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.............................. 5 2.2 Hölder, Minkovsky........................ 6 2.3......................................... 6 2.4........................................ 7 3 Fourier 9 3.1 L p Cc............................... 9 3.2 L 1 Fourier................................. 1 3.3 L 2 Fourier................................. 14 4 (Characteristic Functions) 16 4.1 Fourier................................ 16 4.2 Lévy.................................... 17 4.3...................................... 18 5 (Distributions) 19 5.1..................... 23 5.2............................ 25 5.2.1..................................... 25 5.2.2................................... 27,, ( ) ( ) ( )

App. of Leb. Integral Theory (S. Hiraba) 1 1. 1.1 1 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 [, ], ; (1) µ( ) =, (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)., f,, {A k } n k=1 F X, a k n n n f = a k 1 Ak = a k I(A k ) = fdµ := a k µ(a k ). k=1 k=1, 1 A (x) = I(A)(x) = 1 if x A, = if x / A. f, {f n }; f n f, fdµ := lim f n dµ. n f n = n2 n k=1 fdµ := lim n k=1 ( k 1 k 1 2 n I 2 n f < k ) 2 n + ni(f n), ( n2 n k=1 ( k 1 k 1 2 n µ 2 n f < k ) ) 2 n + nµ(f n)., f + = f = max{f, }, f = ( f) = max{ f, } f ± 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).

App. of Leb. Integral Theory (S. Hiraba) 2 1.1 (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. 1.1. µ B n Lebesgue, dx m = m(dx),,. 1.2 N = {N n ; B B n ; m(b) =, 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.,,, Borel, ( ),,, σ-filed,. 1.1,. 1.2 2 Fubini f, f 1, f 2, (X, F, µ). µ(lim f n f) = 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)) [ f 1 f 2 f n f], µ-a.e., i.e., f n f, µ-a.e. fdµ = lim f n dµ. n 1.1 (Fatou (Fatou s Lemma)) f n, 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).

App. of Leb. Integral Theory (S. Hiraba) 3 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,. 1.5 t, f t = f t(x). t, f t f t (t t ), µ-a.e. h L 1 (X, F, µ), U(t ): t ; f t h ( t U(t )), µ-a.e., f t L 1 (X, F, µ) f t dµ = lim f t t tdµ. 1.2 t f(t) t,. f(t) f(t ) (t t ) {t n}; t n t, f(t n) f(t ) (n ). 1.6 [a, b] f iemann, Lebesgue, b f(x)dx = a [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) η : A B F G η(a B) = µ(a)ν(b). 1.3 η (product measure), µ = µ ν. (X Y, F G, µ ν) (product measure space). 1.8 (Fubini ) (1) f, 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

App. of Leb. Integral Theory (S. Hiraba) 4 Lebesgue. fdxdy = dx fdy = dy fdx. 2 1.2 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 = g f, (f g) h = f (g h)(= f g h ). 1.3. 1.4 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. g Cc, f g, (f g) (n) = f g (n). f, g Cc, (f g) (n) = f (k) g (n k) ( k n).

App. of Leb. Integral Theory (S. Hiraba) 5 2 L p 1 p, p L p. Hölder, Minkovsky, L p. (X, F, µ), f (F), f F. (, F, f F, f,.) 2.1 L p = L p (X, F, µ), 1 p : L p - (1) 1 p <, ( L p = L p (X, F, µ) := {f F : f p < } (, f p := f p dµ) 1/p ), f L p p, L p -. (2) p =, L = L (X, F, µ) := {f F : 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. 2.1 Banach, Hilbert X K = or C, : X [, ] x, y X, a K : (1) x = x =, (2) ax = a x, (3) x + y x + y. (X, ) X {x n } Cauchy lim x n x m =. m,n X Cauchy {x n }, i.e., x X; x n x, X (complete), (X, ) Banach, X (inner product), : X X K; (1) x, x, = x =, (2) x, y = y, x, (3) x, ay + z = a x, y + x, z. (X,, ). x = x, x, Hilbert. 2.1 Cauchy.

App. of Leb. Integral Theory (S. Hiraba) 6 2.2 (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 2.1 1 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 = or g q = f =, µ-a.e. or g =, µ-a.e.,, fg =, µ-a.e.. f p > g q >. a, b ab ap p + bq. b q φ(a) = φ (a) = a = b 1/(p 1),, φ(b 1/(p 1) ) = φ(a). 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,,. 2.3 2.2 (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. Minkowski,, N N f n = lim f N n lim f n p = f n p <. N p p n=1 n=1 n=1 n=1

App. of Leb. Integral Theory (S. Hiraba) 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 2. 2.5 2.6 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 >, X ε := { f > f ε} µ(x ε ) >, f p µ(x ε ) 1/p ( f ε). 2.4 2.2 (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 (2) f n, f F. f n f µ- ; lim f n f p =. n f n f in µ def ε >, lim n µ( f n f ε) =. f n f, µ-a.e. def µ(f n f) = ( { µ ε >, N; n N, f n f < ε } ) c = c µ { f n f < 1/k} = k 1 N 1 n N µ { f n f 1/k} = k 1 N 1 n N k 1, µ { f n f 1/k} = 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.. 2.8 µ σ- =.

App. of Leb. Integral Theory (S. Hiraba) 8 ( ) µ, f n f, µ-a.e. k 1, lim { f n f 1/k} =. N µ n N 2.9 f n, f F. ε > 1 p <, µ( f n f ε) 1 ε p f n f p dµ., L p - =. 2.1 2.11 L 1 -.. ( ) 2.1. [, 1] Lebesgue, ( ) 1,. 2.11.,. 2.3 (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 ) = ) µ(lim f nk f) =. 2.12.

App. of Leb. Integral Theory (S. Hiraba) 9 3 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 }. (A A.) 3.1 1 p <. Cc L p, i.e., f L p, {f n } Cc ; f f n p (n ). ( ) f1 [ n,n], supp f., f = 1 B ; B B, B <. (, f = f + f, f ±,, L p f ± < a.s., 1 f n ±, a.s. 1/2 n, supp f, p 1, L p.) Lebesgue ε >, K;, G; ; K B G; G \ K < ε. ϕ = ϕ ε C c (); ϕ 1, ϕ = 1 on K, ϕ = 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., f L 1 L 2, f n C c L 1, L 2,. [Lebesgue B B; B <, ε >, K;, G; ; K B G; G\K < ε], A (a, b], σ(a) = B. B B, Lebesgue m(b), ε >, A n A; B A n, m(b) 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 := G n. B c, F B ; m(b \ F ) < ε/4., K F, m(f \ K) < ε/4,. [ K;, G; ; K G, ϕ Cc (); ϕ 1, ϕ = 1 on K, ϕ = on G c ] x 1,..., x n K, r 1,..., r n > ; K n k=1 B(x k, r k ) n k=1 B(x k, 2r k ) G ( ), ψ k Cc () (k = 1,..., n) ψ k 1, ψ k = 1 on B(x k, r k ), ψ k > on B(x k, 2r k ), ψ k = on B(x k, 2r k ) c ( ). Ψ = n k=1 ψ k, Φ = n k=1 (1 ψ k), Ψ = on ( n k=1 B(x k, 2r k )) c, Ψ > on n k=1 B(x k, 2r k ), Φ = on n k=1 B(x k, r k ). ϕ := Ψ/(Ψ + Φ).

App. of Leb. Integral Theory (S. Hiraba) 1 3.2 K: cpt, G: open; K G, x 1,..., x n K, r 1,..., r n > ; K n k=1 B(x k, r k ) n k=1 B(x k, 2r k ) G. 3.3 < r < <, ψ C c (); ψ = 1 on B(, r), ψ > on B(, ), ψ = on B(, ) c. f(t) = e 1/t 1 (, ) (t) C, f(t) = t. lim t f(t) = 1. ψ(x) = f( 2 x 2 )/{f( 2 x 2 ) + f( x 2 r 2 )}. 3.2 1 p <. f L p lim f(x + h) f(x) p dx =. h ( ) ε >, g Cc ; f g p < ε/2. f h (x) := f(x + h), f h g h p = f g p., θ (, 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 ),. 3.2 L 1 Fourier 3.1 f L 1, Ff(z) f(z) := 1 e izx f(x)dx f (Fourier transform). 3.1 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).

App. of Leb. Integral Theory (S. Hiraba) 11 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)2 + 1 2 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 /2. 3.4 3.5 (3.1) θ C. e x2 dx = π, g t (x)dx = 1. 3.4 t >, e t x / Fourier t/(π(t 2 +z 2 )): Poisson., F ( ) 1 e t x t (z) = π(t 2 + z 2 ) ( ) = 1 e izx 1 e t x dx = 1 e izx 1 e t x dx, 1 2 2,,. = 2 1 e t x cos(zx)dx = t π(t 2 + z 2 ). 3.6. ( ) 3.5 f L 1,. (1) f := sup f(z) f 1 /. z (2) f. (3) (iemann-lebesgue) lim z f(z) =, i.e, f C ();.

App. of Leb. Integral Theory (S. Hiraba) 12 ( ) (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 (h ), f L 1, Lebesgue, (3) z sup f(z + h) f(z) 1 e ihx 1 f(x) dx (h ). z f(z) = e iz(x+π/z) f(x)dx = e izx f(x π/z)dx, 2 3 2 f(z) = e izx (f(x) f(x π/z)) dx. 3.2, f(z) = 1 2 f(x) f(x π/z) dx ( z ). 3.2 f L 1, F 1 f(x) f(x) := 1 e izx f(z)dz f (Fourier inverse transform). 3.6 (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, a.e.. 3.2 Gauss g t (x) := 1 t e x2 /(2t), x lim t g t (x) =. f L 1, lim t (f g t ) = f in L 1, ( ) lim t g t (x) = (x ) α >, x, lim v v α e vx2 =. 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.

App. of Leb. Integral Theory (S. Hiraba) 13 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 (t )., 3.2 y, f(x ty) f(x) dx (t ),, 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 /2 / = 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),, f g t = F 1 ( fĝt). lim t (f g t ) = f in L 1, lim t ĝ t (z) = 1/ Lebesgue lim t F 1 ( fĝt) = F 1 f = f (. )., f = f, a.e. ( 2.7 ) (2) h = f g, ĥ = f ĝ =, (1) h = ĥ =, a.e. 3.7 lim t 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) =, f L 1 (f )(z) = iz f(z). ( ) (1), Lebesgue. h, e ix(z+h) e ixz h = i x h h e ix(z+s) ds

App. of Leb. Integral Theory (S. Hiraba) 14, e ix(z+h) e ixz h x h h e ix(z+s) ds = x. (2),, 1. 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, lim x xm f (n) (x) =. f (Schwartz ) (rapidly decreasing function). C c S. f(x) = e x2 S. 3.8 f S 1 p, m, n, h(x) = x m f (n) (x) L p. 3.3 f S, f, f S, f = f = f. ( ) m, n. 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) ( z ),, zn ( f) (m) (z) ( 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,.,, Cc S L 1 L 2 L 1, L 2,. 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.

App. of Leb. Integral Theory (S. Hiraba) 15 [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 (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 (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 (n )., f n f m 2 = f n f m 2 (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 2 3.9 L 2 - f. [ ] g n L 1 L 2 ; f g n 2, 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 ĝ f 2 = lim ĝ n f n 2 =. ĝ = f, a.e. 3.8 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.

App. of Leb. Integral Theory (S. Hiraba) 16 4 (Characteristic Functions) f L 1 = L 1 (, B, dx) Fourier F 1 f(z) f(z) = 1 e izx f(x)dx (z ), f 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() e izx µ(dx)., L 1 Fourier, Fourier. Fourier. 4.1 Fourier µ = µ(dx) ( d, B d ) ( 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() = 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 ). 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). 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 () x 2 L 1 (dµ). ( ) (1) L 1 Fourier. (2) h, ( ). (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

App. of Leb. Integral Theory (S. Hiraba) 17. lim h ψ h () = φ (), Fatou, φ () = lim h ( ) 2 sin(hx/2) µ(dx) h/2 x 2 µ(dx). 4.1 (4.1) lim h ψ h () = φ (). 4.2 Lévy L 1 Fourier.,. 4.2 (Lévy ) µ φ, µ({a}) = µ({b}) =. µ((a, b)) = 1 T lim T T µ((a, b)) = 1 T lim T T e iza e izb iz e iza e izb φ(z)dz. iz φ(z)dz 1 [µ({a}) + µ({b})]. 2 ( ) z, (e iza e izb )/iz (b a), Fubini T T e iza e izb φ(z)dz = iz T µ(dx) T z J(T, x, z, b) T J(T, x, a, b) = 2 sin(x a)z T dz 2 z, ( ) sin z z dz = π 2, lim J(T, x, a, b) = T π sin z, sin J(T, x, a, b) 4 z lim J(T, x, a, b)µ(dx) = π T. sin zx dz = z e iz(x a) e iz(x b) dz. iz (x < a or b < x) π (x = a or x = b) (a < x < b). sin(x b)z dz. z π/2 (x > ) (x = ) π/2 (x < ). dz. Lebesgue 1 {a,b} (x)µ(dx) + 1 (a,b) (x)µ(dx).

App. of Leb. Integral Theory (S. Hiraba) 18 π sin z 4.2 J(T, x, a, b) 4 dz., z. ( T ) T sin t T ( ) dt = e tu du sin tdt = t ( ) x > T >, sin t dt = π t 2 T du e tu sin tdt, T sin xt T x dt = t sin z π z dz sin z z dz. 4.3 ( ) µ = ν. µ, ν φ µ, φ ν, φ µ = φ ν ( ) (a, b); µ({a}) = µ({b}) = ν({a}) = ν({b}) = 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. 4.3 µ, µ({a}) > a. 4.3 µ n, µ, µ n µ. µ n µ def f C b (), µ n (f) µ(f). µ(f) = fdµ.. ) 4.4 φ n, φ µ n, µ. µ n µ φ n φ ( 4.5 φ n µ n. φ; φ n φ ( ), φ µ: ; φ µ. φ n φ ( ).

App. of Leb. Integral Theory (S. Hiraba) 19 5 (Distributions),,,,.,,.,.,,,. Schwartz.,,,, Fourier,,,., D := Cc ()., ϕ supp ϕ = {ϕ } ( ),,,. f, T (ϕ) := f(x)ϕ(x)dx (ϕ D). T D, f T = T f. f C 1, f (x)ϕ(x)dx = f(x)ϕ (x)dx = T ( ϕ ), T (ϕ), T T. f C 1, T (ϕ) := T ( ϕ ) T f T. f, a, b C, f + af = b on, at f (ϕ) := a(x)f(x)ϕ(x)dx, T = T f T + at = b on D.,. [ f(x) = e a(x)dx ] e a(x)dx b(x)dx + C,.,,. (,.,..) D := Cc (). ( ϕ D,.) ϕ n ϕ in D def K : cpt; n 1, supp ϕ n K, k, ϕ (k) n ϕ (k) n ϕ (k). ϕ (k), i.e.,, ϕ n D K,,. 5.1 T : D C (Schwartz ) (distribution) def T D, i.e., T (aϕ + bψ) = at (ϕ) + bt (ψ) (a, b C, ϕ, ψ D), : ϕ n in D T (ϕ n ). D.

App. of Leb. Integral Theory (S. Hiraba) 2 K, ϕ D K. def ϕ D, supp ϕ K,, 5.1 T D T D, K :, m 1; k m, ϕ n D K ; ϕ (k) n T (ϕ n )., : K :, m 1, C > ; ϕ D K, T (ϕ) C ϕ m., ϕ m := m k= ϕ(k). [ ] ( ),,. ( ), ( ),. K: ; m 1, C >, ϕ m,c D K, T (ϕ m,c ) > C ϕ m,c m. C = m ϕ m = ϕ m,c, T (ϕ m ) > m ϕ m m. ψ m := ϕ m /(m ϕ m m ) ψ m D K T (ψ m ) > 1. k, m k ψ (k) m (x) 1 m ϕ (k) m (x) 1 (m ). ϕ m m, ψ m in D, T (ψ m ), T (ψ m ) > 1., k m, ϕ n D K ; ϕ (k) n T (ϕ n ). 5.1 f L 1 or L 2 T f D. f T f 1-1,, L 1 L 2 D., f L 2. T f (ϕ) f 2 ϕ 2 f 2 supp ϕ ϕ., T (ϕ) D., 1-1, T f =, D = Cc L 2 : dense, ϕ n D; ϕ n f L 2. = T f (ϕ n ) = f, ϕ n f, f = f 2, f = a.e.. f L 1, T f (ϕ) f 1 ϕ, T f., 1-1,,, e ixz C b ( ), Cc, Lebesgue, f =, Fourier, f = a.e.., a < b, 1 (a,b) Cc, f1 (a,b) dx =., f1 A d = A, σ-filed, 1 Borel filed B 1., A + n = {f 1/n}, = A fdx + A+ n n /n, {f > } = A + n A + n =, {f < } =,, f = a.e.. T n T in D def ϕ D, T n (ϕ) T (ϕ) 5.2.. T n T in D T n T in D. ; T (ϕ) = T (ϕ ). 5.1 (i) [Dirac δ ] T = δ δ(ϕ) = ϕ().. δ (n) (ϕ) = ( 1) n ϕ (n) (),.

App. of Leb. Integral Theory (S. Hiraba) 21 (ii) [Heaviside H(x) = 1 {x } ] T = T H. H = δ., T (ϕ) = H(x)ϕ(x)dx = T (ϕ) = ϕ(x)dx ϕ (x)dx = ϕ() = δ(ϕ), T = T f, T f, T = f., f, f. H = δ. 5.1 (i) (d/dx λ)(h(x)e λx ) =?.) (ii) sgn x (= 1 if x <, = 1 if x > ). [δ]. [2δ]. (i) T f (ϕ) = f, ϕ, T = H(x)e λx, T (ϕ) = H(x)e λx, ϕ = ( (ii) T (ϕ) = T (ϕ ) = e λx ϕ (x)dx = ϕ() + λ ϕ dx + e λx ϕ(x)dx = δ(ϕ) + λ H(x)e λx, ϕ ) ϕ dx = ( ϕ() ϕ()) = 2ϕ(). p(x) = a x n + a 1 x n 1 + + a n (a j C), D ; DT = T, S, T D. 5.3 S = p(d)t = a T (n) + a 1 T (n 1) + + a n T., T D, S D ; S = T.,., T D ; T = T. ϕ C c, Kϕ(x) C c K(ϕ ) = ϕ ( )., S S(ϕ) = T (Kϕ),., S (ϕ) = S(ϕ ) = T (K(ϕ )) = T (ϕ), i.e., S = T. Kϕ, (, ϕ (, x], ϕ, K(ϕ) Cc.,), ρ Cc ρ(x)dx = 1, x Kϕ(x) = K 1 ϕ(y)dy; K 1 ϕ(x) = ϕ(x) ρ(x) ϕ(y)dy, K 1 ϕ, Kϕ Cc, K 1 (ϕ ) = ϕ, K(ϕ ) = ϕ. (. supp K 1 ϕ = supp ϕ supp ρ ϕ (y)dy =.), ρ,, ρ = ρ δ : (Friedrichs), i.e., ρ δ, ρ δ dx = 1, supp ρ δ = [ δ, δ].,. ρ δ (x) = (C/δ) exp[ 1/(1 x 2 /δ 2 )]1 ( δ,δ)(x)., T =, T (ϕ) = T (K 1 ϕ) + T (ρ) ϕ(y)dy.

App. of Leb. Integral Theory (S. Hiraba) 22 K 1 ϕ = (Kϕ), T (K 1 ϕ) = T (Kϕ) =., T (ϕ) = T (ρ) ϕ(y)dy = T T (ρ) (ϕ)., T = T (ρ):. δ = p(d)e E D.. 5.4 p(d)t = S E; p(d)e = δ., S D, S = p(d)t T = S E. p(d)e = E + a 1 E + a 2 E = δ, z(x) z + a 1 z + a 2 z =, z() =, z () = 1, E(x) = H(x)z(x). 5.2. n,, z(x), E (x) = z()δ + H(x)z (x) = H(x)z (x) E (x) = z ()δ + H(x)z (x) = δ + H(x)z (x). n, E(x) = H(x)z(x)., S, T D (T S)(ϕ) := T y (S z (ϕ(y + z))) S z z., S z (ϕ(y + z)) y D, S,., S, supp S = ( {G ; open, ϕ D; supp ϕ G, S(ϕ) = }, supp δ (n) = {} (n ),, supp ϕ (supp S) c,, supp S supp ϕ = S(ϕ) =. ) c 5.3. [n. x x supp ϕ, ϕ (n) () =, δ (n) (ϕ) = ( 1) n ϕ (n) () =, x / supp δ. / supp δ, ϕ D, δ (n) (ϕ) =, ϕ (n) ().] T D, ϕ D, T ϕ(x) = T y (ϕ(x y)). 5.5 T D. (i) ϕ D, T ϕ C, D n (T ϕ) = T D n ϕ = D n T ϕ. (ii) T δ = δ T = T. δ a ; δ a (ϕ) = ϕ(a), δ a δ b = δ a+b. δ H = δ.

App. of Leb. Integral Theory (S. Hiraba) 23 (iii) S D; supp S, S z (ϕ( + z)) C c, T S = S T. D n T = T (D n δ), D n (T S) = D n T S = T D n S. S, T, U D 2, U (T S) = (U T ) S. (i) x,, n = 1. T ϕ(x) = T y (ϕ(x y)), ψ h (y) := (ϕ(x + h y) ϕ(x y))/h, ψ(y) := ϕ (x y). θ (, 1); ψ (k) h (y) ψ(k) (y) = 1 h (ϕ(k) (x + h y) ϕ (k) (x y)) ψ (k+1) (x y) = hϕ (k+2) (x + θh y) h ϕ (k+2)., h, k, ψ (k) h ψ (k)., T (ψ h ) T (ψ)., D(T ϕ) = T Dϕ. T y (ϕ (x y)) = T y ( x ϕ(x y)) = T y ( y ϕ(x y)) = (T ) y (ϕ(x y))., T Dϕ = DT ϕ. n. (ii) T y (δ z (ϕ(y + z))) = T y (ϕ(y)). δ y a(δ z b (ϕ(y + z))) = ϕ(a + b) = δ a+b(ϕ). (δ ) y (H z (ϕ(y + z))) = (δ ) y ( ϕ(y + z)dz) = ϕ (z)dz = ϕ() ϕ( ) = ϕ() = δ(ϕ). (iii) S z (ϕ( + z)), y; S z (ϕ(y + z)). S z, z supp S supp ϕ(y + ), y + z supp ϕ, i.e., y supp ϕ supp S., supp S z (ϕ( + z)) supp ϕ supp S: bdd., (i). T S = S T S, T, U D 2, U (T S) = (U T ) S,. D n T = T (D n δ), (i) D n δ ϕ = ϕ (n),, D n δ ϕ(x) = D n δ y (ϕ(x y)) = ( 1) n δ y ( n y ϕ(x y)) = δ y (ϕ (n) (x y)) = ϕ (n) (x),, (D n T ϕ)(x) = (T D n ϕ)(x) == (T (D n δ ϕ))(x) == ((T D n δ) ϕ)(x). (T ϕ)() = T y (ϕ( y)), x =, D n T (ϕ( )) = (T D n δ)(ϕ( )),, D n T (ϕ) = (T D n δ)(ϕ). D n (T S) = D n T S = T D n S,, D(T S) = (T S) Dδ = T (S Dδ) = T DS., T (S Dδ) = T (Dδ S) = (T Dδ) S = DT S., T = S E, p(d)t = p(d)(s E) = S p(d)e = S δ = S 5.1 ϕ S def def ϕ C, m, n, x n ϕ (m) (x) <, i.e., ϕ(x) C m.n x n. ϕ C, m, n, lim x x n ϕ (m) (x) =. Cc = D S C. S ϕ k in S def m, n, x m ϕ (n) (x). Fourier. (.) ϕ(ξ) = Fϕ(ξ) := e iξx ϕ(x)dx.

App. of Leb. Integral Theory (S. Hiraba) 24 5.6 (i) ϕ S, ϕ S,,. (ii) ϕ(x) = e ax2 (a > ), ϕ(ξ) = ( π/a)e ξ2 /(4a) (iii) ϕ S,. ϕ(x) = 1 ϕ = ψ. Fourier ( ) ˇϕ(x) = F 1 ϕ(x) = 1 e iξx ϕ(ξ)dξ ϕ(ξ)e iξx dξ., ϕ = ψ ( = 1 ϕ( x) = 1 ) ϕ( )(x), ϕ = ˇ ϕ. ϕ( x) = ϕ(x). (i). L 1 Fourier. ϕ k in S., ( iξ) m ϕ (n) (ξ) = ( 1) m F(d m x (( ix) n ϕ(x))(ξ), sup ξ m 1 ϕ(n) (ξ) 1 + x 2 (1 + x2 ) d m x (( ix) n ϕ(x) dx sup(1 + x 2 ) d m x (( ix) n dx ϕ(x)) x 1 + x 2. ϕ = ϕ k,, sup x (1 + x 2 ) d m x (( ix) n ϕ k (x)) (k )., ϕ k in S. 5.7 ϕ, ψ S, ϕψdx = ϕ ψdx, ϕψdx = 1 ϕ ψdx, ϕ ψ = ϕ ψ, 1 ϕψ = ϕ ψ 2 Parseval. ψ = ϕ ϕ 2 = ϕ 2. Fubini, ϕ(x)ψ(x)dx = dxψ(x) e ixy ϕ(y)dy = dyϕ(y) e ixy ψ(x)dx = ϕ(y) ψ(y)dy., ψ ψ,, ψ ˇψ = ψ/() 2. e ixξ ϕ ψ(x)dx = dxe ixξ ϕ(x y)ψ(y)dy = dye iyξ ψ(y) e i(x y)ξ ϕ(x y)dx, 3. ϕ ψ = F 1 ( ϕ ψ), ϕ, ψ ϕ, ψ, F 1 ( ϕ ψ) = () 2 F 1 (ϕ( )ψ( )) = ϕψ,. ; (Tempered distributions) T S : S ; C >, m, n 1; ϕ S, T (ϕ) C (1 + x ) k ϕ (l) (x). k m, l n S D. f L 1 or L 2 T f S. f T f 1-1,, L 1 L 2 S. (.) f L 1, T f (ϕ) f 1 ϕ,. f L 2,, ϕ(x) (1 + x )ϕ(x) /(1 + x ) L 2, ϕ 2 (1 + x )ϕ(x) 1/(1 + x ) 2,

App. of Leb. Integral Theory (S. Hiraba) 25 T f (ϕ) f 2 ϕ 2 C (1 + x )ϕ(x),. 1-1, D. T S, T T (ϕ) := T ( ϕ) (ϕ S). T ξ (ϕ(ξ)) := T x ( ϕ(x)). Fourier Ť Ť (ϕ) := T ( ˇϕ),,.. 5.8 (i) δ = 1, δ = iξ, δ (n) = (iξ) n (n), δ a = e iaξ (iξ) n. (ii) T S, ˇ T = Ť = T, T T S. (iii) T = T =,, f L 1, f = f = a.e. (i) δ (n) a (ϕ) = δ a (n) ( ϕ) = ( 1) n δ a ( ϕ (n) ) = ( 1) n ϕ (n) (a) = (iξ) n e iξa ϕ(ξ)dξ. (ii) Fourier, T T cals 1-1 onto., T n, ϕ S ϕ, T n (ϕ) = T n ( ϕ)., T n. f L 2 T f S, Fourier T f S., T f L 2. 5.9 (Plancherel ) f L 2 f L 2, f 2 = f 2., L 2 Fourier,. f(ϕ) = T f (ϕ) = T f ( ϕ) = f( ϕ) f 2 ϕ 2 = f 2 ϕ 2. Parseval. Cc S L 2, Cc : dense in L 2, S., f L 2, iesz, f L 2,, f 2 f 2.., ˇf = f( )/(), ˇf 2 = f 2 /(), f 2 = ˇf 2 () 3/2 ˇf 2 = f 2 f 2,. 5.1 T S, (i) T (n) = (iξ) n T, (ii) ( ix)n T = ( T ) (n), (iii) 1 = δ. 5.2 5.2.1, t, x u(t, x), : t = / t, x 2 = 2 / x 2 c t u = γ xu. 2, c, γ. u(, x) = u (x).

App. of Leb. Integral Theory (S. Hiraba) 26 5.11 ( ) c t u = γ xu, 2 u(, x) = u (x) S ( ), c u(t, ) = u v(t, ); v(t, x) = 2 4γt e cx /(4γt) v., x, t,,, (c t γ 2 x)v =, v(, x) = c t û = γξ 2 û, û(, ξ) = û (ξ). e γξ2 t/c = û(t, x) = û (ξ)e γξ2 t/c c 2 4γt F(e cx /(4γt) )(ξ),, u S, ϕ ψ = ϕ ψ, ( c û (ξ)e γξ2t/c = 4γt F(u (x) e cx 2 c /(4γt) )(ξ) = F 4γt,, c u(t, x) = 4γt u (y)e c(x y)2 /(4γt) dy = u v(t, )(x). ) u (y)e c(x y)2 /(4γt) dy (ξ), u,, u S.,, u(t, ) = u v(t, )., v, (c t γ 2 x)v =, (c t γ 2 x)u = u (c t γ 2 x)v =., t, v(t, x) δ x. f t (x) = ( t) 1 e x2 /(2t), f t (x) δ (t ) x, F t (x) := f s (y)dy, F t (x) f t (y)dy = 1., t, x > 1 F t (x) = 1 x/ t 1/ t e y2 /2 dy f 1 (y)dy = 1. x < 1 (,( 1/ t) (x/ t) (x) 1,, Lebesgue, F t (x) = 1 1 (,( 1/ t) (x/ t) (x)e y2 /2 dy (t ). F t (x) H(x). Lebesgue, F t H in S,, f t δ in S. Cauchy, t u = p( x )u, u(, x) = u (x). p(z) = a z n + a 1 z n 1 + + a n (a j C)., u = δ, E(t, x)., u D ;, u(t, x) = u (x) E(t, x)., t u = u t E = u p( x )E = p( x )(u E) = p( x ), u(, x) = u E(, x) = (u δ)(x) = u (x).

App. of Leb. Integral Theory (S. Hiraba) 27 5.2.2 t. 5.12 ( ) t 2 u(t, x) = xu(t, 2 x), u(, x) = u (x), t u(, x) = u 1 (x). u, u 1 L 1, u(t, x) = 1 2 [u (x + t) + u (x t)] + 1 2. d Alembert. x+t x t u 1 (y)dy u =, u = δ E(t, x),, E(t, x) = I( x < t)/2., t >, x E = 1 2 (δ t δ t ), 2 xe = 1 2 (δ t δ t)., t E = 1 2 (δ t + δ t ), 2 t E = 1 2 (δ t + δ t). E., u 1, u = u 1 E, u 1 L 1, u(t, x) = v 1 (t, x) := 1 2 u L 1, v(t, x) := 1 2 x+t x t t t u 1 (x y)dy = 1 2 x+t x t u 1 (y)dy. u (y)dy, v (t, x) := t v(t, x) := 1 2 [u (x + t) + u (x t)], v u(, x) =, t u(, x) = u (x), ( t 2 x)v 2 = t ( t 2 x)v 2 = v (, x) = t v(, x) = u (x), t v (, x) = [ x u (x) x u (x)]/2 =, u = v + v 1.