2001 (Lebesgue Integral Theory) Riemann ( [a, b]),,,, 0. Riemann.,,,. ( ), (. ) Lebesgue. Riemann,,,, Lebesgue.,,,,,,. (measure theory),, ( ),., σ- (σ

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1 200 (Lebesgue Itegral Theory) Riema ( [a, b]),,,, 0. Riema.,,,. ( ), (. ) Lebesgue. Riema,,,, Lebesgue.,,,,,,. (measure theory),, ( ),., σ- (σ-field) (measure),., Lebesgue.,, (probability theory), (evet), ( (probability measure) ),. (,, ),. (,.).,. () [a, b], (2) f(x) S( ) =S(f; ), s( ) =s(f; ). (4) f(x) [a, b] Riema f(x)dx. a 2. Riema. b [f: coti. o [a, b] f(x)dx is well-defied.] a 3.,. [f : coti., f f o I f: coti. o I.] 4.,. b b [f : coti., f f o [a, b] lim f dx = fdx. a b a

2 Lebesgue Itegral 2 (Measurable sets ad Measures) [ ] X F () F (2) F c F (3), 2, F F σ- (σ-field) σ- (3) (3 ), B F B F,, σ- σ-field σ() σ-field F σ-field (X, F) (measurable space), F (measurable set) {,X}, {,, c,x} ( X ), 2 X ( ) σ-field ( ). σ-., F σ-, B, F F X, B, \ B, B := ( \ B) (B \ ),,. lim = lim sup :=, lim = lim if := F. N N N N (lim = if sup, lim = sup if.) [ ] (X, F) µ = µ(dx) :F [0, ] (measure) () µ( ) =0 (2), 2, F (disjoit, i.e., i j = if i j) µ( )= µ( ) (σ- ) (X, F,µ) (measure space) [ ] X O σ-field σ(o) Borel field B = B(X) X = R B = B(R ) Borel field 2. X = R, C B = σ(c) (X, F,µ) measure space 3., B, F (a) B µ() µ(b) ( ) (b) B,µ() < = µ(b \ ) =µ(b) µ() (c) µ( ) µ( ) (σ- ) (d) = µ( ) = lim µ( ) (e), N ; µ( N ) < = µ( ) = lim µ( ) (f) µ(lim if ) lim if µ( ) (g) µ( ) < = µ(lim sup )=0(Borel-Catelli ). [ ] (X, F,µ) µ(x) < (fiite measure) µ(x) = (probability measure) µ(x) =, 2, F; µ( ) < ( ) = X µ σ- (σ-fiite measure) [ ] µ ( ) (X, F,µ) (Ω, F,P) ω Ω, P = P (dω) F (evet)

3 Lebesgue Itegral 3 2 (Examples of Measure Spaces) (X, F,µ). Lebesgue (Lebesgue measure) X = R, F = B = k= (a k,b k ]( a k b k ) µ() = k= (b k a k ) µ Lebesgue dx m = m(dx) 2 (coutig measure) X F =2 X µ() = ( ) if is a fiite set, µ() = if otherwise 3 δ- (Dirac measure) X F =2 X x X µ() =ifx, µ() =0ifx/ X µ = δ x 4 (discrete measure) X = {x } F =2 X µ = p δ x (p > 0) (Ω, F,P) p k = P ({k}) ( P = k p kδ k ) 5 Ω={0,, 2,,}, 0 <p< ( ) p k = p k ( p) k,k=0,, 2,,. k 6 Poisso Ω={0,, 2, },λ>0 7 Ω=N, 0 <p< p k = e λ λk,k=0,, 2,. k! 8 Ω=(a, b), F = B (a, b) <a<b< 9 Cauchy Ω=R, F = B P () = m R,a>0 p k = p( p) k,k=, 2,. P () = /(b a). 0 N(m, v) Ω=R, F = B P () = exp 2πv a π a 2 +(x m) dx. 2 [ ] (x m)2 dx. 2v m R ( ),v >0( ) d N(m, v) Ω=R d, F = B d [ det Q P () = exp ] (2π) d/2 2 (x m)q t (x m) dx. m R d, v : d d,q= v. P (Ω) =.

4 Lebesgue Itegral 4 3 (Measurable Fuctios) [ ] (X, F), R = R {+, }. f : X R, F- ( ) {f a} := {x X : f(x) a} F ( a R). (X, F) =(R, B) f : R R Borel- ( ), Borel.. f : X R () f F- (2) {f >a} F ( a R) (3) {f a} F ( a R) (4) {f <a} F ( a R) 2. [ a R] [ a Q]. 3. f,g, f ( =, 2, ) F-, ( ) α R. () αf (2) f + g (3) fg (4) /f (5) f (6) sup f (7) if f (8) lim sup f (9) lim if f (0) lim f 4. f,g F-, f,f g, f g ( ). f g(x) = max{f(x),g(x)}, f g(x) = mi{f(x),g(x)}. 5. Borel. [ ] () X (x) =(x ), 0(x/ ), (= I (x) ) (defiig fuctio). (2) f : X R, a,,a R X {,, }, f(x) = a k k (x) = k= a k I k (x) f (simple fuctio). [ ] f : X [0, + ] {f } f = lim f. 6.,. f (x) = 2 2 k=0 k= k 2 I [k/2,(k+)/2 )(f(x)) + 2 I [2, ](f(x)). 4 Lebesgue (Lebesgue Itegrals) (X, F,µ) f = f(x) fdµ = fdµ = f(x)µ(dx). X X [ ] f = i= a i i : fdµ = fdµ = f(x)µ(dx) := a i µ( i ). X X i= well-defied f. ( f = m m j= b j Bj a i µ( i )= b j µ(b j ). {C i,j := i B j } i=. ) j= [ ] f,g ()(2), (3). () (f + g)dµ = fdµ+ gdµ (2) α 0 αfdµ = α fdµ (3) f g fdµ gdµ.

5 Lebesgue Itegral 5 [ ] f {f } f = lim f.. fdµ := lim f dµ. [ ] well-defied, {f }. m g = b j Bj g f gdµ lim f dµ gdµ = gdµ < j=. j; b j > 0,µ(B j)=, µ(g >0) <, 0< ɛ(< b j), = {f > g + ɛ}( X) ( = {g >0}).. f : X R f + := f 0, f := (f 0). f = f + f, f = f + + f. [ ] f : X R F-. f + dµ, f dµ f fdµ := f + dµ f dµ. f + dµ, f dµ f (itegrable). 2. f. f dµ = f + dµ + f dµ (( ),.) f f: itegrable f dµ <. [ ] f, F. f f fdµ := fdµ L - (L -fuctio). { L = L (X, F,µ):= f; f (X, F,µ), i.e., (L = L (X) =L (dµ) =L (X, dµ).)., f,g. } f dµ < f = g, µ-a.e. µ(f g) =0. (a.e.=almost everywhere,. 0.) 3.. (,, (5) (6) (7) (8), (9). (0) ) () F,µ() =0= fdµ =0 (2) afdµ = a fdµ ( a R) (3) f,g L (X, F,µ)= (f + g)dµ = fdµ + gdµ (4) f g, µ-a.e. = fdµ gdµ f = g, µ-a.e. = fdµ = gdµ (5) f 0,µ-a.e. fdµ =0= f =0,µ-a.e. (6) fdµ =0 ( F)= f =0,µ-a.e.

6 Lebesgue Itegral 6 (7) f L (X, F,µ)= f <,µ-a.e. (8) g L (X, F,µ) f g, µ-a.e. = f L (X, F,µ) (9) fdµ f dµ (0) f =Ref + iimf : X C ( i = ) Re f, Imf f fdµ = Refdµ + i Imfdµ. ( ) () (3),,. (2) a 0, a =, a<0. (3) (f + g) + + f + g =(f + g) + f + + g + (4) f g (µ-a.e. ) fdµ > gdµ <, (3) f =0. f g, µ-a.e. := {f g} (), (3). (5) := {f /},. (6) := {f 0} f + =0,µ-a.e.. f. (7). 5 (Covergece Theorems) f,f,f 2, (X, F,µ) µ(lim f f) =0 f f f f, µ-a.e. (a.e. almost everywhere µ = P P -a.s. a.s. almost surely ). f f, µ-a.e. f dµ fdµ ( ) X =(0, ), F = B(0, ),µ(dx) =dx f 0( ) f dx = 2.,. (,, ) [ ] [ (Mootoe Covergece Theorem)] [0 f f 2 f f ( )], µ-a.e. fdµ = lim f dµ. ( ) f. f {f,k } k= ; lim k f,k = f g k := sup{f,k : k},g := lim k g k k f,k g k f k f k, g = f, g k [ ] [Fatou (Fatou s Lemma)] f 0,µ-a.e. ( =, 2, ) lim if f dµ lim if f dµ. ( ) g := if{f k : k } [ ] [Lebesgue (Lebesgue s Covergece Theorem)] h L (X, F,µ) [ f h ( =, 2, ) f f ( )],µ-a.e. f L (X, F,µ)

7 Lebesgue Itegral 7 fdµ = lim f dµ. Lebesgue (Domiated Covergece Theorem) µ(x) < h Lebesgue (Bouded Covergece Theorem) ( ) h f h, µ-a.e. f + h h f Fatou s lemma 3. µ(dx): a fiite measure o (R, B) i = lim e ix/ µ(dx) =µ(r) R 4. f L ([0, ), B([0, )),dx) t 0 L(t) := e tx f(x)dx ( ) t 0, {t } [0, ); t t L(t ) L(t) 5. xf(x) L ([0, ), B([0, )),dx) L(t) t>0 C ( ) x 0, 0 < h <t/2= e (t+h)x e tx h = x h e (t+s)x ds h 0 xe tx/2 x. 6. µ(dx): a fiite measure o (R, B), i = z R F (z) = e izx µ(dx) [0, ) x L (R, B,µ) C [ ] [ (bsolute Cotiuity)] f L (X, F,µ) µ() 0= fdµ 0. ( ) MCT { f f := f dµ = lim fdµ f dµ = ( f f )dµ + ( f ) ( f >) f dµ f dµ X ( f f )dµ + µ(). Riema Lebesgue ( ) [ ] [a, b] f Riema Lebesgue b f(x)dx = a [a,b] f(x)m(dx). Riema Lebesgue m Lebesgue ( ) [a, b] 2 Riema (Darboux ) g,h g, h h f g

8 Lebesgue Itegral 8 Lebesgue f Riema h = f = g, m- a.e. Lebesgue f [ ] X M [ M M] [ M M] ( ) m() ( σ-field ) [ ] [ (Mootoe Class Theorem)] X m() =σ(). ( ) m() m() M c := { X; c m()} M c m(). m() c m(). m() M := {B X; B m()} M m(), B m() B m() M m() M. m() B, B B m() B M M m() M 6 (Product Measure Spaces) Lebesgue 3 ( Fatou,,,,,..). f 0,µ-a.e. ( =, 2, ) f dµ = f dµ 2. f dµ < f f dµ = f dµ. 3. f L (X, F,µ) { } F fdµ = fdµ. (X j, F j,µ j )(j =,,) ( σ-.) [ ] () X X = ( j F j,j =,,) (2) F F := σ(x X ) (X X, F F ) (product measurable space) [ ] µ j,j =,, σ- (X X, F F ) µ : = ( j F j,j =,,) µ() =µ ( ) µ ( ). [ ] µ (product measure) µ = µ µ (X X, F F,µ µ ) (product measure space) 4. (F F 2 ) F 3 = F F 2 F 3 ( σ(σ(f F 2 ) F 3 )=σ(f F 2 F 3 )) =2

9 Lebesgue Itegral 9 (X, F,µ), (Y, G,ν) σ- F G x := {y Y :(x, y) } (x X), y := {x X :(x, y) } (y Y ) 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- [ ] µ, ν σ- F G () x X x G x ν( x ) F- (2) y Y y F y µ( y ) G- (3) ν( x )µ(dx) = µ( y )ν(dy). 5. σ- (R, B,µ) Lebesgue (R, 2 R,ν) = {(a, a) :a R} ν( x )µ(dx) µ( y )ν(dy) R R 6. () (2) σ() =F G. (3). ( ) µ, ν M M m() = σ() =F G µ, ν σ- X F X, Y G Y ; µ(x ),ν(y ) < X Y ( ) F G µ ν() = ν( x )µ(dx) = µ( y )ν(dy) (X Y, F G) σ- 7. µ, ν (X, F) σ- F = σ() µ = ν F µ = ν [ ] [Fubii ] (X, F,µ), (Y, G,ν) σ- f F G- f(x, y) x- f x, y- f y G- F- () f 0,µ ν-a.e. x f x dν F-, y f y dµ G- fdµ ν = dµ fdν = dν fdµ. (2) f f dµ ν, dµ f dν, dν f dµ

10 Lebesgue Itegral 0 fdµ ν = dµ fdν = dν fdµ. ( ) f(x, y) = (x, y) ( F G) 8. {a ij } Fubii a ij < + i= j= a ij < + = a ij = a ij. j= i= i= j= j= i= 9., B F G x X ν( x )=ν(b x ) µ ν() =µ ν(b) 0. f L (X, F,µ),g L (Y, G,ν) fg L (X Y, F G,µ ν) fgdµ ν = fdµ gdν X Y. f(x, y) = x2 y 2 (x 2 + y 2 ) 2 0 dy 0 X f(x, y)dx = π 4, dx Fubii 0 Y 0 f(x, y)dy = π 4

11 Lebesgue Itegral 7 L p - (L p -spaces, Covergece Notio) (X, F,µ) M(X, F,µ) [ ] L p = L p (X, F,µ), p : L p - () p< ( L p = L p (X, F,µ):={f M(X, F,µ): f p < } ( f p := f p dµ) /p ) f L p p L p - (2) p = L = L (X, F,µ):={f M(X, F,µ): f < }, ( f = ess.sup f := if{α : f α, µ a.e.}: f ) f L L - ( f f <,µ-a.e. ) (3) p L p - (orm) L p [ ] p, q, /p +/q =.,,. () [Hölder ] f L p,g L q fg L fg f p g q. (2) [Mikowski ] f,g L p, f + g p f p + g p. ( ) p =, <p< () fg =0,µ-a.e. = {fg 0} µ() > 0 log a, b > 0 p log a + ( a log b log q p + b ), i.e., a /p b /q a q p + b q a = f p / f p dµ, b = g q / g q dµ (2) q = p/(p ), i.e., /p +/q = f + g p dµ f f + g p dµ + g f + g p dµ f + g p L q Ho lder /q =/p. [ ] (L p, p )( p ) µ-a.e. p Baach ( ) p =2 f,g = fgdµ L 2 (L 2,, ) Hilbert ( ) f,g = fgdµ ( f g f = g, µ-a.e. f L p [f ] L p / [f] p = f p (L p /, p) Baach (L p, p) Baach ) X K = R or C : X [0, ] x, y X,a K () x =0 x = 0, (2) ax = a x, (3) x + y x + y. (X, ) X {x } Cauchy lim x x m =0. m, X Cauchy {x }, i.e., x X; x x 0, X (complete) (X, ) Baach, X (ier product), : X X K; () x, x 0, = 0 x = 0, (2) x, y = y, x, (3) x, ay + z = a x, y + x, z.

12 Lebesgue Itegral 2 (X,, ) x = x, x Hilbert 2. Cauchy 3. (X, ) complete {x } X; x <, x X ( ) ( ) {x } Cauchy {x j }; x j x j+ < 2 j x j ( ) {f } L p ; f < f := f, µ-a.e. f L p f k = lim f lim f p = f p <. p p k= k= f <, µ-a.e., f = f µ-a.e. f p = f f < p p f L p 4. µ(x) <, p<q, L p (X, F,µ) L q (X, F,µ). 5. f,g R f g f g; f g(x) := f(x y)g(y)dy () f,g L f g f g, (2) f L,g L 2 f g 2 f g 2 [ ] () p<. f,f L p (X, F,µ) f f p or L p - ; f f i L p lim f f p =0. (2) f,f M(X, F,µ). f f µ- ; f f i µ ɛ >0, lim µ( f f ɛ) =0. (, µ = P i µ i P i pr. ) 6. µ = 7. f,f M(X, F,µ). ɛ>0 p<, µ( f f ɛ) ɛ p f f p dµ., L p - = L ( ) 8. [0, ] Lebesgue ( ) [ ] () p L p - (2) f f i µ ( )= {f k } {f }; f k f, µ-a.e. (k ). ( ) (2) µ({ f k f /2 k }) < /2 k Borel-Catelli ( -3 (g)) µ( k ) < µ(lim sup k )=0) µ(lim f k f) =0 0. k= k= R

13 Lebesgue Itegral 3 8 (Probability Theory) /2 ( ) ( ) /2 ( ) 0 (σ-field) ( ) (Ω, F,P) (probability space) F (evet) P = P (dω) (probability measure) X = X(ω) (radom variable) (mea) (expectatio) E[X] := XdP = X(ω)P (dω) F E[X; ] :=E[X ] (variace) V (X) := E[(X EX) 2 ]=E[X 2 ] E[X] 2 [ ] () X,...,X (idepedet) a,...,a R P (X a,...,x a )=P (X a ) P (X a ). (2) X,X 2,... X,...,X [ ] X,...,X (),..., B P (X,...,X )=P (X ) P (X ). (2) Borel f,...,f. E[f (X ) f (X )] = E[f (X )] E[f (X )]. /2 ( ) X 0 ( ) E[X ]=/2 X k= [ ] [ (Law of Large Numbers)] X,X 2,... EX = m sup V (X ) < ( ) lim X k = m, a.s., i.e., P lim (X k m) =0 =. k= k=

14 Lebesgue Itegral 4 sup E[X] 4 < ( sup E[X] 4 < ) X X m m = 0, i.e., E[X ]=0 ( ) 4 X k 0 Hölder k= E[X 2 ] (E[X 4 ]) /2 ( ) 4 E X k = E[Xk]+ 4 k= k= k= i j, i,j ( ) 4 ( E ) 4 X k = 4 E X k P ( lim k= ) X k =0 = k= E[Xi 2 ]E[Xj 2 ] 2 sup E[Xk] 4 k 0 2 sup E[Xk] 4 < k (X k m) = k= X k m 0 k= ( ) ( ) ( ) ( Gauss )..,,. [ ] [ (Cetral Limit Theorem)] X,X 2,...,. EX = m R, V (X )= v (0, ). a b ( lim P a ) b (X k m) b = e x2 /(2v) dx. 2πv k= a, (X k m) 0, v N(0,v). k=

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...