I (Analysis I) Lebesgue (Lebesgue Integral Theory) 1 (Seiji HIRABA) 1 ( ),,, ( )

I (Analysis I) Lebesgue (Lebesgue Integral Theory) 1 (Seiji HIRABA) 1 ( ),,, ( )

1 (Introduction) 1 1.1... 1 1.2 Riemann Lebesgue... 2 2 (Measurable sets and Measures) 4 2.1 σ-... 4 2.2 Borel... 5 2.3... 6 2.4... 7 3 (Measurable Functions) 9 4 Lebesgue (Lebesgue Integrals) 11 4.1 Lebesgue... 11 4.2 Lebesgue... 12 5 (Convergence Theorems) 14 6 (Complete Measure Spaces) 16 6.1... 16 6.2 Riemann... 16 6.3... 16 7 (Exchange Theorems of Integral Order) 18 7.1... 18 7.2... 18 7.3... 19 7.4 Fubini... 20 8 L p -, (L p -spaces, Convergence Notion) 22 9 (Outer Measures and Extension Theorem of Measures) 25 9.1... 25 9.2... 26 10 (Differentials of Measures) 29 10.1 Lebesgue-Stieltjes... 29 10.2 Radon-Nikodym... 29 11 (Probability Theory) 31

Lebegue Integral 1 1 (Introduction) 0, [0, 1] 1., (Measure Theory), Lebesgue, ( ) ( )., Riemann, Lebesgue.,, Lebesgue (Lebesgue s Convergence Theorem) Fubini (Fubini s Theorem).,,. (Probability Theory),, Lebesgue Riemann,,. 1.1 ( ) (measure)., 1 0, [0, 1] 1. (.) 1= [0, 1] = {x} =0. x [0,1] [a, b] (a<b) b a. 1 0,, 1=. {x} [x, x +1/n] =1/n 0(n ) {x} =0.,.. 1.1 n 1 [x, x +1/n] ={x}. 1.2 N ( ) (countable), Q, R.,. ( ) ( ) Lebesgue, L,. (1) L, (2) A L A c L, (3) A n L,, 2,... n 1 A n L.

Lebegue Integral 2. (4) [a, b] L(a <b), (5) A L; A =0 B A, B L, L, m([a, b]) = b a (a <b), m : L [0, ] Lebesgue. (1) m( ) =0 (2) A n L m( n 1 A n)= n 1 m(a n). m(a) = A m = m(dx) =dx.,,.,. 1.2 Riemann Lebesgue Lebesgue, Riemann. 1.1 [a, b] (a<b) f(x), = {x k } n ; a = x 0 <x 1 < <x n = b, S( ) = sup [x k 1,x k] f(x k x k 1 ), s( ) = inf f(x k x k 1 ) [x k 1,x k]. inf S( ) = sup s( ), f(x)dx, f Riemann a, f Riemann. x k = a +(b a)k/n b a f(x)dx = lim n b 1 f(x k 1 )(x k x k 1 ) = lim n n f (a + kn (b a) ). Riemann 1, 2 f,., f.,,,. 1.3. f(x) [0, 1], 0, 1 Riemann [0, 1] Q =0. [0, 1] Q c =1. 1 1.. Lebesgue. 1.4 [0, 1] Q =0 Lebesgue. Lebesgue f,, f 1 ([(k 1)/2 n,k/2 n ))..

Lebegue Integral 3 f : R R {± } a R, {f a} = {x R; f(x) a} L, Lebesgue., Lebesgue., f 0, fdx = R f(x)dx := lim n ( n2 n k 1 2 n { k 1 2 n f< k } 2 n + n {f n} )., f + = f 0 = max{f, 0}, f =( f) 0 = max{ f,0} f ± 0 f = f + f, f = f + + f, f + dx, f dx, Lebesgue, fdx =. f L 1 = L 1 (R). f + dx f dx f dx <, f 1.1 Riemann Lebesgue. Lebesgue, Lebesgue. 1.2 (Lebesgue ) f n,f, f n f a.e., h L 1 ; f n h a.e. f n dx fdx. R d Lebesgue, Lebesgue, Fubini. 1.3 (Fubini ) R 2 f(x, y), (1) f 0 a.e. f(x, y)dxdy = dy f(x, y)dx = dx f(x, y)dy. R 2 R R R R (2) f f(x, y) dxdy, dy f(x, y) dx, dx f(x, y) dy R 2 R R R R,, f(x, y)dxdy = dy f(x, y)dx = dx f(x, y)dy. R 2 R R R R,. Riemann,, Lebesgue. Lebesgue (R, L,dx) (X, F,µ)=(, σ, ). fdµ = f(x)µ(dx). X

Lebegue Integral 4 2 (Measurable sets and Measures),,,., X, 2 X. 2.1 σ- 2.1 X F, i.e., F 2 X (1) F (2) A F= A c F (3) A 1,A 2, F = A n F σ- (σ-additive class) σ- (σ-field). (3) (3 ) A, B F= A B F. σ- ( ). {,X}, {,A,A c,x} (A X ), 2 X ( ) σ-field ( ). 2.1 F, A, B, A 1,A 2, F. F. n n X, A B, A \ B, A B := (A \ B) (B \ A)( ), A k, A k. A B =(A c B c ) c, A \ B = A B c, A B. n A k =( n Ac k )c. 2.2 A σ-. (1) X {A X : A A c ( )} n A k. (2) X = R, a b, (a, b] n (a k,b k ], b = (a, ), a = b. [ ] (1) A, B, A c,b c,a B,(A B) c = A c B c,,. σ-, X = N, A n A, A n / A. (2), 2, A = n (a k,b k ], A c = n (a k,b k ] c. 1 m n, J m = {(j 1,...,j m ); 1 j 1 < <j m n} n n m A c = (b j, ) (,a jk ] (b j, ), j=1 m=1 j j k (j 1,...,j m) J m 1 (max b j, ), n m (,a jk ] (b j, ) = m=1 j j k (j 1,...,j m) J m n m=1 (j 1,...,j m) J m σ-, (0, 1) = n 1 (0, 1 1/n]. ( max j j k b j, min,...,m a j k ].

Lebegue Integral 5 2.3 F. {A n } F {B n } F. {B n } (disjoint), n 1, n A k = n B k 2.1 X A, σ-field F 0. 2.2 F 0 σ(a), A σ-field. F σ-field, (X, F) (measurable space), F (measurable set). [ 2.1 ], F := {F : F A σ-field}, F 0 = F,. (1) F, i.e., A σ-field. (2) F 0 σ-field. (3) F 0 (1) 2 X F. (2) F F, F, F 0. A F 0 F F,A F A c F. A c F 0. A n F 0 A n F. (3) A F 0 (2), F 0 F. F F, F 0 F., F 0 A σ-field. F 1 A σ-field, F 1 F, F 0 F 1. F 1 F 0 = F 1.. 2.2 Borel 2.3 X, O σ-field σ(o) Borel field, B(X). X = R n B n = B(R n ) n Borel field. n =1, B = B 1. 2.4 X = R, C. B 1 = σ(c). O σ(c), C σ(o). 2.5 A 1 := {(a, b); a<b }, A 2 := {[a, b) : <a<b }, A 3 := {[a, b] : <a<b< }, A 4 := {(a, b] : a<b< }, A 5 := {(,r]:r Q}, σ(a i )=B 1, i =1, 2, 3, 4, 5. ( ): 1 ( ). O σ(a 1 ). [ ] A 1 O. σ(a 1 ) σ(o) =B 1. ( ) σ(a 1 )=B 1. σ(a 1 ) A 2, σ(a 2 ) A 3, σ(a 3 ) A 4, σ(a 4 ) A 1 i =1, 2, 3, 4 OK. A 5,, a<b, r n <s n r n a, s n b, (a, b) = (r n,s n ]= ((,s n ] \ (,r n ]) n 1 n 1, σ(a 5 ) A 1, A 5 A 4 B 1.

Lebegue Integral 6 2.3 R = R {± }, + =,, : a R ( ) a ± = ±, a = (a >0), = (a <0), 0 = 0=0, a/ =0.. / ( ). / = 1/ = 0=0.,. 2.4 (X, F). µ = µ(dx) :F [0, ] (measure) (1) µ( ) =0 (2) A 1,A 2, F (disjoint, i.e., A i A j = if i j) µ( A n )= µ(a n ) (σ- (σ-additivity). (X, F,µ) (measure space). (X, F,µ) measure space. 2.6 A, B F.. (1) A 1,,A n F disjoint = µ( n A k)= n µ(a k)( ). (2) A B = µ(a) µ(b) ( ). (3) A B,µ(A) < = µ(b \ A) =µ(b) µ(a). (4) µ(a B) =0= µ(a) =µ(b) =µ(a B) =µ(a B). (5) µ(a B) < = µ(a B) =µ(a)+µ(b) µ(a B). (1) σ- A n+1 = A n+2 = =. (2). (3) B =(B \ A) (B A). (4), (5) A B =(A \ B) (A B) (B \ A). {A n }, lim A n = n 1 A n (if A n ), = n 1 A n (if A n ). lim A n = lim sup n A n := N 1 n N A n,, 2, lim A n. {a n } lim A n = lim inf A n := n N 1 n N A n lim a n := lim N inf n N a n = sup N 1 inf a n, n N lim a n := lim sup a n = inf sup a n. N n N N 1 n N 2.7 A 1,A 2, F.. (1) µ( n A n) n µ(a n) (σ- ) (2) A n = µ( n A n) = lim µ(a n ) (,, ) (3) A n, µ(a 1 ) < ( N 1; µ(a N ) < ) = µ( n A n) = lim µ(a n ).

Lebegue Integral 7 (4) [ N 1; µ(a N ) < ]. (5) µ(lim A n ) lim µ(a n ) (6) µ( n 1 A n) < ( N 1; µ( n=n A n) < ) = µ(lim A n ) lim µ(a n ) (7) [ N 1; µ( n=n A n) < ]. (8) n µ(a n) < = µ(lim A n )=0(Borel-Cantelli ). ( ) (4), (7) X = N, X = R Lebesgue. (1) 2.3 B n. (2) B n = A n \ A n 1 (A 0 = ). (3) B n = A 1 \ A n. (5) {a n }, lim a n := lim N inf n N a n,, N 1, n N, µ(a n ) µ( n N A n), inf n N µ(a n ) µ( n N A n). B N = n N A n (N ) N lim µ(a n ) = lim inf µ(a n) lim µ( N n N N n N A n )=µ( N 1 n N A n )=µ(lim A n ). (6) A = n 1 A n B n = A \ A n. (8) [Borel-Cantelli ] µ( A n ) µ(a n ) < C N = n N A n (N ),, σ-, 0 µ(lim A n )=µ( N 1 n N A n ) = lim µ( A n ) lim N n N N n N µ(a n )=0. 2.5 (X, F,µ). µ(x) <, (finite measure), µ(x) =1 (probability measure). µ(x) = A 1,A 2, F; µ(a n ) < ( n) A n = X µ σ- (σ-finite measure). [ ] µ ( ), (X, F,µ) (Ω, F,P),. ω Ω, P = P (dω), F (event). 2.4 (X, F,µ). 1 (counting measure) X:, F =2 X, µ(a) = A (A ) ifa is a finite set, µ(a) = otherwise. 2 δ- (Dirac measure) X:, F =2 X, x X. µ(a) =1ifx A, µ(a) =0if x/ X µ = δ x. 3 (discrete measure) X = {x n }:, F =2 X. µ = n p nδ xn (p n > 0)

Lebegue Integral 8 4 B n Lebesgue (Lebesgue measure on B n ) X = R n, F = B n. A = n (a k,b k ]( a k b k ), µ(a) = n (b k a k ) µ. B n Lebesgue, dx m = m(dx)., Lebesgue,,, B n Lebesgue. 9,. (Ω, F,P). p k = P ({k}) ( P = k p kδ k ) 5 Ω={0, 1, 2,,n}, 0 <p<1 ( ) n p k = p k (1 p) n k,k=0, 1, 2,,n. k 6 Poisson Ω={0, 1, 2, },λ>0 p k = e λ λk,k=0, 1, 2,. k! 7 Ω=N, 0 <p<1 p k = p(1 p) k 1, k =1, 2,. 8 Ω=(a, b), F = B 1 (a, b) ( <a<b< ) P (A) = A /(b a). 9 Cauchy Ω=R 1, F = B 1, m R,a>0 a 1 P (A) = π a 2 +(x m) 2 dx. 10 N(m, v) Ω=R 1, F = B 1 P (A) = A A ] 1 (x m)2 exp [ dx. 2πv 2v m R : (mean),v >0: (variance). 11 d N(m,V) Ω=R d, F = B d, x R d, t x ( ). [ det Q P (A) = exp 1 ] t (x m)q(x m) dx. (2π) d/2 2 A m R d :, V : d d ; (covariance), Q = V 1 2.8, P (Ω) = 1.

Lebegue Integral 9 3 (Measurable Functions) 3.1 (X, F), R = R {+, }. f : X R: F- ( ) def {f a} := {x X : f(x) a} F ( a R). f F. (X, F) =(R, B) (B 1 Borel field) f : R R Borel- ( ), Borel. 3.1 f : X R : (1) f F- (2) {f >a} F ( a R) (3) {f a} F ( a R) (4) {f <a} F ( a R) 3.2 [ a R] [ a Q]. 3.3 f : X R F- f 1 (B) :={f B} F ( B B) {f =+ }, {f = } F. f, F- f 1 (B) F. {f a} = f 1 ([,a]) = {f = } f 1 ((,a]) ( ). {f = ± } F. B = σ({(,a]; a R}) ( 2.5) D = {A R; f 1 (A) F}: σ-field, {(,a]; a R}. 3.4 f,g, f n (n =1, 2, ) F-, ( ) : α R. (1) αf (2) f + g (3) fg (4) 1/f (5) f (6) sup f n (7) inf f n (8) lim f n (9) lim f n (10) lim n f n (1) α>0, =0,< 0. (2) {f <a g} = r Q {f<r<a g}. (3) f 2 F, fg = {(f + g) 2 +(f g) 2 }/2. (4) {1/f a} = {f >0,af 1} {f<0,af 1} (1). (5) { f a}, a 0,< 0. (6) {sup f n a} = {f n a} (7) inf f n = sup( f n ) (8) lim f n = inf N 1 sup n N f n 3.5 f,g F-, f,f g, f g ( ). f g(x) = max{f(x),g(x)}, f g(x) = min{f(x),g(x)}. f g, f g f ± g 3.6 Borel. f : R R f 1 (O 1 ) O 1. D = {A R; f 1 (A) F}: σ-field, O 1. 3.7. F R σ-field. F-, F B (f(x) =x) 3.2 (1) A X 1 A (x) =1(x A), 0(x/ A), 1 A A (defining function). (2) f : X R, a 1,,a n R X {A 1,,A n },

Lebegue Integral 10 f(x) = a k 1 Ak (x) f (simple function). 3.8 ( ) f : X [0, + ] {f n } f = lim n f n. f : X [0, + ]: F-, {f n }: ; 0 f n f. f n (x) = 2 2n k 1 2 n 1 {(k 1)/2 n f<k/2 n }(x)+2 n 1 {f 2n }(x). 3.9 (R, B) f(x) =x 2, f 1,f 2,f 3. 3.3 (X, F,µ). (1) µ(n) =0 N F µ-. µ- N A N µ-. (2) X P (x) {x X : P (x) } µ-, P (x) P, µ a.e.. f = g, µ-a.e. def µ(f g) =0. def f n f, µ-a.e. µ(f f) =0. def lim f n exists, µ-a.e. µ(lim f n lim f n )=0. M = M(X) =M(X, µ) =M(X, F,µ):={f : X R; F- }. 3.10 M(X, F,µ) f g def f = g, µ-a.e., (1) f g = af ag ( a R), (2) f 1 g 1 f 2 g 2 = f 1 + f 2 g 1 + g 2., {f M(X, F,µ): f < +,µ-a.e.}/.

Lebegue Integral 11 4 Lebesgue (Lebesgue Integrals) 4.1 Lebesgue (X, F,µ) f fdµ = X fdµ = 4.1 f = n i=1 a i1 Ai, : n fdµ = fdµ = f(x)µ(dx) := a i µ(a i ),, a i 1 Ai dµ := X X i=1 X i=1 f(x)µ(dx). a i µ(a i ). 4.1 well-defined, f. m m f = b j 1 Bj a i µ(a i )= b j µ(b j ). j=1 i=1 j=1 i=1 ( ) {C ij = A i B j }. 4.2 f,g. (1) (f + g)dµ = fdµ + gdµ (2) α 0= αfdµ = α fdµ (3) f g = fdµ gdµ ( ) (1), (3). 4.2 f, {f n }, f = lim n f n ( 3.8).. fdµ := lim f n dµ. n 4.1 g = Z well-defined, {f n }. mx j=1 b j1 Bj ; g f Z Z gdµ lim n Z f ndµ ( ). (1) gdµ < ( µ(g >0) < ) (2) gdµ = ( j 0; b j0 > 0,µ(B j0 )= ). b 0 = min{b j > 0} (b 0 =0 g =0, b 0 > 0 ), 0 < ε<b 0. {f n >g ε} X {g >0} = {g b 0}, A n = {f n >g ε} {g>0}. Z Z Z X X f ndµ f n1 An dµ (g ε)1 An dµ = (b j ε)µ(b j A n) (b j ε)µ(b j {g>0}) j;b j >0 (1) = R gdµ εµ(g >0), (2) (b j0 ε)µ(b j0 )=. j;b j >0 4.3 f : X R, f + := f 0, f := (f 0). f = f + f, f = f + + f. 4.3 f : X R F-. f + dµ, f dµ, f, fdµ := f + dµ f dµ. f + dµ, f dµ, f (integrable).

Lebegue Integral 12 4.4, (X, F,µ) f,. f dµ = f + dµ + f dµ (( ),. ), f, f : integrable f dµ <. 4.4 f: (X, F,µ), A F. 1 A f, f A fdµ := 1 A fdµ A. L 1 - (L 1 -function). { L 1 = L 1 (X) =L 1 (X, µ) =L 1 (X, F,µ):= f M(X, F,µ):f, i.e., } f dµ < 4.2 Lebesgue., (X, F,µ) f,g. 4.5 A F,µ(A) =0= fdµ =0 4.6 afdµ = a A fdµ ( a R) 4.7 f,g L 1 (X, F,µ)= (f + g)dµ = fdµ + gdµ fdµ, gdµ > or < (f + g)dµ > or <,. 4.8 A, B F A B = = fdµ = fdµ + fdµ A B A B 4.9 f g, µ-a.e. = fdµ gdµ 4.10 f = g, µ-a.e. = fdµ = gdµ 4.11 4.12 4.13 f 0,µ-a.e. fdµ =0= f =0,µ-a.e. A fdµ =0 ( A F)= f =0,µ-a.e. f L 1 (X, F,µ)= f <,µ-a.e. 4.14 4.15 g L 1 (X, F,µ) f g, µ-a.e. = f L 1 (X, F,µ) fdµ f dµ

Lebegue Integral 13 4.16 ( ) f =Ref + iimf : X C (, i = 1) Ref, Imf f, fdµ = Refdµ+ i Imfdµ.. ( 4.15). 4.5 4.10,,. 4.6 a 0, a = 1, a<0. 4.7 : 4.8 4.7. (f + g) + + f + g =(f + g) + f + + g + 4.9. f g fdµ > gdµ <, 4.7 f =0. f g, µ-a.e. A := {f g} 4.5, 4.8. 4.10 4.9. 4.11. A n := {f 1/n}. 4.12. A := {f 0} f + =0,µ-a.e.. f. 4.13.. 4.16. 4.15. z = z e i arg z z = e i arg z z., θ R, ) Re (e iθ fdµ = Re ( e iθ f ) dµ f dµ, θ = arg fdµ.

Lebegue Integral 14 5 (Convergence Theorems) f,f 1,f 2, (X, F,µ). µ(lim f n f) =0 f n f, f n f, µ-a.e.. a.e. almost everywhere. ( µ = P P -a.s., a.s. almost surely. ) 5.1 f n f, µ-a.e., f n dµ fdµ,. ( ) X =(0, 1), F = B(0, 1),µ(dx) =dx, f n 0( ) f n dx =1. 5.1 ( (Monotone Convergence Theorem)) [0 f 1 f 2 f n f], µ-a.e., i.e., 0 f n f, µ-a.e. fdµ = lim f n dµ. n ( ) X 0 = {0 f n f} (f,f n f1 X0,f n 1 X0 ) µ-a.e., f n.. f n, {f n,k } ; lim k f n,k = f n. g k := max{f n,k : n k}, g := lim k g k, n k f n,k g k f k f k,n, g = f, g k. 5.1 (Fatou (Fatou s Lemma)) f n 0,µ-a.e. ( n 1) lim inf ndµ lim inf n n f n dµ. ( ) µ-a.e.. g n := inf{f k : k n} MCT. 5.2 (Lebesgue (Lebesgue s Convergence Theorem)) f n f, µ-a.e. h L 1 (X, F,µ); f n h ( n 1), µ-a.e., f L 1 (X, F,µ) fdµ = lim f n dµ. n Lebesgue (Dominated Convergence Theorem). µ(x) < h Lebesgue (Bounded Convergence Theorem). ( ) µ-a.e., f, Fatou s lemma f L 1. h f n h, µ-a.e. f n + h h f n Fatou s lemma. 5.2 µ(dx): a finite measure on (R, B) i = 1, lim e ix/n µ(dx) =µ(r) n R. 5.3 f L 1 ([0, ), B([0, )),dx) t 0, L(t) := e tx f(x)dx, [0, ). ( ) t 0, {t n } [0, ); t n t, L(t n ) L(t). ( )

Lebegue Integral 15 L(t) at t 0 t 0, {t n } [0, ); t n t, L(t n ) L(t). 5.4 xf(x) L 1 ([0, ), B([0, )),dx) L(t) t>0 C 1. ( ) x 0, 0 < h <t/2= e (t+h)x e tx h = x h h 0 e (t+s)x ds xe tx/2 x. 5.5 µ(dx): a finite measure on (R, B), i = 1. z R, F (z) = e izx µ(dx),, n 1, x n L 1 (R, B,µ) C n. ( x n L 1 k n, x k L 1. Hölder,) 5.3 ( (Absolute Continuity)) f L 1 (X, F,µ), µ(a) 0= fdµ 0. A ( ) { f n := f n ( f n) ( f >n) MCT f dµ = lim n f n dµ, : fdµ f dµ = ( f f n )dµ + A A A A f n dµ ( f f n )dµ + nµ(a). X

Lebegue Integral 16 6 (Complete Measure Spaces) 6.1 6.1 (X, F,µ). (1) µ(n) =0 N F µ-. µ- N, A N µ-. (2) µ-, µ. 6.1 (X, F,µ), (X, F, µ), µ µ. (1) F := {A N : A F,N N} F σ-field (2) A F,N N, µ(a N) :=µ(a), (X, F). (3) µ. 6.2 (X, F, µ) (X, F,µ). 6.3 (Lebesgue ) m = m(dx) =dx B n Lebesgue,(R n, B n,m) (R n, B n, m), L n := B n Lebesgue, Lebesgue. m m, (R n, L n,m) Lebesgue. [ ],. 6.2 Riemann Riemann Lebesgue. 6.1 [a, b] f Riemann, Lebesgue, b f(x)dx = a [a,b] f(x)m(dx)., Riemann, Lebesgue m Lebesgue. ( ) [a, b] 2 n, Riemann (Darboux )., g n,h n g, h, h f g, Lebesgue f Riemann. h = f = g, m-a.e. Lebesgue f. 6.3 (R n, L n,m) Lebesgue. B n L n 2 Rn,,, 3.

Lebegue Integral 17 6.2 A R n, [A L n,m(a) =0 2 A L n ]. L n 2 Rn.. ( Lebesgue.) 6.1 A L n x R n, x + A L n, m(x + A) =m(a). [ 6.2 ] (1) ( ) 2 A L n = m(a) =0. A.. E R n ; R n = (q + E) ( disjoint union). (1) q Q n q Q n,a (q + E) L n, A = q Q A (q + E). q n Qn, B := A (q + E) A, B L n. r Q n,r+ B L n. B q + E, {r + B} r Q n. C = r Q n ; r 1 (r + B) Ln, >m(c) = m(r + B) = m(b) = m(b). r Q n ; r 1 r Q n ; r 1 m(b) =0,, m(a) =0. (1). x, y R n x y x y Q n. C., C C x C C, E := {x C ; C C}. C = x C + Q n R n = (x C + Q n )= {x + q} = (q + E). C C x E q Q n q Q n,. (1). ( ) 2 Rn = L n, m(r n )=0.. 6.3 L n B n. B n = ℵ,. Cantor Lebesgue 0, L n, 2 R., L n = 2 R > R = ℵ = B n.

Lebegue Integral 18 7 (Exchange Theorems of Integral Order) 7.1 7.1 X M [A n M n A n M] [A n M n A n M]. (. ) A A, m(a). ( σ-field. ) 7.1 σ-, M,, σ-. 7.1 ( (Monotone Class Theorem)) A X m(a) =σ(a). ( ) σ-,. m(a). A m(a). M c := {A X; A c m(a)} M c m(a). A m(a) A c m(a). A, B m(a) A B m(a) A m(a), M A := {B X; A B m(a)}, M A m(a), A, B m(a) A B m(a). M A m(a). A A A M A m(a) M A., A A,B m(a) A B m(a). A m(a) B A, A, B A B m(a),, B M A. A M A, m(a) M A. 7.2, Lebesgue, 1 3. 7.2 f n 0,µ-a.e. (n =1, 2, ) f n dµ = f n dµ 7.3 7.4 f n dµ < f n, f n dµ = f n dµ. f L 1 (X, F,µ). {A n } F S fdµ = An A n fdµ. (X j, F j,µ j )(j =1,,n). ( σ-. ) 7.2 (1) X 1 X n A A = A 1 A n (A j F j,j =1,,n), A. (2) F 1 F n := σ(x 1 X n ), (X 1 X n, F 1 F n ) (product measurable space).

Lebegue Integral 19 7.3 7.2 µ j,j =1,,n σ-. (X 1 X n, F 1 F n ) µ : A = A 1 A n (A j F j,j =1,,n) µ(a) =µ 1 (A 1 ) µ n (A n ).. 7.3 µ (product measure), µ = µ 1 µ n. (X 1 X n, F 1 F n,µ 1 µ n ) (product measure space). 7.5 (F 1 F 2 ) F 3 = F 1 F 2 F 3. (, σ(σ(f 1 F 2 ) F 3 )=σ(f 1 F 2 F 3 )). σ(f 1 F 2 ) F 3 σ(f 1 F 2 F 3 ). A 3 F 3 1, G = {B X 1 X 2 ; B A 3 σ(f 1 F 2 F 3 )}, G F 1 F 2 ; σ-field, σ(f 1 F 2 ) G. n =2. (X, F,µ), (Y, G,ν) σ-. A F G, A x := {y Y :(x, y) A} (x X), A y := {x X :(x, y) A} (y Y ), A x-, y-. (X Y, F G) f f x : y f(x, y) (x X), f y : x f(x, y) (y Y ), f x-, y-. 7.1 µ, ν σ-. A F G, : (1) x X, A x G x ν(a x ) F-, (2) y Y, A y F y µ(a y ) G-, (3) ν(a x )µ(dx) = µ(a y )ν(dy). 2,. 7.6, σ- : (R, B,µ) Lebesgue, (R, 2 R,ν), A = {(a, a) :a R} ν(a x )µ(dx) µ(a y )ν(dy) R R 7.7 A.. (1). (2) σ(a) =F G. (3) A.

Lebegue Integral 20 ( ) A. µ, ν, M M m(a) =σ(a) =F G. µ, ν σ- X n F X, Y n G Y ; µ(x n ),ν(y n ) < X n Y n, n. ( ) A F G, (µ ν)(a) := ν(a x )µ(dx) = µ(a y )ν(dy). (X Y, F G) σ-,. A. 7.8 µ, ν (X, F) σ-, F = σ(a) A. A, µ = ν F, µ = ν. 7.4 Fubini 7.3 (Fubini ) (X, F,µ), (Y, G,ν) σ-. f F G-. f(x, y) x- f x, y- f y G-, F-, : (1) f 0,µ ν-a.e. x f x dν F-, y f y dµ G-, fd(µ ν) = dµ fdν = dν fdµ. (2) f. f d(µ ν), dµ f dν, dν f dµ,, fd(µ ν) = dµ fdν = dν fdµ. ( ) ν(a x )= 1 Ax (y)ν(dy) = 1 A (x, y)ν(dy), µ(a y )= 1 A y(x)µ(dx) = 1 A (x, y)µ(dx) Y Y X X µ ν ( ), f(x, y) =1 A (x, y) (A F G).. 7.9 2 {a ij } Fubini. a ij < + a ij < + = a ij = a ij. i=1 j=1 j=1 i=1 i=1 j=1 j=1 i=1

Lebegue Integral 21 7.10 A, B F G. x X ν(a x )=ν(b x ) µ ν(a) = µ ν(b). 7.11 f L 1 (X, F,µ),g L 1 (Y, G,ν) fg L 1 (X Y, F G,µ ν), fgdµ ν = fdµ gdν X Y X Y. 7.12 f(x, y) = x2 y 2 (x 2 + y 2. ) 2 1 0 dy 1 0 f(x, y)dx = π 1 4, dx Fubini 0 1 0 f(x, y)dy = π 4

Lebegue Integral 22 8 L p -, (L p -spaces, Convergence Notion) (X, F,µ), M(X, F,µ). 8.1 L p = L p (X, F,µ), 1 p : L p - (1) 1 p<, ( L p = L p (X, F,µ):={f M(X, F,µ): f p < } (, f p := f p dµ) 1/p ), f L p p, L p -. (2) p =, L = L (X, F,µ):={f M(X, F,µ): f < }, (, f = ess.sup f := inf{α : f α, µ a.e.}: f ), f L, L -. ( f f <,µ-a.e. ) (3) p L p - (norm), L p. 8.1 1 p, 1 q, 1/p +1/q =1., 1,. (1) [Hölder ] f L p,g L q fg L 1 fg 1 f p g q. (2) [Minkowski ] f,g L p, f + g p f p + g p. ( ) p =1,. 1<p<. (1) fg =0,µ-a.e.. A = {fg 0} µ(a) > 0. log, a, b > 0 1 p log a + 1 ( a log b log q p + b ), i.e., a 1/p b 1/q a q p + b q a = f p / f p dµ, b = g q / g q dµ A A, A. (2) q = p/(p 1), i.e., 1/p +1/q =1 f + g p dµ f f + g p 1 dµ + g f + g p 1 dµ f + g p 1 L q Ho lder, 1 1/q =1/p. 8.1. 8.2 (L p, p )(1 p ) µ-a.e., p, Banach ( ). p =2 f,g = fgdµ L 2, (L 2,, ) Hilbert ( )., f,g = fgdµ. ( 3.10, f L p [f ] L p /, [f] p = f p (L p /, p) Banach.,, (L p, p) Banach. ) X K = R or C, : X [0, ] x, y X, a K : (1) x =0 x = 0, (2) ax = a x, (3) x + y x + y. (X, )

Lebegue Integral 23 X {x n} Cauchy Ω lim xn xm =0. m,n X Cauchy {x n}, i.e., x X; x n x 0, X (complete), (X, ) Banach, X (inner product), : X X K; (1) x, x 0, = 0 x = 0, (2) x, y = y, x, (3) x, ay + z = a x, y + x, z. p (X,, ). x = x, x, Hilbert. 8.2 8.3 Cauchy. (X, ) complete {x n } X; x n <, x n X. ( ) ( ) {x n } Cauchy {x nj }; x nj x nj+1 < 2 j, x nj. ( ) {f n } L p ; n f n < f := n f n, µ-a.e. f L p. f k = lim f n lim n p p n f n p = f n p <. n f n <, µ-a.e., f = n f n µ-a.e.. f p = f n f n < p p f L p. 8.4 µ(x) <, 1 p<q, L p (X, F,µ) L q (X, F,µ). 8.5 f,g R. f g f g; f g(x) := f(x y)g(y)dy, (1) f,g L 1 f g 1 f 1 g 1, (2) f L 1,g L 2 f g 2 f 1 g 2. 8.2 (1) 1 p<. f n,f L p (X, F,µ) f n f p or L p - ; R f n f in L p lim n f n f p =0. (2) f n,f M(X, F,µ). f n f µ- ; f n f in µ ε >0, lim µ( f n f ε) =0. n (, µ = P in µ in P in pr.. ) 8.6 µ =.

Lebegue Integral 24 8.7 f n,f M(X, F,µ). ε>0 1 p<, µ( f n f ε) 1 ε p f n f p dµ., L p - =. 8.8 8.9 0 L 1 -. 0. ( ) 8.8. [0, 1] Lebesgue, ( ) 1, 0 0. 8.9. 0, 0. 8.3 (1) 1 p, L p -,.. (2),. f n f in µ (n )= {f nk } {f n }; f nk f, µ-a.e. (k ). ( ) (2) µ({ f nk f 1/2 k }) < 1/2 k, 1-11 (h) ( µ(ak ) < µ(lim sup A k )=0) µ(lim f nk f) =0. 8.10.

Lebegue Integral 25 9 (Outer Measures and Extension Theorem of Measures) 2 Lebesgue,. ( ), σ-field, Borel field., σ-field.. 9.1 (X, F 1,µ),(X, F 2,ν), F 1 F 2. ν F1 = µ, i.e., F 1 ν = µ µ ν, ν µ. 9.1 (Carathéodory ) A X. µ 0 : A [0, ] ( ) µ 0 ( ) =0 A σ-, i.e., A n A(n =1, 2,...) disjoint A n A = µ 0 ( A n )= µ 0 (A n ), (X, σ(a)) µ µ A = µ 0. µ 0 A σ- : {X n } A; µ 0 (X n ) <, X n = X µ ( σ-,)., (outer measure), σ(a). 9.1. µ 0 A, ( ),, ( ) Lebesgue. 9.2 (Lebesgue ) A = n (a k,b k ]( a k b k,k =1, 2,...,n), µ(a) = n (b k a k ) (R n, B n ) µ. Lebesgue, dx m = m(dx). ( ) n =1. (n 2. ) (a, b] ( a b ) A, ( 1-2 (b)). A = n i=1 (a i,b i ] A, m 0 (A) = n i=1 (b i a i ), m 0 : A [0, ] µ 0 ( ) =0., (c, d] = (c j,d j ]( ) = m 0 ((c, d]) = m 0 ((c j,d j ]) j=1 j=1

Lebegue Integral 26... <c<d<. 0 < ε < d c, [c + ε, d] j=1 (c j,d j + ε/2 j ) [c + ε, d] N; (c+ε, d] [c+ε, d] N j=1 (c j,d j +ε/2 j )., d (c+ε) =m 0 ((c+ε, d]) m 0 ( N j=1 (c j,d j + ε/2 j )) N j=1 (d j + ε/2 j c j ) < j=1 (d j c j )+ε, ε>0 d c j=1 (d j c j ), i.e., m 0 ((c, d]) j=1 m 0((c j,d j ]). c = or d =, m 0 ((c, d] =, L>0,,, m 0 (( L, L] (c, d]) = m 0 ( j 1 (( L, L] (c j,d j ])) = j 1 m 0 (( L, L] (c j,d j ]) j 1 m 0 ((c j,d j ]). L m 0 (( L, L] (c, d]), j 1 m 0((c j,d j ]= = m 0 ((c, d]). 9.2 9.1 µ :2 X [0, ] : (1) µ ( ) =0, (2) A B µ (A) µ (B) (3) A n 2 X,, 2,... µ ( A n ) µ (A n ). A X µ - : Y X, µ (Y ) µ (Y A)+µ (Y A c ). ( (1), (3),, =. ) 9.1 µ X F = {A X; A µ - } σ-field, µ (X, F ). 9.2 F. F, A F A c F. A, B F A B F., µ (Y (A B)) = µ (Y (A B) A)+µ (Y (A B) A c )=µ (Y A)+µ (Y A c B) µ (Y ) = µ (Y A)+µ (Y A c ) = µ (Y A)+µ (Y A c B)+µ (Y A c B c ) = µ (Y (A B)) + µ (Y (A B) c ). ( 9.1 ) A n F,, 2,... Y X. µ (Y (A n A n+1 )) = µ (Y (A n A n+1 ) A n )+µ (Y (A n A n+1 ) A c n) = µ (Y A n )+µ (Y A n+1 ), µ (Y k A n)= k µ (Y A n ). k A n F µ (Y )= k k µ (Y A n )+µ (Y ( A n ) c )

Lebegue Integral 27.,,, k, µ (Y ) µ (Y ) k µ (Y A n )+µ (Y ( A n ) c ) µ (Y A n )+µ (Y ( A n ) c ). σ- A n F. F σ-field. Y = A n µ ( A n ) µ (A n ), σ- µ F, σ-,. ( ) I. Y X, { } µ (Y ) := inf µ 0 (A n ); A n A,Y A n, µ X, A F. (1), (2). (3). A n X, n =1, 2,.... ε>0, {A n,k } A; A n k A n,k k µ 0(A n,k ) µ (A n )+ε/2 n. µ µ ( A n ) n k µ 0 (A n,k ) µ (A n )+ε n, ε 0 (3). A A. Y X, ε > 0, {B n } A; Y B n µ0 (B n ) µ (Y )+ε. µ (Y A)+µ (Y A c ) {µ 0 (B n A)+µ 0 (B n A c )} = µ 0 (B n ) µ (Y )+ε. ε>0 Y X A µ -, i.e., A F. σ(a) F, µ F, σ-, σ(a). II. µ 0 A, σ- µ = µ 0 on A. A A. µ µ (A) µ 0 (A). ε>0, {A n } A; A n 1 A n n 1 µ 0(A n ) < µ (A)+ε. µ 0 µ 0 (A) =µ 0 ( (A A n )) = µ 0 (A A n ) µ 0 (A n ). µ 0 (A) <µ (A)+ε, ε>0,. III. µ := µ σ(a),. µ 0 A, σ-,,. ( 7.8. ) 9.3 (X, F,µ ). 9.4, σ(a) F, µ 0 σ-, σ(a) =F, i.e., (X, σ(a),µ) (X, F,µ ). 9.2 Lebesgue m m 0 m, m - Lebesgue., Lebesgue Lebesgue (R n, B n,m).

Lebegue Integral 28 [ ] X, 2 X X, A 2 X, F := σ(a). µ 0 on A, i.e., µ 0 ( ) =0, Y X, { } µ (Y ) := inf µ 0 (A n ); A n A,Y A n. A F def µ (Y ) µ (Y A)+µ (Y A c ) ( Y X) σ-field, A F, i.e., F = σ(f F, µ (X, F ) ; µ = µ 0 on A. µ := µ F. 9.3 9.4 [ 9.3 (X, F,µ ) ] µ - µ, µ,, µ (A) =0 A F. µ (A) =0. µ µ (Y A) µ (A) =0,µ (Y A c ) µ (Y )., µ (Y A)+µ (Y A c ) µ (Y ) ( Y X). A F. [ 9.4 ], σ(a) F, µ 0 σ-, σ(a) =F. F = σ(a), N µ-. (1) F F ( ) N F. A N def N F; A N,µ(N) =0, µ µ µ (A) µ (N) =µ(n) = 0, i.e., µ (A) =0. µ (Y A)+µ (Y A c ) µ (Y )( Y X). A F. (2) µ 0 σ- F F µ 0 µ σ-, µ. A F, A 0 F,C N; A = A 0 C. A c F, µ, n 1, B n σ(a) =F; A c B n,µ (A c ) µ(b n ) <µ (A c )+1/n. B = B n A c B F, µ(b) =µ (A c ). µ µ (A) =µ (X) µ (A c )=µ (X) µ (B) =µ (B c ). A 0 := B c F A 0 A, µ (A) =µ(a 0 ), C := A \ A 0 C N., µ (C) = µ (A) µ(a 0 ) = 0,, n 1, C n F; C C n,µ(c n ) <µ (C)+1/n =1/n. N := C n F C N µ(n) = 0, i.e., C N. µ 0 A σ- µ σ-. ( ), ( ). {B n } F ; B n = X, µ (B n ) <. B n A. (.)

Lebegue Integral 29 10 (Differentials of Measures) R,,.,. 10.1 Lebesgue-Stieltjes (R, B) 1 Borel. 10.1 µ σ- on (R, B),,. F (x):, µ((a, b]) = F (b) F (a). F (x), µ on (R, B). 10.1 F (x) µ Lebesgue-Stieltjes, df (x)., fdµ fdf = f(x)df (x). R [ 10.1 ] µ((0,x]) (x >0) F (x) = 0 (x =0) µ((x, 0]) (x <0)., A, A = n (a k,b k ] A( ), µ 0 (A) := (F (b k ) F (a k )), A, i.e., µ 0 ( ) =0., Lebesgue, A, σ-, µ. 10.2 Radon-Nikodym (X, F,µ). ( Lebesgue.) ν (X, F). ν µ. 10.2 (1) ν µ; ν µ, (absolute conti.) (2) ν µ; ν µ (singular) def E F; µ(e) =0,ν(E c )=0. A def A F; µ(a) =0,ν(A) =0. f L 1 (dµ),f 0, dν = fdµ, i.e., ν(a) = fdµ ν µ.

Lebegue Integral 30 E F, ν 1 := µ E,ν 2 := µ E c ν 1 ν 2. 10.2 (X, F), µ σ-. ν on (X, F),. (1) (Lebesgue ) ν 1,ν 2 ; ν = ν 1 + ν 2,ν 1 ν 2,ν 1 µ.. (2) (Radon-Nikodym ) ν 1 µ f L 1 (dµ); f 0,dν 1 = fdµ. f µ-a.e.. Radon-Nikodym f dν 1 /dµ, ν 1 µ Radon-Nikodym. ( ) µ., g G def g M(X, F,µ); g 0, gdν ν(a) ( A F), α := sup { gdµ; g G } ( ν(x)). n 1, g n G; g n dµ > α 1/n. f n := max{g 1,...,g n } f n G., A F, A k = A {f n = g k }, f n dµ = f n dµ = g k dµ ν(a k )=ν(a). A A k A k f n G. f n f, f G. f g n, fdµ = α<. f 0, f L 1 (dµ). dν 1 := fdν, ν 2 := ν ν 1. ν 1 ν 2., Hahn (Hahn-Jordan ). λ = λ 1 λ 2, Hahn E F; λ 0onE, λ 0onE c., λ n := ν 2 µ/n, n 1 E n F; λ n 0onE n, λ n 0onEn. c A F ν(a) ν 1 (A)+ν 2 (A E n ) ν 1 (A)+ 1 n µ(a E n)= (f + 1 n 1 E n )dµ. f = f +1 En /n f G. α fdµ = fdµ + 1 n µ(e n)=α + 1 n µ(e n). A A µ(e n )=0. N = n 1 E n µ(n) =0. n 1,N c E c n ν 2 (N c )=0. ν 1 ν 2. 0 ν 2 (N c )=λ n (N c )+ 1 n µ(n c ) 1 µ(x) 0. n µ 0.

Lebegue Integral 31 11 (Probability Theory)., 1/2 (n ), ( ) 1/2 n,., ( ) 0.,.,,,.,. (σ-field), ( ). (Ω, F,P), (probability space), A F (event), P = P (dω) (probability measure). X = X(ω) (random variable), (mean) (expectation), E[X] := XdP = X(ω)P (dω). A F E[X; A] :=E[X1 A ]. (variance) V (X) :=E[(X EX) 2 ]=E[X 2 ] E[X] 2.,. (1) X 1,...,X n (independent) a 1,...,a n R, P (X 1 a 1,...,X n a n )=P(X 1 a 1 ) P(X n a n ). (2) X 1,X 2,... n, X 1,...,X n. X 1,...,X n : (1) A 1,...,A n B, P (X 1 A 1,...,X n A n )=P(X 1 A 1 ) P(X n A n ). (2) Borel f 1,...,f n, E[f 1 (X 1 ) f n (X n )] = E[f 1 (X 1 )] E[f n (X n )]. 1.. 1/2 ( ).,, n X n 1, 0 ( ) E[X n ]=1/2, n 1 X n. n, n. :

Lebegue Integral 32 [ (Law of Large Numbers)] X 1,X 2,..., EX n = m, sup V (X n ) <. : ( ) 1 1 lim X k = m, a.s., i.e., P lim (X k m) =0 =1. n n n n. sup E[X 4 n ] <. ( sup E[X 4 n] < ) X n X n m, m = 0, i.e., E[X n ]=0. ( n ) 4 X k, 0, Hölder, E[X 2 ] (E[X 4 ]) 1/2, ( n ) 4 E X k = E[Xk]+ 4 E[Xi 2 ]E[Xj 2 ] n 2 sup E[Xk] 4 k i j,1 i,j n, ( ) 4 ( E 1 n ) 4 X k 1 = n n 4 E X k. P ( lim n 1 n ) X k =0 =1. 1 n 2 sup E[Xk] 4 < k 0, 1 n (X k m) = 1 n X k m 0 (n ) n, ( n ),.,,,, ( ), ( Gauss ).,.,,. [ (Central Limit Theorem)] X 1,X 2,...,. EX n = m R, V (X n )=v (0, ). : a b ( lim P a 1 n ) b 1 (X k m) b = e x2 /(2v) dx. n n 2πv a

Lebegue Integral 33, 1 n n (X k m) 0, v N(0,v). 1. σ-field 2. 3. Kolmogorov {X n = X n (ω)},2,..., {X t = X t (ω)} t 0. (Brownian motion),, A. Einstein, N. Wiener, Wiener.,,.,,,.,. ( Einstein,,. ),,,.,.,.,,,,,,.,,,.

Lebegue Integral 34 1 (X, F,µ). 1. X = {x n }, F =2 X,µ= a n δ xn (a n > 0). X f fdµ,. 2. A n F,, 2,, µ(a n ) < µ(lim sup A n )=0. n 3. (0, 1) X f(t, x), t (0, 1) x, µ-a.e. x X, t. F (t) := f(t, x)µ(dx).. (a) h L 1 (X, F,µ); sup t f(t, ) h, µ-a.e. F (t). X (b) f(t, x) µ-a.e. x X, t, t (0, 1), δ = δ(t) > 0( ), g t = g t (x) L 1 (X, F,µ)(x ); F (t). f(t + s, x) f(t, x) sup g t (x), s ( δ,δ);s 0,t+s (0,1) s µ(dx)-a.e. 4. ([0, 1], B([0, 1]),µ) Lebesgue, ([0, 1], 2 [0,1],ν), f(x, y) =1 if x = y, f(x, y) =0ifx y. ( ) ( ) f(x, y)µ(dx) ν(dy) f(x, y)ν(dy) µ(dx) [0,1] [0,1] [0,1] [0,1] f Fubini 5. (X, F,µ)=(R, B(R),dx) Lebesgue. f L 1,g L 2 f g 2 f 1 g 2. ( ): f (x y)g(y) = f (x y) 1/2 f (x y) 1/2 g(y) Hölder, Fubini. Fubini, Hölder Z R g(y x)g(z x) dx g 2 2. )

Lebegue Integral 35 2 (X, F,µ), 1 p<, ( µ ) p L p = L p (X, F,µ). { ( } 1/p L p = f; f p = f(x) µ(dx)) p <. L = L (X, F,µ) ( µ ), f =ess.sup f. ess.sup f = min{m 0; f M, µ-a.e.}. 1. f,f n L 1 f n f, µ-a.e.. f n 1 f 1 = f n f in L 1, f n dµ f dµ = f n f dµ 0 Lebegue. fi fi ( ) fi f n f f n fi f f n f =( f n f f n )+ f n. 2. (X, F,µ)=(R, B(R),dx) 1 Lebesgue. R f, supp f = {x R; f(x) 0} f (support). ( compact ) C c = C c (R), Cc = C c C compact. (a) Cc, Cc, 1 p, L p. ( ) f Cc, supp f supp f C c L p. (b) R f g Cc, h(x) f g(x) = f(y)g(x y)dy h C, n h (n) (x) =f g (n) (x) = f(y)g (n) (x y)dy. R R ( ) n =1. 3. (X, F,µ)=(R, B(R),dx) 1 Lebesgue. f L 1,g L 2 f g 2 f 1 g 2. Z ( ) f (x y)g(y) = f (x y) 1/2 f (x y) 1/2 g(y) Hölder, Fubini. Fubini, Hölder g(y x)g(z x) dx g 2 2. ) 4. (X, F,µ) µ,, µ(x) <. 1 p<q ( p/q f p f q µ(x) 1/p 1/q, i.e, f p p = f p dµ f dµ) q µ(x) 1 p/q. ( µ 1 p<q L p L q. µ.) ( ) p = q/p(> 1), q ;1/p +1/q =1 Hölder.