdµ+ b gdµ.. A, B. (i) f n f + = max{f, 0} 47 f + dµ = lim f n dµ, X = A, B, (33) f n dµ = f n dµ+ f n dµ X X A B f + dµ = f + dµ + f + dµ. f = min{f,0} f ± A B

34 2 34 (ii) af+ bg 41 af f a f ± a a a a = b =1 f, g, f + g F = {x f(x) 0, f(x)+g(x) 0, g(x) < 0} f F F f F g g n, f + g f n + g n f n =(f n + g n ) g n f 47, (32) ( ) fdµ= lim f n dµ = lim (f n + g n ) dµ + ( g n ) dµ = (f + g) dµ + ( g) dµ. F F F F F F f, g 0 fdµ<, 0 ( g) dµ <, (0 ) F (f + g) dµ = (i) F F fdµ ( g) dµ <, F (f + g) dµ = fdµ+ F F F F gdµ. F 50 f g µ({x f(x) g(x)}) =0 g fdµ= gdµ.. µ Lebesgue 32 g. 49 F F 48 fdµ= F F fdµ= F gdµ=0 gdµ+ fdµ F Lebesgue (Ω, F) = (R N, F N ) (Lebesgue ) Riemann Riemann Riemann Riemann Lebesgue ([ 1 (pp.79, )]Seizo) (Ω, F,µ)=(R N, F N,µ N ) Lebesgue R N f Lebesgue Riemann. Lebesgue I L, Riemann I R f 0 f ± n N n = k n j=1 f n = nj (k n =2 nn ) x nj nj a nj = f(x nj ) b nj = inf{f(x) x nj } k n j=1 b nj χ nj f n n f δ n = max a nj b nj lim δ n =0, lim f n = f 1 j k n nj Riemann Lebesgue µ N ( nj ) I R = lim a nj µ N ( nj ), I L = lim f n dµ. j

35 2 35 I R I L = a nj µ N ( nj ) f n dµ j j a nj b nj µ( nj ) δ n µ() 0, n. I R = I L. 7. (Ω, F,µ) Ω = R {± }, F = {B C B B 1,C 2 {± } }, F/F f : Ω Ω 52 Fatou ([, 13.2 (p.82)]) {f n } f 1 dµ > lim f n(x) =f(x), x, + f fdµ= lim f n dµ.. {f n } f = n f n f fdµ= n f n dµ.. f 1 f n, f F = {x n; f n (x) = } = {x f 1 (x) = } 48 F f, f n F n = {x f n (x) = } ( F) F n f m (x) =f(x) =, m n, µ(f n ) > 0 n F = µ(f n )=0 F n µ(f ) n µ(f )=0 48 F 0 F f, f n n g k = f k f k 1, k =2, 3, 4,, f n = f 1 + g k, f = f 1 + g k, g k 0. g k G kn, n =1, 2, 3, h k1 = G k1, h kn = G kn G k,n 1, n =2, 3, 4,, h kn h kn = g k, g k dµ = lim N G kn dµ = lim N k=2 N h kn dµ = (32) ((32) ) (36) f N dµ = f 1 dµ + N k=2 g k dµ = f 1 dµ + N k=2 h kn dµ. h kn dµ. k=2

36 2 36 N f f 1 = h kn f f 1 k=2 k=2 N h kn 49, 47, (32) fdµ= (f f 1 ) dµ + = h kn dµ + f 1 dµ. k=2 (36) f 1 dµ = lim N N N k=2 h kn dµ + f 1 dµ 53 { n } = n f fdµ= n fdµ.. [, 13.3 (p.88)] 54 { n } µ( n fdµ= lim fdµ. n. [, 13.3 (p.89)] n )=0 f 7.2. Fatou 55 f n 0 lim inf f n dµ lim inf f n dµ.. g n (x) = inf f k(x) ( f n ) {g n } lim inf f n k n 52 lim inf f n dµ = lim g n dµ = lim inf g n dµ lim inf f n dµ. 13. (i) 52 {f n } bound bound lim sup f n dµ lim sup f n dµ. (ii) [, p.92, 1] f n 56 (Lebesgue, [, 13.6 (p )]) f n φ f n (x) φ(x), x, n N, lim f n dµ = ( lim f n) dµ.

37 φ ± f n 55 lim f n dµ lim inf f n dµ lim sup f n dµ lim f n dµ.. bound [, p.92, 1] (Ω, F,µ)=(R, F 1,µ 1 ) Lebesgue = (0, 1], f n (x) = x 1 χ [(2n) 1,n 1 ](x), {f n } 0 f n (x) dx = log 2 0. (0,1] n 57 ([, 13.6 (p.93)]) t ( a, a) f t : Ω lim f t (x), x, φ t 0 f t (x) φ(x), x, t ( a, a) lim f t dµ = (lim f t ) dµ. t 0 t 0. t n 0 {t n } (, [, 13.7 (p.93)]) f n f n (x) dµ < µ({x f n (x)}) =0 x f n (x) f n dµ = ( f n ) dµ.. 52 f n dµ = f n dµ <. φ(x) = f n (x) ( 0), F = {x Ω φ(x) = } φdµ< 48 µ(f )=0. 48 F F (0 ) φ(x) <, x, f n (x) g n (x) = n f k (x) g n (x) φ(x), x, k=1. Riemann [, 47 (p.157, p.159)] f n f n Riemann Riemann (i) m, n N cos 2n (πm!x) [0, 1] Riemann Lebesgue lim ( lim m cos2n (πm!x)) x Riemann

38 2 38 (ii) I =[0, 1] f n 0 [0, 1] J =(a, b) (a +2 n (b a),b 2 n (b a)) 1, 0 1 f f n χ I\ 0 1 Riemann ( 43) ( 41) Lebesgue Lebesgue Riemann 59 (, [, (p.91)]) µ() < f n {f n } M>0 f n (x) M, x, n N, f n lim f n lim f n dµ = ( lim f n) dµ.. φ = Mχ f α (a<α<b) x f α (x) α φ f α α (x) φ(x), x, a<α<b, f α dµ α d f α f α dµ = dα α dµ. 14. lim δ 1 (f α+δ f α )= f α a< δ 0 α α<b 0 <δ<b α δ 1 f α+δ (x) f α (x) φ(x), x. 57 δ 1 (f α+δ f α ) δ 1 (f α+δ f α ) dµ = lim δ 0 f α α dµ. 8. Fubini 18 (Ω, F,µ) Ω x P (x) P (x) x Ω A = {x Ω P (x) } A F µ(a) =0 P (x) P (x), (µ ) a.e., (µ ) a.e. x Ω P (x) P (x)

39 Fubini (X, F X,µ X ), (Y,F Y,µ Y ), (Ω, F,µ) X, Y (Ω, F, µ) ( 3) Ω =X Y = {(x, y) x X, y Y } X Y F F, F X, F F Y, σ- ( 25) µ µ X µ Y (13) µ( F )=µ X () µ Y (F ) F µ X, µ Y σ ,16 (37) X n F X,µ X (X n ) <, n X n = X, (38) Y n F Y,µ Y (Y n ) <, n Y n = Y, X n, Y n, n Z +, Ω n = X n Y n (39) Ω n F, µ(ω n ) <, n Ω n =Ω. Ω σ Fubini Ω [] x = {y Y (x, y) }, [] y = {x X (x, y) }, (28) (Ω, F ) Ω x µ Y ([] x ) X Ω (Fubini ( )) (i) F x X [] x F Y µ Y ([] ) : X Ω F X - µ Y ([] x ) dµ X (x) =µ() µ() < µ Y ([] x ) <, µ X a.e.. (ii) f : Ω Ω F- x X ( f(x, ) : Y Ω ) F Y - f(,y) dµ Y (y) : X Ω F X - f(x, y) dµ Y (y) dµ X (x) = Y X Y f(ω) dµ(ω) Ω (iii) f : Ω Ω F- µ X a.e. x X f(x, ) : Y Ω Y f(,y) dµ Y (y) : X Ω X ( ) Y f(x, y) dµ Y (y) dµ X (x) = f(ω) dµ(ω) 44 X Y Ω (iv) x, X y, Y statement f f : Ω Ω X ( Y X ) ( ) f(x, y) dµ Y (y) dµ X (x) = f(x, y) dµ X (x) dµ Y (y) = f(ω) dµ(ω) Y X Ω (Ω, F, µ) F σ- F a.e. σ- 43 [, p.103] 44 [, pp ]

40 f : Ω Ω F- ( ) f(x, y) dµ Y (y) dµ X (x), X Y Y ( X ) f(x, y) dµ X (x) dµ Y (y), (f ) 61 Ω f(ω) dµ(ω), 62. f 61 f f ± f 61 f 63 (Fubini ( )) X, Y 61 (Ω, F,µ) (Ω, F, µ) (i) F µ X a.e. x X [] x F Y µ Y ([] ) : X Ω F X - µ Y ([] x ) dµ X (x) = µ() µ() < µ Y ([] x ) <, X µ X a.e.. (ii) f : Ω Ω a.e. F- µ X a.e. x X ( f(x, ) : Y ) Ω F Y - f(,y) dµ Y (y) : X Ω F X - f(x, y) dµ Y (y) dµ X (x) = Y X Y f(ω) d µ(ω) Ω (iii) f : Ω Ω F- µ X a.e. x X f(x, ) : Y Ω Y f(x, y) dµ Y (y) : X Ω X ( ) Y f(x, y) dµ Y (y) dµ X (x) = f(ω) d µ(ω) X Y Ω (iv) x, X y, Y statement f : Ω Ω ( ) ( ) f(x, y) dµ Y (y) dµ X (x) = f(x, y) dµ X (x) dµ Y (y) = f(ω) d µ(ω) X Y Y X Ω 64 f : Ω Ω F- ( ) f(x, y) dµ Y (y) dµ X (x), X Y Y ( X ) f(x, y) dµ X (x) dµ Y (y), Ω f(ω) d µ(ω), (f ) X = Y = R Lebesgue [, pp ] f(x, y) µ- f µ- Fubini 61, 63 ± Lebesgue

41 2 41 Riemann ( ) 45 ( ) f(x, y) dµ Y (y) dµ X (x) X Y [, p.106 ( 1)] X = Y =(0, 1], Ω = X Y, f(x, y) = x2 y 2 y (x 2 + y 2, F (x, y) = arctan(x/y) F (x, y) = ) 2 x x 2 + y 2, F (x, y) = x y x 2 + y 2, f(x, y) = 2 F. f ( 43) x y ( ) 1 ( 1 2 ) F 1 [ ] y=1 f(x, y) dµ Y (y) dµ X (x) = x y dy y dx = x 2 + y 2 dx = π 4. X Y ( Y X ) f(x, y) dµ X (x) dµ Y (y) = [ x x 2 + y 2 ] x=1 x=0 dy = y=0 1 1+y 2 dx = π 4, f A = {(x, y) Ω x y}, B = {(x, y) Ω x<y}, f + A = f A, f + B =0,f A =0,f B = f B ( 1 ) 1 ( x ) 1 f + (ω) dω = f + x (x, y) dy dx = f(x, y) dy dx = Ω x 2 dx =, + x2 1 ( 1 ) 1 ( y ) 1 f (ω) dω = f y (x, y) dx dy = f(x, y) dx dy = y 2 dy =, + y2 Ω f , 63 f ( ) Fubini Ω σ- 19 Ω M (40) (41) A n M, n =1, 2, 3,, A 1 A 2 A 3 = n A n M, n =1, 2, 3,, A 1 A 2 A 3 = n A n M, A n M, 65 Ω A A M[A] A C A M[A] C M[A] A

42 Ω M σ-. σ- (40) (41) M n A n M, n N, B n = A k B 1 B 2, B n M. (40) A n = B n M. n n k=1 67 J Ω M[J ]=σ[j].. M[J ] 66 M[J ], σ[j ], M[J ]= σ[j ] M 1 = {A Ω A c M[J ]} 46 A J A c J M[J ] M 1 J. A n M 1, ( n N, ) A 1 A 2 A 3, A c n M[J ], n N, A c 1 A c 2, c (41) A n = A c n M[J ], A n M 1, M 1 (40) n n n (41) M 1 J M 1 M[J ]. M 1 M[J ] M[J ] M 2 = {A Ω B J = A B M[J ]} A, B J A B J M[J ] M 2 J. M 1 M 2 M 2 M[J ]. M 2 M[J ] J M 3 = {A Ω B M[J ]= A B M[J ]} M 3 J. M 1 M 3 M 3 M[J ]. M 3 M[J ] Fubini (Ω, F,µ) σ (X, F X,µ X ), (Y,F Y,µ Y ), (Ω, F, µ) 68 J def = { n i F i i F X F i F Y, i =1, 2,,n, n N} F = σ[j ] i=1. 21 F = σ[j ] (i) M = { Ω [] x F Y, x X} 68 J M 47. { n } ( ) n = {[ n ] x } ( ) n n [ n ] x =[] x. [] x F Y M. M n n 67 (M ) M[J ]=σ[j ]=F. F x X [] x F Y. 46 [, p.98] 47 [, pp ]

43 2 43 k N M k = { F µ Y ([ Ω k ] x ) x F X - } Ω k (39) M k F ( ) n = { n } [ n Ω k ] x ( ) n n [ n Ω k ] x =[ Ω k ] x 2 n n (42) lim µ Y ([ n Ω k ] x )=µ Y ([ Ω k ] x ), Ω k k F M k. F k µ Y ([ Ω k ] x ) x F X -. k x X [ Ω k ] x [ Ω k ] x =[] x. 2 k µ Y ([] x ) = lim µ Y ([ Z k ] x ). µ Y ([] x ) F X - k k N M k = { F X µ Y ([ Ω k ] x ) dµ(x) =µ( Ω k )} M k F (42) φ(x) =µ Y ([Ω k ] x ) µ(ω k ) < k F µ Y ([] x ) dµ(x) =µ(). X (ii) f F ( 40) 52 (iii) f = f + f a.e. (iv) Fubini ([, 3 (p.105)]) F, µ() =0, µ X a.e. x [] x F Y, µ Y ([] x )=0. x, X, y, Y. (39) Ω σ 32, 33 F, µ(f )=0 F F 61 µ Y ([F ] x ) dµ X (x) =µ(f )=0. µ Y ([F ] x ) 0 µ Y ([F ] x )=0, X µ X a.e.. [] x [F ] x µ Y µ Y ([F ] x )=0 x [] x F Y µ Y ([] x )=0 70 f : Ω Ω µ-a.e. 48 F- F- g f + g 0, µ({ω Ω f(ω) g(ω)}) =0. f + F- f n = k n j=1 a nj χ nj [, (p.65), 4 (p.105)] a.e.

44 2 44 (43) nj F, Ω= k n j=1 nj. (39) Ω σ 32, 33 n, j F nj nj, µ( nj F nj )=0 k n F nj F F = F F nj (43) Ω F k n j=1 µ(ω F )=0. g n = k n j=1 ( nj F nj ) F nj j=1 a nj χ Anj, A nj = F nj F F,j=1, 2,,k n, n N, g n F- n Ω nj x F F x A nj A nj nj j = j, g n F = f n F. x Ω F g n (x) =0. g = lim g n Ω F- F- { f + (x), x F, g(x) = 0, x Ω F. f + = f, µ-a.e. f + g 0 g (i) F (39) Ω σ 32, 33 F, µ( F )=0 F F 61 x X [F ] x F Y µ Y ([F ] x ): X Ω F X - µ Y ([F ] x ) dµ X (x) =µ(f ) X µ( F )=0 69 µ X a.e. x [ F ] x F Y, µ Y ([ F ] x )=0. x [F ] x [ F ] x = {x X y Y ;(x, y) F ( F )} =[] x, [] x F Y, µ Y ([] x )=µ Y ([F ] x ) F X - µ Y ([] x ) dµ X (x) =µ(f )= µ(f )= µ() (ii) 70 F- g f + g 0, F = {ω Ω f(ω) g(ω)} µ(f )=0 f = f +, µ-a.e.,. g 61 x X g(x, ( ) : Y Ω ) F Y - g(,y) dµ Y (y) : X Ω F X - g(x, y) dµ Y (y) dµ X (x) = Y X Y g(ω) dµ(ω) Ω 69 µ X a.e. x [F ] x F Y, µ Y ([F ] x )=0. F y 2 [F ]x F Y [F ] x f [F ]x F Y - x y Y [F ] x F f(x, y) =g(x, y). Y [F ] x f(x, ) F Y - f(x, ) Y F Y - [F ] x g(x, ) 50 f(x, y) dµ Y (y) = g(x, y) dµ Y (y) X F X - Y Y ( ) µ X a.e. x 50 f(x, y) dµ Y (y) dµ X (x) = X Y X

45 2 45 X Ω ( ) g(x, y) dµ Y (y) dµ X (x) = g(ω) dµ(ω) f = g, µ-a.e., Y Ω g(ω) dµ(ω) = f(ω) dµ(ω) Ω (iii) f = f + f a.e. (iv) 9. [ ] (i) [, 12.6 (p.83)] f R N Lebesgue ɛ 0 g (f(x) g(x)) dx <ɛ (ii) Lebesgue [, 12.7 (pp.84 85)] f R N Lebesgue y R N f( + y), f( ) f(x + y) dx = RN f( x) dx = RN f(x) dx. RN Lebesgue 50 (iii) [, 12.8 (p.85)] f R N Lebesgue lim f(x + y) f(x) dx =0. y 0 RN (iv) [, 12.5 (p.82), 20.3 (p.153)] f g fg inf x f(x) c sup x g(x) c (a, b) R F ( ) φ b a φdf = φ(a)(f (c) F (a)) + φ(b)(f (b) F (c)), a<c<b, c F, φ (a, b) R f φ b a φ(x) f(x) dx = φ(a) c a f(x) dx + φ(b) b c f(x) dx, a < c < b, c F, φ straightforward Stieltjes 14.3 f g dµ = c g dµ, 50 93

46 σ ( 11) ( 13) (Ω, F) Φ: F R {± } + ± σ Φ( A n )= Φ(A n ), A n F, n N, ± 71 Φ(A) A (V (Ω)) 0 µ(a) µ(ω). (i) Φ Φ + R {+ } ±Φ (ii) P (Ω) = 1 (iii) [, p.120] [, 3 (p.132)] σ (iv) ± Φ(A + B) =Φ(A)+Φ(B) = (?) ± Feynman σ (v) ( 11.2) (vi) 71 ( 10.2) (signed measure) (vii) n n Φ(φ) =0, Φ( A i )= Φ(A i ) Φ(A) < Φ(B) < A B i=1 i=1 Φ(A B) =Φ(A) Φ(B) Φ Φ(B)+Φ(A B) =Φ(A) ill-defined 2

47 3 47 (44) A 1 A 2 = Φ( A n ) = lim Φ(A n), (45) Φ(A 1 ) < +, A 1 A 2 = Φ( A n ) = lim Φ(A n), 2 A B Φ(A) Φ(B), Φ( A i ) Φ(A i ), 2 Φ i=1 i= Hahn support disjoint 10.2 (Ω, F) Φ: F R {+ } 10.1 Φ Φ 21 F V () = V (Φ; ) = sup{φ(a) A, A F} Φ V () =V (Φ; ) = inf{φ(a) A, A F} V () =V (Φ; ) = V (Φ; ) + V (Φ; ) sup, inf ± Φ V () A = (46) V () Φ() V () A = (47) V () 0 V () (48) V () = V () V () (49) V () max{ V (), V ()} = sup{ Φ(A) A, A F}. (50) V ( Φ; ) = sup{φ(a) A, A F}= V (Φ; ). 71 (Jordan ) V, V V (Ω) < (Ω, F) Φ = V + V V (48) Φ Φ (50), (48) V (Ω) <, V (Ω) <

48 (i) V (Ω) > (ii) V σ (47) V (iii) Φ = V + V Φ (48) V (i) (51) V (Ω) = 72 V (A) = B A, A, B F, V (B) = V (A B) =.. Φ V (B)+V(A B) = inf Φ(C) + C B inf D A B Φ(D) inf Φ() =V (A). A Φ(X 0 ) <, V (X 0 )= X 0 F Φ(A) < V (A) > Φ A 0 F M def =Φ(A 0 ) <. V (A 0 ) >. (51) 72 V (A c 0 )=. B 1 A c 0 F Φ(B 1 ) 1. A 1 = A 0 + B 1 Φ(A 1 ) M 1(< ). A ( 0 A 1 A 2,Φ(A n ) M n, n Z +, F (44) ) Φ A n = lim Φ(A n)= Φ (52) X 0 F; Φ(X 0 ) <, V(X 0 )=. (53) X n+1 X n,v(x n )=, > Φ(X n ) n, n Z +. X n F, n Z +, X 0 (52) X n (53) V (X n )= Φ(X n ) < (54) Φ() Φ(X n ) n 1(< ) X n F V (X n )= 72 V () = V (X n ) =. V () = X n+1 =, V (X n ) = X n+1 = X n (54) (53) n +1 X n+1 = X n > Φ(X n+1 ) n +1 Φ(X n+1 )= Φ(X n )+Φ() (54) n 1 > (53) X n F, n Z +,( ) (45), (52), (53) Φ X n = lim Φ(X n) =. Φ(X 0 ) < X 0 X n ( n=0 n=0 ) Φ X n < (51) V (Ω) > 51 [, pp ]

49 3 49 (ii) F, = n, n F, n N, Φ σ V V () = inf{φ(a) A, A F}= inf{ Φ(A n ) A, A F} V ( n ). ɛ>0 V ( n ) > n N A n n F Φ(A n ) <V( n )+2 n ɛ m n n m = A n A m = = n A n V () Φ( A n )= Φ(A n ) < V ( n )+ɛ. V () = V ( n ) V σ (iii) F A F A Φ(A)+Φ( A) =Φ() Φ(A)+V () Φ() Φ(A) + V (). V () > V () +V () Φ() Φ() V ()+ V () Φ= V + V ( ) 73 V A n <, A k = A k = lim A n Φ( lim A n)= k=n k=n lim Φ(A n) ( ) Φ V (Ω) < V A n < 74 (Hahn ) V (H) =0 V (H c )=0 V(H) < H F Φ : F R {+ } µ µ(h) < H F Φ() =µ( H c ) µ( H) µ Φ V H H Φ µ(ω) < V (Ω) <. n N Φ(A n ) V (Ω) + 2 n A n F 71 V (55) V (A n ) V (Ω) V (A n )+2 n 2 n, V (46) (56) V (A n c )= V (Ω) + V (A n ) Φ(A n )+2 n + V (A n ) 2 n H = k=n 0 V (H c ) A k n N A c k=n A k c (56) 2 k =2 n+1. k=n 52 [, p.125] Φ + V V

50 3 50 V (H c )=0 V V (H) < A k n 0 V (H) = lim V ( k=n A k ) lim inf V (A n )=0. Φ() =µ( H c ) µ( H) Φ 71 Φ Φ() =Φ(H c )+Φ(H ) = V (H c )+V (H c )+ V (H )+V (H ) = V (H c ) V (H c ) V (H )+V (H ) =V (H c ) V (H ). k=n V (H c ) = V (H ) =0 n 75 F V () = sup{ Φ( j ) n N, j=1 n j =, j=1 j F, j =1, 2,,n}. n. = j j V ( j ) Φ( j ) (49) A = = j V j=1 n n V () sup{ Φ( j ) n N, j =, j=1 j=1 j F,j=1, 2,,n} 74 F Φ() =V ( H c ) V ( H) H F 1 = H c, 2 = H Φ( 1 )=V( 1 ), Φ( 2 )= V( 2 ) n n V () =V ( 1 )+V ( 2 )= Φ( 1 ) + Φ( 2 ). V () sup{ Φ( j ) n N, j =, j F, j =1, 2,,n} 53 j=1 j=1 11. (Ω, F) µ µ (Ω, F,µ) 22 F F σ- (Ω, F) Φ: F R {± } µ (absolutely continuous) F µ() =0 = Φ() =0 Φ µ (singular) µ(s) =0, S F F, S = = Φ() =0 53 sup { j } sup

51 3 51 F Φ() =Φ( S). (i) Φ Φ µ Φ Φ µ Φ µ 11.2 (ii) µ F Φ F µ F µ F = F µ σ µ F \F Radon Nikodým ( 12) 12 [ ] µ µ F 54 µ µ Φ, Ψ a, b aφ+bψ ± 76 ([, 2 (p.129)]) Φ Φ 0. µ( 0 )=0 0 F 0 = F Φ() =0 A F B = A 0, C = A c 0 Φ(C) =0 µ(b) µ( 0 )=0 Φ Φ(B) =0 Φ(A) =Φ(B)+Φ(C) =0 77 Φ V Φ V. Φ µ() =0 F Φ() =0 A F A µ(a) =0 Φ(A) =0 V () =V () =0 V () =0 V V µ() = 0 F V () = 0 (49) Φ() V () =0 Φ Φ µ(s) =0 S F S = F Φ() =0 A F A A S = Φ(A) =0 V () =V () =0 V () =0 V V µ(s) =0 S F S = F V () =0 (49) Φ() V () =0 Φ 78 Φ ɛ>0 V (X) < X F δ = δ(ɛ, X) > 0 F µ() <δ X Φ() <ɛ. (i) Φ V (Ω) < X (ii) V (X) < Ω =N, F =2 Ω, µ(n) =2 n,φ(n) = S F S F

52 3 52 (iii) [, 18.2]. Φ ɛ>0 V (X) < X F, n F, n N n X, µ( n ) < 2 n Φ( n ) ɛ n 0 = k k=n µ( 0 ) µ( k ) k=n µ( k ) < 2 1 n, n N, k=n µ( 0 )=0 Φ 77 V V ( 0 )=0 V ( 0 )=V( 0 )=0 ( ) k ( X) V k < 71 V k=1 2 (46) 0= V ( 0 ) = lim V ( k ) lim sup V ( n ) lim sup Φ( n ), 55 k=n 0= V ( 0 ) = lim V ( k=n k ) lim sup k=1 V ( n ) lim sup Φ( n ). lim Φ( n)=0. Φ( n ) >ɛ n 79 Φ: F R {+ } ɛ>0 = (ɛ) F µ() <ɛ V ( c ) <ɛ. Φ V F µ() =0 A = A F V (A) =0 (ɛ) µ() <ɛ V ( c ) <ɛ ɛ>0 = (ɛ) F µ() <ɛ V ( c ) <ɛ n N µ( n ) < 2 n V ( c n ) < 2 n n F 0 = k k=n µ( 0 ) µ( k ) k=n µ( 0 )=0. 0 c = µ( k ) < 2 1 n, n N, k=n k=n 0 V ( c 0 ) = lim V ( k=n c k V 2 k c ) lim inf V ( n c )=0. F 0 = 0 c V () =0 (49) Φ() =0 µ( 0 )=0 Φ 55 [, 18.2 (p.128)] V [, p.128]

53 (Ω, F,µ) f : Ω Ω Ω µ Φ() = fdµ Φ: F R Φ f f Φ σ µ 57 Radon Nikodým ( 12) µ σ Dirac delta (R N, F N,µ N ) N Lebesgue w n Ω, n =1, 2, 3,, Ω p n R, n =1, 2, 3,, p n < Φ : F R Φ() = x p n n Φ p n > 0, n N, Φ p n =1 Φ Lebesgue σ Φ Lebesgue Cantor [, p.42 43] (R, F 1,µ 1 ) Lebesgue [0, 1] R (1/3, 2/3) 1/3 1/3 3 2 n 3 n 2 n 1 [0, 1] Cantor n I n 2 n 1 3 n (Lebesgue ) σ 2 n 1 3 n =1, µ 1 () =1 1=0 Cantor Lebesgue x [0, 1] x G =[0, 1] \ = φ : I n G [0, 1] [0, 1] R I 1 φ =1/2, I φ =2 2, φ =32 2, I n 2 n 1 0 φ =(2k 1)2 n φ G [0, 1] φ Cantor (a, b] [0, 1] Φ((a, b]) = φ(b) φ(a) Φ 2 Φ F 1, G Φ() = 1, Φ(G) =0 Φ Lebesgue ( 2.3.2) Cantor φ φ [0, 1] 57 [, 13.4 (pp.89 90)] 48 78

54 Radon Nikodým 12 (Ω, F,µ) σ- F F, Φ: F R {+ } Φ Φ Φ σ ( 4.2) Ω = X n X n F Φ(X n ) <, n N, 12.1 σ µ Φ σ 12.2 µ σ σ 59 µ F Φ F Φ σ (58) α α (59) F Φ Ψ =Φ F Φ σ (58) [, 29] σ Lebesgue 80 (Lebesgue ) σ Φ: F R {+ } µ F Ψ F Φ =F +Ψ Φ F, Ψ. Φ=F +Ψ=F +Ψ F F =Ψ Ψ 76 0 F = F,Ψ =Ψ 71 Φ Φ σ Φ(X n ) <, X n (X n, F 2 Xn ) Φ Φ : F R + (57) F def = {F : F R + ; σ µ, F () Φ(), F}, (58) α def = sup F (Ω) F F α 0 α Φ(Ω) < F {F n } lim F n(ω) = α F : F R {+ } (59) F () = sup{ F n ( n ) n F, n N, n }, F, 58 [ ] [, 18] 18.3 (p ) µ σ 18.4(i)(ii) µ = (kaiseki.jnk ) 59 [, 18 (pp )] 18.4 (iii) ; rev [, p.130]; ; rev (58), (57) [ ] µ (57) Radon Nikodým

55 3 55 { n } = { } sup F () 0, F, F µ() =0 n, n F, n N, µ( n )=0 F n F F n ( n )=0 n n F (59) F () =0 F, n F, n N, = n, F (59) ɛ>0 n,m,(n, m) N 2, n m n,m n, n N, F ( n ) ( F m ( n,m )+2 n ɛ)= F m ( n,m )+ɛ F ()+ɛ n n m m n (n, m) (n,m ) n,m n,m = n,m, F m F m n σ (60) F ( n ) F (). n, { n } ɛ>0 F (59) Ẽm F, m N, Ẽ m m F () F m (Ẽm)+ɛ = F m ( ( n Ẽm)) + ɛ = m m n n F m ( n Ẽm)+ɛ F ( n )+ɛ m n Ẽm = n n, F m ( F) σ (59) n F ( n ) F (). (60) F σ Ψ=Φ F F n F F n Φ Φ σ n n, ( n F, n N, ) = n F n ( n ) n Φ( n )=Φ( n n ) Φ() F () Φ(), F Ψ Φ, F σ Ψ σ F 0 Ψ(Ω) Φ(Ω) < Ψ Ψ µ F (57) F F α (58) F (Ω) α (59) n =Ω, m =, m n, F (Ω) F n (Ω) n F (Ω) lim F n(ω) = α (61) F (Ω) = α. (62) Ψ() = sup{ψ(n) N, N F, µ(n) =0}, F 81 Ψ 0 Ψ() Ψ(), F, F µ() =0, F, Ψ() =Ψ() 81. F (62) N = Ψ() Ψ( ) =0 N, N F, Ψ Ψ(N) Ψ() (62) Ψ() Ψ(). Ψ Ψ µ() =0 (62) N = Ψ() Ψ() Ψ() =Ψ(). σ = n ɛ>0 (62) B, µ(b) =0 B F

56 3 56 Ψ() Ψ(B)+ɛ = Ψ(B n )+ɛ Ψ( n )+ɛ. Ψ σ B n n,0 µ(b n ) µ(b) =0 µ (62) = n ɛ>0 (62) B n n, µ(b n )=0, B n F, n N, B n Ψ( n ) ( Ψ(B n )+2 n ɛ)=ψ( B n )+ɛ Ψ()+ɛ. σ Ψ( n )= Ψ() Ψ Ψ = Ψ Ψ 81 (57) Ψ F Ψ σ Ψ Ψ Φ µ() =0, F, 81 Ψ() =Ψ() Ψ() =0 F (57) F + Ψ F +Ψ=Φ F + Ψ F (58) (61) F (Ω) F (Ω) + Ψ(Ω) α = F (Ω) Ψ(Ω) = 0 Ψ 0 Ψ() = Ψ(), F. Ψ µ(b n )=0,B F, n N, {B n } lim Ψ(B n) = Ψ(Ω) B 0 = B n F σ µ(b 0 ) = 0. Ψ Ψ(Ω) Ψ(B 0 ) lim sup Ψ(B n ) = Ψ(Ω) Ψ(B 0 ) = Ψ(Ω) < B 0 c, F Ψ() =0 µ(b 0 )=0 Ψ µ Radon Nikodým 82 (Radon Nikodým) µ σ φ =Ω 0 Ω 1 Ω 2 Ω 3, Ω n =Ω, Ω n F Ω n F 61 F : F R {± } σ µ (i) F f : Ω R {± } 0 f<, a.e., F () = fdµ, F. (ii) Ω n F, n N, f a.e. f f = f, a.e. f F µ (Radon Nikodým ) f = df µ-a.e. dµ df 0 dµ 61 F (Ω n) Ω n µ(ω) < [, 18.4] n=0

57 F F σ F : F R + (i) µ f F σ- F = F def = σ[f, {Ω n n N}] k N Ẽ F, Ẽ Ω k Ω k 1, Ẽ = (Ω k Ω k 1 ) F 83. F = F (63) F = F def = { k (Ω k Ω k 1 ) k F,k N} k=1 F F F F {Ω n } σ- F Ω n, n N, (63) k =Ω,1 k n, k =, n<k, F, F, k =, k N, F σ- F F F ( c n (Ω n Ω n 1 )) = c n (Ω n Ω n 1 ) F, ( ) ( ) m,n (Ω n Ω n 1 ) = m,n (Ω n Ω n 1 ) F, m=1 m=1 F F F def F 1 = {φ : Ω 1 R + { } F φdµ F (), F}, Ω 1 def α 1 = sup φdµ ( F (Ω) < ), φ F Ω 1 1 φ 1 dµ = α 1 φ 1 F 1 Ω 1 k 2 F k 1, α k 1, φ k 1 F k, α k def F k = {φ : Ω k R + { } F φ Ωk 1 = φ k 1, φdµ F (), F}, Ω k def α k = sup φdµ ( F (Ω) < ), φ F Ω k k φ k dµ = α k φ k F k 63 φ k F k Ω k (64) φ k Ωk 1 = φ k 1,k N f F - f F F f F - 63 µ σ [, (p.132)] Ω =[0, 1], µ =, F = µ 1, χ 0 F k

58 k N F k, α k well-defined, φ k 84. k 1 k =1 { φ k 1, on Ω k 1, φ = F 0, on Ω k Ω k 1, k F k α k well-defined. α k α k 1 α k lim φk,n dµ = α k φ k,n F, n N, φ k = sup φ k,n Ω k F Ω k n 1 φ k 64 φ k,n F φ k Ωk 1 = φ k 1 (65) φ k dµ F (), F, Ω k φ k F φ k dµ α k. φ k φ k dµ φk,n dµ Ω k Ω k Ω k n n φ k dµ α k. Ω k (66) φ k dµ = α k, Ω k φ k 84 def (65) φ k,n = max φ k,m, n N, φ k,n n 1 m n lim φ k,n = φ k lim φ k,n φ k ω Ω k, ɛ>0 m n m φ k (ω) φ k,m (ω)+ɛ φ k,n (ω)+ɛ lim φ k,n(ω)+ɛ. Ẽ k,n,m def = {ω Ω k Ω k 1 φ k,n (ω) = φ k,m (ω), φ k,n (ω) > φ k,l (ω), 1 l m} F n Ẽ k,n,m =Ω k Ω k m n Ẽk,n,m = k,n,m (Ω k Ω k 1 ) m=1 k,n,m F k,n,m k,n,m m 1 l=1 k,n,l k,n,m k,n,m =, m m, φ k Ωk 1 = φ k,n Ωk 1 = φ k 1 n φ k,n dµ = φ k 1 dµ + Ω k Ω k 1 m=1 n = φ k 1 dµ + ( k,n,m) c Ω k 1 m m=1 F ( ( n k,n,m ) c )+ F ( k,n,m ) m = F (). m=1 k,n,m (Ω k Ω k 1 ) k,n,m Ω k φk,m dµ φ k,m dµ Ω =N, µ({n}) =1,F = {, 2N, 2N 1, N}, F (2N) =F (2N 1) = 1/2, Ω k Ω k 1 φ k F - unique φ k,n n φ k sup φ k,n α k φ k,n sup F () F F F F µ- Ω k 0 φ 0 σ µ Radon Nikodým

59 3 59 φ k 1 F k 1, φ k,m F k F σ 52 φ k,m F k φ k dµ = lim φ k,n dµ = lim max φk,m dµ F (), F. Ω k Ω 1 m n k Ω k (65) F f : Ω R + { } f Ωk = φ k f F - (64) f well-defined f Ω k Ω k+1, k N, 54, (65), (67) fdµ= lim k φ k dµ F (), F. Ω k F () = fdµ, F, Ψ() =F () fdµ, F, Ψ Ψ σ f 0 (67) F () Ψ() 0, F, Ψ µ() =0, F, F () =0 Ψ() =0 Ψ µ Ψ µ 76 Ψ 0 85 n n F 1 n µ() Ψ(), n, F = µ( n )= n n F 1 n µ() Ψ(), n, F, µ( n ) > 0. g(ω) =f(ω)+ 1 n χ n (ω) g : Ω R + { } 65 g F F gdµ= fdµ+ 1 n µ( n) F ( n c )+F ( n )=F (). n c fdµ+ fdµ+ψ( n ) n 0 <µ( n ) = lim µ( def n Ω k ) k 1 µ( n Ω k ) > 0 k 0 = min{k k 1 µ( n Ω k ) > 0} µ( n Ω k0 1) = 0 g(ω) = f(ω) + 1 n χ n\ω (ω) g g k0 1 Ω = f k0 1 Ω = φ k0 1 k 0 1 g F k0. α k0 gdµ α k0. (66) Ω k0 α k0 gdµ= fdµ+ 1 Ω k0 Ω k0 n µ(ω k 0 n )=α k0 + 1 n µ(ω k 0 n ) >α k0 Ψ n N G n = 1 n µ Ψ Ψ G n G n : F R {+ } 74 n F 65 g(ω) =f(ω)+ 1 n f(ω) χ n (ω) f 0 Ψ µ Ψ F 1 n f 1 n F

60 3 60 { G n () 0, G n () 0, n, F, n c, F, 85 µ( n )=0 0 = n n µ µ( 0 ) = lim µ ( n m=1 m ) µ( m )=0. m=1 c 0, F, c n G n () 0, Ψ() 1 n µ() n Ψ() =0 Ψ (i) f Radon Nikodým density (ii) Ω n F F = F f Ω R + F () = Λ n = {ω Ω f(ω) > f(ω)+ 1 n } F 66 1 n µ(λ n) (f f) dµ = F (Λ n ) F (Λ n )=0 Λ n µ(λ n )=0. ( ) µ(f > f) =µ {f > f + 1 n } µ(λ n )=0. f dµ, F, f< f µ(f f) = σ µ σ 82 [, (p.132)] Ω =[0, 1], F = F = F 1 [0,1] [0, 1] Lebesgue µ = F = µ 1 Lebesgue (Ω, F,P) B F σ- Λ F D Λ = {ξ : Ω R B-, P(Λ Θ) = ξdu} Θ 82 D Λ ξ 0, D Λ ξ i, i =1, 2, ξ 1 = ξ 2, P -a.e, P -a.e. 23 D Λ / B Λ P (Λ B) 67 D Λ P (Λ B) version version version ω Ω P (Λ B)(ω) Λ F P ( B)(ω) F P (Θ i ) > 0, i =1, 2,, Θ i =Ω, ={Θ i i =1, 2, } B = σ[ ] 66 F F F = F 67 [, pp ], [, p.26]

61 3 61 (68) P (Λ B)(ω) = P (Λ Θ i), ω Θ i, P (Θ i ) P (Λ B) D Λ D Λ B a.e. ω Λ F support Θ i Θ i Λ B P ( B)(ω) (68) 0 (68) [ ] Λ ={X x} x X ([, p.169]) X (Ω, F,P) B F σ- µ : Ω B 1 R (i) µ(ω, ) a.e.-ω B 1 (ii) A B 1 µ(,a) P (X 1 (A) B) version µ µ(ω, A) =µ(ω, A), P -a.e. ω, A B (Ω, B,P) Ω=N, B = {φ, 2N, 2N 1, N}, P( ) =P ( ) = 1 2. Ω (Ω, F,µ) F =2 Ω, µ({n}) =1,n N, f(1) = f(2) = 1, f(n) =0, n 3, 2 f f Ω µ- P () = f(x)dµ(x), B, µ P f f(3) = f(4) = 1/3, m/n 1.2 f(1) = (1 p 2 ), f(2) = p P p 2 )2 2, f(6) = (1 p 2 ) , p P\{2,3} p P\{2} 68 nrealsn Wiener Banach Malliavin calculus calculus [Cacoullos, p.33, Q155, p.155] 1 Q155 1 Radon Nikodým [Cacoullos, p.33, Q155, p.155] (1 70 Radon Nikodým ([, p ], [, p.165], [, 112 (p.420)], [, p.335 (iii)], [, p.204]) µ(ω) < Radon Nikodým

62 ( [, 32, 35]) f [a, b] R (i) f F (x) = ) x (ii) F f F (x) =F (a)+ a fdx [a, b] f (F = f x a dx f [ 87 ] [, 19], [, 125 (pp )], [, 32 34], [, 27,28 ], ([ ], [ ]) ([ ]) ([ ]) R N R 24 [a, b] R F : [a, b] R F ɛ>0 δ = δ(ɛ) > 0 n N n n n {(a k,b k ] k =1, 2,,n} (b k a k ) <δ F (b k ) F (a k ) <ɛ k=1 n =1 n = 88 f [a, b] R Lebesgue (i) f F (x) = F (x) =f(x) x a k=1 f(x) dx a.e. x [a, b] (ii) F g F (x) F (a) = x a g(x) dx, x [a, b], x F (x) =f(x), a.e. x [a, b], F (x) =F (a)+ f(x) dx, x [a, b],. F F F F OK a.e. a R Jordan [, 19] F 2.2 Φ((a, b]) = F (b) F (a) Φ Radon Nikodým [, 32]

63 [a, b] F (a, b] Lebesgue f F (x) F (a) = x a f(x) dx, x (a, b] Lebesgue R N Lebesgue Vitali 72 a.e. Radon Nikodým [, 122 (p.453)], [, 34.4], [, p.219] Radon Nikodým R N n R x a.e. x dual space Radon Nikodým Lebesgue Stieltjes 25 (Ω, F) Φ : Ω R f : Ω R {± } Φ V f dv < F fdφ def = fd V fdv Φ f Lebesgue Stieltjes Ω 19 V, V V + V = V fd V + fdv f dv < f d V <, f dv <, fd V, Ω Ω Ω fdv, fdφ Lebesgue Lebesgue Stieltjes Lebesgue 90 ( ) (Ω, F,µ) Φ : Omega R µ ( 82) a.e. φ Φ() = φdµ, F, f Ω Φ V fφ Ω µ fdφ= fφdµ, F. 72 R N 0 0 [, 123 (pp )], [, 33], [, p.212] [, 4 (p.141)]

64 3 64. Φ= V V, f = f + f Radon Nikodým φ f R R Stieltjes [, pp ] 14.3 [, pp ] Stieltjes 26 [a, b] F : [a, b] R M>0 [a, b] k a = x 0 <x 1 < <x k 1 <x k = b, (k N) F (x j ) F (x j 1 ) M 91 ([, 19.1 (p.134)]) [a, b] ( 13) 92 ([, pp ]) [a, b] F F 1, F 2 F = F 1 F 2 F F i I R F I 92 F 1, F 2 F = F 1 F 2 Φ i ((a, b]) = F i (b) F i (a) (Ω, F i, Φ i ) 2.2 F 1 F 2 ( B 1 ) Φ=Φ 1 Φ 2 F V F V F = F 1 + F 2 V Φ ((a, b]) = V F (b) V F (a) V Φ B 1 σ f : Ω R {± } f dv Φ < F Ω fdφ def = fd V F fdv F fdf F f Lebesgue Stieltjes Lebesgue Stieltjes 93 ( ) I =[a, b] F 88 φ F (x) F (a) = b a f(x) df (x) = x a b a φ(x) dxm x [a, b], f(x) φ(x) dx. F = φ, a.e.-x, φ(x) > 0, a.e., F F 1 b a f(x) dx = b a f(f (x)) F (x) dx, 73 [, 20.1 (pp )] j=1

65 Riemann-Stieltjes 94 F f Riemann Stieltjes Lebesgue Stieltjes F f Riemann Stieltjes [, pp ] [, pp ] 95 (i) (a, b) F φ b φ(x) df (x) =F (b)φ(b) F (a)φ(a) b a a F (x) dφ(x), x (a, b). (ii) (a, b) f Lebesgue φ b φ(x) f(x) dx = F (b)φ(b) F (a)φ(a) b a a x F (x) = b a a F (x) dφ(x), x (a, b). f(x) dx + F (a). F (a) φ(x) F (x) φ (x) dx Riemann-Stieltjes

66 Lebesgue (Lebesgue) R n -Lebesgue Ω ( 1.2.4) R N Lebesgue (i) 2 Lebesgue ( 15.1) Lebesgue (Lebesgue ) Lebesgue 2 R (ii) Lebesgue ( ) 2 R ( ) (iii) 74 R N Lebesgue ( ) Lebesgue R Lebesgue 2.2 Lebesgue (R, F 1,µ 1 ) F 1 77 µ = µ 1 µ(f ) > 0, F [0, 1] F A 96 ([, 2 4 (p.262)]) R N C O ℵ 1. C c O 1:1 G O G = U(x nk,r mk ) x n r m U U 1 :1- G O {(n k,m k ) k N} Z 2N Z 2N = ℵ N 0 = ℵ1 O ℵ 1 O O ℵ 1 k=1 96 A ℵ 1 96 ℵ 1 A = {F 0,F 1,,F ξ, } ξ<γ γ ℵ 1 97 F ξ A x ξ,x ξ F ξ ξ η x ξ, x ξ, x η, x η. F 0 Lebesgue ξ <λ(γ) ξ X λ = {x ξ,x ξ ξ<λ} ℵ 1 F λ \ X λ ℵ 1 λ [, p.108] 77 Fubini [, pp ] Fubini

67 x ξ x ξ Lebesgue µ µ () =µ ( ) = Lebesgue µ () =µ ( ) µ ([0, 1] \ ) =µ([0, 1] \ ) =1 µ() =1 µ (). µ () > 1/2 G G µ(g) 1 µ(g) < 1 [0, 1] G µ([0, 1] G) > 0 [0, 1] G A. 97 ξ x ξ [0, 1] G [0, 1] G. G µ(g) 1 16 µ () 1 F Lebesgue Fubini Lebesgue R N Fubini 61, 63 f : Ω Ω F ( F)- R 2 F 2 x y [, pp ] Lebesgue Lebesgue 20 Lebesgue Lebesgue R N Lebesgue 98 ( [, 7.1 (pp.35 36), 21.2 (p.155), 21.3 (p.157)]) Ω=R N (i) Lebesgue F N, x R N + x def = {y R N y x } F N µ N ( + x) =µ N (). (ii) Lebesgue (R N, F,µ) ( F)( x R N ) + x F, µ( + x) =µ(), [0, 1] N F, µ([0, 1] N )=1 F F N µ N F = µ. (iii) Lebesgue uclid T R N (T = def = {y R N y }) T F N µ N (T)=µ N (). 99 ([, 12.7 (p.84)]) f R N Lebesgue f(x) dx RN y R N f(x + y), f( x) Lebesgue f(x + y) dx = f( x) dx = f(x) dx.

68 Lebesgue Lebesgue Lebesgue Lebesgue Lebesgue 98 Lebesgue A [0, 1), A F N, (Vitali 1905) 78 [, pp.49 51] 98 A Lebesgue 2 RN. 1 2 (Banach Tarski 79 ) µ(b(o, 1)) = µ(b(o, 2)) σ Jordan Jordan 81 m (A) = inf{m(f ) F J N : F A} Lebesgue m - σ Borel σ[j N ] σ Baire [, p.116] Borel Baire R [, 441D Baire ] R N Borel Lebesgue Cantor ( 2.3.2) C 0 Lebesgue Lebesgue 2 C Lebesgue 2 C C Cantor 82 Borel Lebesgue ( 100) 78 Lebesgue [, p.87] [Lebesgue, ] 82 [Lebesgue, 6 (p.12)]. 83 [Lebesgue, ]

69 4 69 Lebesgue f : [a, b] R + {(x, y) R 2 a x b, 0 y f(x)} R 2 Lebesgue a<b f : (a, b] R + ( 5.1.3) S = {(x, y) R 2 a< x b, 0 <y f(x)} R 2 b S F 2 ( 2) f(x)µ 1 (dx) =µ 2 (S) <f(x) L, a<x b, Γ R R ( 3) Γ(A) = inf (d n c n )(b n a n ) A I= n (an,bn] (cn,dn] J2 n N S I =(a, b] (0,L] 11 Γ(S)+Γ(S c I) =(b a)l n N S = {(x, y) k k +1 <f(x), a < x b, 0 <y f(x)} n n 0<k nl {(x, y) k k +1 k +1 <f(x), a < x b, 0 <y n n n } 0<k nl = {x k k +1 k +1 <f(x), a < x b} (0, n n n ]. S 0<k nl 0<k nl {x k n <f(x) k +1 n, a < x b} (0, k n ]. f x R Lebesgue µ 1 ({x k n 0<k nl Γ(S) < µ 1 ({x k n lim 0<k nl 0<k nl Γ(S c I) = lim 0<k nl <f(x) k +1 n, a < x b}) k n µ 1 ({x k n µ 1 ({x k n <f(x) k +1 n, a < x b}) k n + b a n, n N. <f(x) k +1 n, a < x b}) k n a Γ(S) <f(x) k +1 n, a < x b}) (L k n ) Γ(S)+Γ(S c I) =L lim µ 1({x 0 <f(x) L, a < x b}) =(b a)l b a f(x)µ 1 (dx) =µ 2 (S) =Γ(S) 84 [Lebesgue, 16,17 (p.23)] 85 [, 12 4, 5 (pp.85 86, 279)], [, 118 (p.439)], [Lebesgue, 18,19 (pp.23 26)]

70 Riemann Riemann (1854) Lebesgue (1902) ( ) ([, 16.1, 16.2 (pp.112, 113)], [, 112 (p.443)]) (i) [a, b] Riemann 0 (ii) f [a, b] Riemann Lebesgue [, 120 (pp )] [ ] Riemann Darboux [, 16 (pp )] Riemann Lebesgue 88 Lebesgue Riemann a.s. Riemann 89 Riemann Lebesgue d f = f 90 dx I =[0, 1] [0, 1] J =(a, b) c φ(x) =2x sin( 1 x ) cos( 1 x ) φ(x a) a c 0 φ(x a) =0 x c c a + d a, b f : [0, 1] [ 3, 3] f(x) =0, x, J =(a, b) [0, 1] \ φ(x a), x (a, a + d), f(x) = 0, x (a + d, b d), φ(b x), x (b d, b), f c Riemann Φ(x) =x 2 sin( 1 x ) F : [0, 1] R + F (x) =0,x, J =(a, b) [0, 1] \ Φ(x a), x (a, a + d), F (x) = Φ(d), x (a + d, b d), Φ(b x), x (b d, b), Lebesgue f(x) c f 0 [0, 1] \ 87 [Lebesgue, ] 88 [, 115 (p.430)] [Lebesgue, 29 (pp.37 39)].