Lebesgue Fubini L p Banach, Hilbert Höld

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1 II (Analysis II) Lebesgue (Applications of Lebesgue Integral Theory) 1 (Seiji HIABA) 1 ( ),,, ( )

2 Lebesgue Fubini L p Banach, Hilbert Hölder, Minkovsky Fourier L p Cc L 1 Fourier L 2 Fourier (Charateristic Functions) Fourier Lévy Schwartz

3 Applications of Lebesgue Integral Theory Lebesgue (X, F, µ) (measure space)., X, 2 X, F 2 X σ (σ-field), i.e., (1) F, (2) A F = A c F, (3) A n F(n 1) = A n F. µ = µ(dx) (measure), i.e., µ : F [0, ], ; (1) µ( ) = 0, (2) A n F : = µ( A n ) = µ(a n ). (measurable function) f : X = {± }. f a, {f a} = {x X; f(x) a} F. f, Lebesgue fdµ = fdµ = X X f(x)µ(dx)., {A k } n k=1 F X n n f = a k 1 Ak (a k ) = fdµ := a k µ(a k ). k=1 f 0, {f n }; 0 f n f, fdµ := lim f n dµ. n f n = fdµ := lim n n2 n k=1 ( n2 n k=1 k 1 2 n 1 {(k 1)/2 n f<k/2 n } + n1 {f n}, k=1 ( k 1 k 1 2 n µ 2 n f < k ) ) 2 n + nµ(f n)., f + = f 0 = max{f, 0}, f = ( f) 0 = max{ f, 0} f ± 0 f = f + f, f = f + + f, f + dµ, f dµ, Lebesgue, fdµ =. f L 1 = L 1 (X, F, µ). f + dµ f dµ f dµ <, f, Lebesgue. B n = B( n ) n Borel field, i.e., n O n σ-field σ(o n ) (= O n σ-field).

4 Applications of Lebesgue Integral Theory (B n Lebesgue (Lebesgue measure on B n )) X = n, F = B n. A = n k=1 (a k, b k ] ( a k b k ), µ(a) = n k=1 (b k a k ) µ µ B n Lebesgue, dx m = m(dx),,. 1.2 N = {N n ; B B n ; m(b) = 0, N B} (m ), (1) L n = L( n ) := {A N : A B n, N N } σ-field. (2) A N L n, i.e., A B n, N N, m(a N) := m(a), ( n, L n ). 1.2 m m, ( )Lebesgue, L n Lebesgue, Lebesgue, ( n, L n, m) Lebesgue. 1.1, Fubini 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. ) 1.3 ( (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 1.1 (Fatou (Fatou s Lemma)) f n 0, µ-a.e. ( n 1) lim inf ndµ lim inf n n f n dµ. 1.4 (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). 1.1 X F- f, f n (n = 1, 2,... ),. f n f, µ-a.e., f n 2 f, µ-a.e., sup f n dµ <, n 1 X f L 1 (X, F, µ),, fdµ = lim f n dµ. n

5 Applications of Lebesgue Integral Theory 3,. 1.5 t, f t = f t(x). t 0, f t f t0 (t t 0), µ-a.e. h L 1 (X, F, µ), U(t 0): t 0 ; f t h ( t U(t 0)), µ-a.e., f t0 L 1 (X, F, µ) Z Z f t0 dµ = lim f t t0 tdµ. 1.2 t f(t) t 0,. 1.6 f(t) f(t 0) (t t 0) {t n}; t n t 0, f(t n) f(t 0) (n ). [a, b] f iemann, Lebesgue, b a f(x)dx = [a,b] f(x)m(dx)., iemann, Lebesgue m Lebesgue. Fubini. 1.7 (X, F, µ), (Y, G, ν) σ-, i.e., {X n } F; X n = X, µ(x n ) <, ν. F G := σ(f G), (X Y, F G) µ : A B F G µ(a B) = µ(a)ν(b). 1.3 µ (product measure), µ = µ ν. (X Y, F G, µ ν) (product measure space). 1.8 (Fubini ) (1) f 0, µ ν-a.e. fd(µ ν) = X Y (X, F, µ), (Y, G, ν) σ-. : X dµ fdν = dν fdµ. Y Y X (2) f. f d(µ ν), dµ f dν, X Y X Y Y dν f dµ X,, fd(µ ν) = dµ X Y X Y fdν = Y dν fdµ. X Lebesgue. fdxdy = dx fdy = dy fdx. 2

6 Applications of Lebesgue Integral Theory Lebesgue m(dx) = dx a [ ] m(a + a) = m(a) ( A + a = A ), i.e, m(dx + a) = m(dx), [ ] m( A) = m(a) ( A = A ), i.e., m( dx) = m(dx),. f(a x)dx = f(y)dy iemann y = a x, dy = dx, i.e., dx = dy. dx = m(dx) = m( dy) = m(dy) = dy.. f(a x)dx = f(a x)m(dx) = f(y)m( dy) f(y)dy (X, F, µ) = (, B(), dx) 1 Lebesgue. f, g, (f g)(x) = f(x y)g(y)dy. f g (convolution)., f, g f, g L 1 (, B(), dx) (f g)(x)dx = f(x)dx g(y)dy ( f, g Fubini ) f L 1 ((, B(), dx). g, g f g (f g) (x) = f(y)g (x y)dy.

7 Applications of Lebesgue Integral Theory 5 2 L p 1 p, p L p. Hölder, Minkovsky, L p. (X, F, µ), M(X, F, µ). 2.1 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. (L p, p ) (1 p ) Banach ( ). p = 2 f, g = fgdµ L 2, (L 2,, ) Hilbert ( )., f, g = fgdµ. (, I,.) 2.1 Banach, Hilbert X K = or C, : X [0, ] x, y X, a K : (1) x = 0 x = 0, (2) ax = a x, (3) x + y x + y. (X, ) X {x n } Cauchy lim x n x m = 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. (X,, ). x = x, x, Hilbert. 2.1 Cauchy.

8 Applications of Lebesgue Integral Theory (X, ) complete {x n } X; x n <, n=1 n=1 x n X. ( ) ( ) {x n } Cauchy {x nj }; x nj x nj+1 < 2 j, x nj. 2.2 Hölder, Minkovsky p, 1 q, 1/p + 1/q = 1., 1,. (1) [Hölder ] fg 1 f p g q. f L p, g L q fg L 1. (2) [Minkowski ] f, g L p, f + g p f p + g p. ( ) p = 1,. 1 < p <. (1) f p = 0 or g q = 0 f = 0, µ-a.e. or g = 0, µ-a.e.,, fg = 0, µ-a.e.. f p > 0 g q > 0. a, b 0 ab ap p + bq. b q ϕ(a) = ϕ (a) = 0 a = b 1/(p 1),, ϕ(b 1/(p 1) ) = 0 ϕ(a) 0. a = f / f p, b = g / g q,. (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 Hölder, 1 1/q = 1/p. 2.3,, (L p, p ) (1 p ) µ-a.e., p, Banach ( ). 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) Banach., (L p, p) Banach. ) ( ) {f n } L p ; n f n < f := n f n, µ-a.e. f L p. n f k = lim f n n k=1 p k=1 p lim n k=1 n f n p = f n p <. k=1

9 Applications of Lebesgue Integral Theory 7 n f n <, µ-a.e., f = n f n µ-a.e.. f p = f n f n < p p n=1 f L p. 2.4 f, g. f g f g; f g(x) := f(x y)g(y)dy n=1, (1) f g 1 f 1 g 1, (2) f g 2 f 1 g µ(x) <, 1 p < q, L p (X, F, µ) L q (X, F, µ). µ(x) <, f L, lim p f p = f. f f, µ-a.e. lim sup p f p f. ε > 0, X ε := { f > f ε} µ(x ε ) > 0, f p µ(x ε ) 1/p ( f i nfty ε) (1) 1 p <. f n, f L p (X, F, µ) f n f p or L p - ; f n f in L p def lim f n f p = 0. n (2) f n, f M(X, F, µ). f n f µ- ; f n f in µ def ε > 0, lim n µ( f n f ε) = 0. f n f, µ-a.e. def µ(f n f) = 0 ( { µ ε > 0, N; n N, f n f < ε } ) c = 0 c µ { f n f < 1/k} = 0 k 1 N 1 n N µ { f n f 1/k} = 0 k 1 N 1 n N k 1, µ { f n f 1/k} = 0 N 1 n N 2.7 (1) 1 p. f n f in L p, f n g in L p, f = g, µ-a.e.. (2) f n f in µ, f n g in µ, f = g, µ-a.e µ σ- =.

10 Applications of Lebesgue Integral Theory 8 ( ) µ, f n f, µ-a.e. k 1, lim { f n f 1/k} = 0. N µ n N 2.9 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 - = L ( ) [0, 1] Lebesgue, ( ) 1, , (1) 1 p <, L p -,.. (2) µ σ-,.. (3),. f n f in µ (n ) = {f nk } {f n }; f nk f, µ-a.e. (k ). ( ) (3) µ({ f nk f 1/2 k }) < 1/2 k, Borel-Cantelli ( µ(a k ) < µ(lim sup A k ) = 0) µ(lim f nk f) = Fourier, (X, F, µ) = (, B, dx) 1 Lebesgue, 1 p <. L p L p (, B, dx). 3.1 L p Cc Cc Cc (). f supp f := {f 0}. (A A.) p <. Cc L p, i.e., f L p, {f n } Cc ; f f n p 0 (n ). ( ) f1 [ n,n], supp f., f = 1 B ; B B, B <. Lebesgue ε > 0, K;, G; ; K B G; G \ K < ε.

11 Applications of Lebesgue Integral Theory 9 φ = φ ε C c (); 0 φ 1, φ = 1 on K, φ = 0 on G c. 1 B φ p dx G \ K < ε. ε = 1/n f n = φ ε. 3.1 f L p f n = f1 [ n,n], f n f in L p. ( ) a, b 0, p 1 (a + b) p 2 p 1 (a p + b p ) ( ) Lebesgue. [Lebesgue ] B B; B <, ε > 0, K;, G; ; K B G; G\K < ε, A (a, b], σ(a) = B. B B, Lebesgue m(b), ε > 0, A n A; B [ A n, m(b) X m(a n) < m(b) + ε/4. A n A, G n A n m(g n) < m(a n) + ε/2 n+2. G := S G n. B c, F B ; m(b \ F ) < ε/4., K F, m(f \ K) < ε/4,. [ K;, G; ; K G, φ Cc (); 0 φ 1, φ = 1 on K, φ = 0 on G c ] x 1,..., x n K, r 1,..., r n > 0; K S n k=1 B(x k, r k ) S n k=1 B(x k, 2r k ) G ( ), ψ k Cc () (k = 1,..., n) 0 ψ k 1, ψ k = 1 on B(x k, r k ), ψ k > 0 on B(x k, 2r k ), ψ k = 0 on B(x k, 2r k ) c ( ). Ψ = P n k=1 ψ k, Φ = Q n k=1 (1 ψ k), Ψ = 0 on ( S n k=1 B(x k, 2r k )) c, Ψ > 0 on S n k=1 B(x k, 2r k ), Φ = 0 on S n k=1 B(x k, r k ). φ := Ψ/(Ψ + Φ). 3.2 K;, G; ; K G, x 1,..., x n K, r 1,..., r n > 0; K S n S k=1 B(x k, r k ) n k=1 B(x k, 2r k ) G < r < <, ψ C c ; ψ = 1 on B(0, r), ψ > 0 on B(0, ), ψ = 0 on B(0, ) c. f(t) = e 1/t 1 (0, ) (t) C, f(t) = 0 t 0. lim t f(t) = 1. ψ(x) = f( 2 x 2 )/(f( 2 x 2 ) + f( x 2 r 2 ) p <. f L p lim f(x + h) f(x) p dx = 0. h 0 ( ) ε > 0, g Cc ; f g p < ε/2. f h (x) := f(x + h), f h g h p = f g p., θ (0, 1); g h (x) g(x) = hg (x + θh), g h g p p = g = sup g. g(x + h) g(x) p dx h g supp g. f h f p f h g h p + g h g p + g f p 2 f g p + h g supp g 2 f g p < ε (h 0),.

12 Applications of Lebesgue Integral Theory L 1 Fourier 3.1 f L 1, Ff(z) f(z) := 1 f (Fourier trasform). 3.1 e izx f(x)dx f L 1. a, b, h(x) = f( x), f(x+a), e ibx f(x) Fourier ĥ(z) = f(z), e iaz f(z), f(z b). g L 1,. F(f g)(z) = f(z)ĝ(z). ( ),,. e izx = e iz(x y) e izy, Fubini, F(f g)(z) = dx e izx f(x y)g(y)dy = dx e iz(x y) f(x y)e izy g(y)dy = dy e izy g(y) e iz(x y) f(x y)dx (Fubini ) = dy e izy g(y) e izx f(x)dx ( ) = f(z)ĝ(z). 3.3 Gauss g t (x) := 1 t e x2 /(2t), Fg t (z) = ĝ t (z) = 1 e tz2 /2. ( ) θ C,. (3.1) e θzx g t (x)dx = e tθ2 z 2 /2 ( θ = i.) θ C.,, θ. θzx x2 2t = 1 2t (x tθz) tθ2 z 2 e θzx g t (x) = e tθ 2 z 2 /2 g t (x tθz). e θzx g t (x)dx = e tθ2 z 2 /2 g t (x tθz)dx = e tθ2 z 2 /2 g t (x)dx = e tθ2 z 2 / (3.1) θ C. e x2 dx = π, g t (x)dx = 1.

13 Applications of Lebesgue Integral Theory t > 0, e t x / Fourier p t (z) := Poisson., p t (x) = F ( ) 1 e t x t (z) = π(t 2 + z 2 ) t π(t 2 + z 2 ). ( ) p t (z) = 1 e izx 1 e t x dx = 1 e izx 1 e t x dx, 2,, p t (z) = e t x cos(zx)dx = t π(t 2 + z 2 ) ( ) 3.5 f L 1,. (1) f := sup f(z) f 1 /. z (2) f. (3) (iemann-lebesgue) lim z f(z) = 0, i.e, f C0 (). ( ) (1). f(z) e izx f(x) dx f(x) dx = f 1. (2) z, h, f(z + h) f(z) e i(z+h)x e izx f(x) dx e ihx 1 f(x) dx. e ihx 1 2 e ihx 1 0 (h 0), f L 1, Lebesgue, (3) z 0 sup f(z + h) f(z) 1 e ihx 1 f(x) dx 0 (h 0). z f(z) = e iz(x+π/z) f(x)dx = e izx f(x π/z)dx, f(z) = e izx (f(x) f(x π/z)) dx. 3.2, f(z) = 1 2 f(x) f(x π/z) dx 0 ( z ).

14 Applications of Lebesgue Integral Theory f L 1, F 1 f(x) f(x) := 1 f (Fourier inverse trasform). 3.6 e izx f(z)dz (1) [ ] f, f L 1 f L 1, F 1 (Ff) = f, a.e., i.e., f = f, a.e. (2) [ ] f, g L 1, f = ĝ f = g, µ-a.e Gauss g t (x) := 1 t e x2 /(2t), x 0 lim t 0 g t (x) = 0. f L 1, lim t 0 (f g t ) = f in L 1, ( ) lim t 0 g t (x) = 0 (x 0) α > 0, x, lim v v α e vx2 = 0. g t dx = 1, y/ t = ỹ, (f g t )(x) f(x) = (f(x y) f(x))g t (y)dy = (f(x ty) f(x))g 1 (y)dy. Fubini, 3.2 Lebesgue f g t f 1 dx f(x ty) f(x) g 1 (y) dy dy g 1 (y) f(x ty) f(x) dx 0 (t 0)., 3.2 y, f(x ty) f(x) dx 0 (t 0),, f(x ty) f(x) dx 2 f 1. Lebesgue. ( 3.6 ) Gauss g t (x) = e x2 /(2t) / t, ĝ t = g t., ĝ t (z) = e tz2 / = g 1/t (z)/ t, (3.1) [ e θzx g t (x)dx = e tθ2 z 2 /2 ] t 1/t θ = i, e izx g 1/t (z)dz = e x2 /(2t), ĝ t (x) = 1 g 1/t (x) = 1 e x2 /(2t) = g t (x). t t (f g t )(x) = (f ĝ t )(x) = f(x y) ĝ t (y)dy 1 = dyf(x y) e izy ĝ t (z)dz = dze izx 1 ĝ t (z) e iz(x y) f(x y)dy = dze izx ĝ t (z) f(z) = F 1 (ĝt f)(x),

15 Applications of Lebesgue Integral Theory 13, f g t = F 1 ( fĝt). lim t 0 (f g t ) = f in L 1, lim t 0 ĝ t (z) = 1/ Lebesgue lim t 0 F 1 ( fĝt) = F 1 f = f ( )., f = f, a.e. (2) h = f g, ĥ = f ĝ = 0, (1) h = ĥ = 0, a.e. 3.7 lim t 0 F 1 ( fĝt) = f. 3.7 f L 1. (1) xf(x) L 1 f C 1, (Ff) (z) = if(xf(x))(z). (2) f C 1, lim x f(x) = 0, f L 1 (f )(z) = iz f(z). ( ) (1), Lebesgue. h 0, e ix(z+h) e ixz h = i x h h 0 e ix(z+s) ds, e ix(z+h) e ixz h x h h 0 e ix(z+s) ds = x. (2),, 1 0. e ixz f (x)dx = [e izx f(x)] x= x= + iz e izx f(x)dx = iz f(z). Cc Fourier, ; Schwartz S = S(). f S def f C, m, n 1, lim x xm f (n) (x) = 0. f (Schwartz ) (rapidly decreasing function). C c S. f(x) = e x2 S. 3.8 f S 1 p, m, n 1, h(x) = x m f (n) (x) L p. 3.3 f S, f, f S, f = f = f. ( ) m, n 1. g(x) = x m f(x), (1), ĝ(z) = F(x m f(x))(z) = i m ( f) (m) (z). h(x) = g (n) (x) = (x m f(x)) (n) S, (2), ĥ(z) = (iz)n ĝ(z) = i m+n z n ( f) (m) (z). iemann-lebesgue ĥ(z) 0 ( z ),, zn ( f) (m) (z) 0 ( z ), f S. f S. f, f L 1, f = f = f. 3.3 L 2 Fourier L 2 L 1, Fourier. L 1 L 2 L 2,.

16 Applications of Lebesgue Integral Theory L 1 L 2 L 2. (Cc ) L 1 Fourier,. 3.4 f L 1 L 2, f, f L 2,, f 2 = f 2 = f 2. ( ) [1st Step] f, g S, f, g = fg dx, f, g = = dz g(z) dx f(x) e izx f(x)dx = e izx g(z)dz = dx f(x) e izx g(z)dz dx f(x)ğ(x) = f, ğ., f, g = f, ğ. g f f = f, f 2 = f 2. [2nd Step] f L 1 L 2, f n S; f n f in L 1, in L 2.. f, f L 2, f n f in L 2, f n f in L 2. f(z) f n (z) f f n 1 0 (n ). Fatou, f L 2. f 2 lim inf n f n 2 = lim inf n f n 2 = f 2 <. f f n 2 lim inf f m f n 2 = lim inf f m f n 2 m m f m f n 2 0 (n ). sup m n [3rd Step] f, g L 1 L 2, [2nd Step] f n, g n S, [1st Step],, f, g = f, ğ. g = f. 3.3 f L 2, f n L 1 L 2 ; f f n 2 0 (n )., f n f m 2 = f n f m 2 0 (m, n ). L 2, f n L 2 -. ( ). Ff f := lim f n in L 2 n, f (L 2 ) Fourier., f n Fourier f n, L 2 - F 1 f f := lim f (L 2 ) Fourier. n f n in L L 2 - f. [ ] g n L 1 L 2 ; f g n 2 0, ĝ = lim ĝ n in L 2. ĝ n f n 2 = g n f n 2 g n f 2 + f f n 2 0 ĝ f 2 = lim ĝ n f n 2 = 0. ĝ = f, a.e.

17 Applications of Lebesgue Integral Theory f, g L 2. (1) [Plancherel ] f, g = f, ğ. f, ĝ = f, ğ = f, g. f 2 = f 2 = f 2. (2) [ ] f = f = f, a.e. ( ) (1). (2) g L 2, f, g = f, g,, g = f f. 4 (Charateristic Functions) f L 1 = L 1 (, B, dx) Fourier F 1 f(z) f(z) = 1 e izx f(x)dx (z ), f 0 µ(dx) = f(x)dx, (, B). e izx f(x)dx = e izx µ(dx). f L 1 f ± L 1 ; f = f + f,. µ := µ/µ(),, i.e, µ() = 1 1 e izx µ(dx) = µ() n e izx µ(dx)., L 1 Fourier, Fourier. Fourier. 4.1 Fourier µ = µ(dx) ( d, B n ) ( µ (distribution) ), ϕ(z) = ϕ µ (z) µ (characteristic function). n ϕ(z) ϕ µ (z) := e i z,x µ(dx) ( z, x = z k x k ). n 4.1 µ ϕ = ϕ µ,. (1) ϕ(0) = 1, ϕ(z) 1, ϕ(z) = ϕ( z). (2) ϕ. n (3) [ ] n 1, α k C, z k (k = 1,..., n), α j α k ϕ(z j z k ) 0. k=1 j,k=1 ( ) (1). (2) L 1 Fourier. (3). n n 2 n α j α k ϕ(z j z k ) = α j α k e i(zj zk)x µ(dx) = α j e izjx µ(dx) 0. j,k=1 j,k=1 j=1 4.1 ϕ,., L 1 (dµ) = L 1 (, B, µ). (1) x L 1 (dµ) ϕ C 1, ϕ (z) = i xe izx µ(dx). (2) ϕ (0) x 2 L 1 (dµ).

18 Applications of Lebesgue Integral Theory 16 ( ) (1) L 1 Fourier. (2) h 0, ( ). (4.1) ψ h (z) = ψ h (z) := (ϕ(z + h) + ϕ(z h) 2ϕ(z))/h 2 ( ) 2 i sin(hx/2) e izx µ(dx) h/2. lim h 0 ψ h (0) = ϕ (0), Fatou, ϕ (0) = lim h 0 ( ) 2 i sin(hx/2) µ(dx) h/2 x 2 µ(dx). 4.1 (4.1) lim h 0 ψ h (0) = ϕ (0). 4.2 Lévy L 1 Fourier.,. 4.2 (Lévy ) µ ϕ,. µ((a, b)) = 1 T lim T T µ({a}) = µ({b}) = 0 e iza e izb iz µ((a, b)) = 1 T lim T T ϕ(z)dz 1 [µ({a}) + µ({b})]. 2 e iza e izb ϕ(z)dz. iz ( ) z 0, (e iza e izb )/iz (b a), Fubini T T e iza e izb ϕ(z)dz = iz z J(T, x, z, b) T J(T, x, a, b) = 2 T T µ(dx) T sin(x a)z T dz 2 z T e iz(x a) e iz(x b) dz. iz sin(x b)z dz. z, 0 sin z z dz = π 2, 0 sin zx dz = z π/2 (x > 0) 0 (x = 0) π/2 (x < 0). lim J(T, x, a, b) = T 0 (x < a or b < x) π (x = a or x = b) (a < x < b).

19 Applications of Lebesgue Integral Theory 17 π sin z, sin J(T, x, a, b) 4 0 z lim J(T, x, a, b)µ(dx) = π T. dz. Lebesgue 1 {a,b} (x)µ(dx) + 1 (a,b) (x)µ(dx). 4.3 ( ) µ, ν ϕ µ, ϕ ν, ϕ µ = ϕ ν µ = ν. ( ) (a, b); µ({a}) = µ({b}) = ν({a}) = ν({b}) = 0 I., [a, b] ; (a n, b n ) I; (a n, b n ) [a, b], µ = ν on I,, µ([a, b]) = ν([a, b]). σ({[a, b]; < a b < }) = B, µ = ν on B. 4.2 µ, µ({a}) > 0 a Schwartz 5.2

App. of Leb. Integral Theory (S. Hiraba) Lebesgue (X, F, µ) (measure space)., X, 2 X, F 2 X σ (σ-field), i.e., (1) F, (2) A F = A c F, (3)

App. of Leb. Integral Theory (S. Hiraba) Lebesgue (X, F, µ) (measure space)., X, 2 X, F 2 X σ (σ-field), i.e., (1) F, (2) A F = A c F, (3) Lebesgue (Applications of Lebesgue Integral Theory) (Seiji HIABA) 1 1 1.1 1 Lebesgue........................ 1 1.2 2 Fubini...................... 2 2 L p 5 2.1 Banach, Hilbert..............................