Lebesgue Riemann Lebesgue Lebesgue Canto

Transcription

1 Lebesgue Riemann Lebesgue Lebesgue Cantor Cantor Besicovitch

2 Fubini Lebesgue Banach Hilbert Lebesgue Borel Lebesgue 1.1 Lebesgue Lebesgue Fourier Lebesgue Fourier Lebesgue X Lebesgue Lebesgue 2 1 Lebesgue, Henri (né au 28 juin, 1875, et décédé au 27 juillet, 1941). 2 H. Lebesgue: ntégrale, longueur, aire 24(23) Lebesgue 2

3 Lebesgue Lebesgue Lebesgue Lebesgue 3 Lebesgue 1.2 Riemann [, 1] f(t) f(t) 1 f(t) dt = { (t, u) ; t 1, u f(t) } (1) 1 f(t) dt = lim f(σ n ) δ n (2) n {t, t 1,, t N } = t < t 1 < < t N = 1 δ n = t n+1 t n σ n t n σ n t n+1 = max t n+1 t n n (2) σ n (2)

4 (2) Riemann Riemann Riemann 1.1 k = 1, 2, d k,n = n 2 k, n =, 1,, 2k, 2 k 2 k k k = 2 k k k d k,n σ k,n, σ k,n d k,n+1 D k = n f(σ k,n ) 1 2 k n f(σ k,n) 1 2 k 1 2 k = n= 1 ( f(σk,n 2 k ) f(σ k,n) ) f(t) Lipschitz L > f(t) f(s) L t s, t, s 1, (3) σ k,n σ k,n 2 k 2 k 1 D k L n= 1 2 2k = L, k, 2k k k σ k,n = n S 2 k k k S k S k S k+1 = 2 k 1 n= S k S k+1 L { ( ) 1 2n 2 k+1 f 2 k+1 f 2 k 1 n= 1 2 2k+2 = L 2 k+2 ( )} 2n k+1 S k = 4 k (S l S l 1 ) + S 1 l=2 1 f(t) dt Lipschitz c 2r > ( r + c, c + r) f(t) osc r (c, f) osc r (c, f) = sup f(s) f(s ) s,s ( r+c,c+r) 4 k 4

5 r > r > ( r + c, c + r ) ( r + c, c + r) osc r (c, f) osc r (c, f) f(t) c osc(c, f) = lim r osc r (c, f) f(t) Lipschitz osc r (f, c) 2rL osc(c, f) = osc(c, f) = c f(t) f(t) f(t) osc(c, f) > c f(t) (f) = { c [, 1] ; osc(c, f) > } [, 1] f(t) Riemann [, 1] f(t) Riemann f(t) [, 1] (f) Lebesgue Lebesgue (1) (2) f(t) = [, 1] f(t) u f(t) > u t + (u, f) = { t ; f(t) > u, } t + (u, f) f(t) u 1.2 f(t) + (u, f) f(t) = t 2, 1 u + (u, f) = ( u, 1], u < 1, u 5 B. Riemann : Gesammelte mathematische Werke, wissenschaflicher Nachlass und Nachträge (ed. R. Narasimhan) Springer-Verlag. 6 H. Lebesgue: Leçons sur l intégration et la recherche des fonctions primitives Gauthier-Villars 7 5

6 + (u, f) (a j, b j ) + (u, f) = j (a j, b j ) + (u, f) + (u, f) = (b j a j ) + (u,f) f(t) dt = j bj a j f(t) dt u + (u, f) = +(u,f) u dt +(u,f) f(t) dt f(t) dt f(t) dt u + (u, f) +(u,f) < v 1 < v < v 1 <, lim n ± v n = ±, (4) + (v 1, f) + (v, f) + (v 1, f) < f(t) < + n= + (v n, f) =, n= + (v n, f) = [, 1] = + (v n, f) \ + (v n+1, f) n= t + (v n, f) \ + (v n+1, f) = v n < f(t) v n v (4) v n f v (t) = v n, t + (v n, f) \ + (v n+1, f) t f v (t) f v (t) 6

7 1.2 v (4) < v 1 < v < v 1 < f v (t) f v (t) v v f v (t) f v (t) < f(t) 1.3 f(t) = t 2 t + 1/4 v N = {2 N, n =, ±1, ±2,, } N = 1, 2,, 1.1 f vn (t) f vn (t) N + (v n, f) \ + (v n+1, f) v n + (v n, f) \ + (v n+1, f) n= f(t) dt v n+1 + (v n, f) \ + (v n+1, f) n= (5) f v (t) f v (t) dt = v n + (v n, f) \ + (v n+1, f) n= {v n } f(t) f(t) {v n } (5) 1 f(t) Lebesgue Lebesgue f(t) w, t < τ w 1, τ t < τ 1 f(t) = w N, τ N 1 t < 1 1 7

8 f(t) t 1: (1) Riemann Lebesgue Riemann Lebesgue Lebesgue Riemann Riemann Lebesgue Lebesgue 1.4 Lebesgue Riemann Lebesgue [, 1] f n (t) n f(t) 1 1 lim f n (t) dt = f(t) dt (6) n f n (t) Riemann f(t) f(t) Riemann f n (t) Lebesgue f n (t) 1 f(t) Lebesgue (6) 8

9 t 2: f n (t) F (t, x) x x 1 F (t, x) dt = 1 F (t, x) dt x 1.3 f n (t) = exp( n sin πt), n = 1, 2,, t 1 < f n (t) 1 f n () = f n (1) = 1 < t < 1 lim n f n (t) = f(t) = lim f n(t) = n 1, t =, < t < 1 1, t = 1 2 Lebesgue 1 lim n e n sin(πt) dt = (7) 4.1 Riemann f n (t) f(t) [, 1] 1 1 1/m f n (t) dt = lim f n (t) dt m 1/m (n : ) 9

10 1 1/m lim f n (t) dt = n 1/m (m : ) 1 f n (t) dt = 1 1/m 1/m f n (t) dt + { 1/m } 1 f n (t) dt + f n (t) dt 1 1/m 2 m 1 lim f n (t) dt n 2 m m (7) 2 (7) Lebesgue Lebesgue {f n (t)} 1 lim lim f n (t) f m (t) dt = m n f(t) 1 lim f n (t) f(t) dt = n Fourier Cantor Cantor = [, 1] = [, 1 3 ], 1 = [ 1 3, 2 3 ], 2 = [ 2 3, 1] 2 C = (C, C 2 ), = C, 2 = C 2 8, 2 C 2 = C C 2 C 2 C = C 1 1

11 C ij = C i C j 2 N C N = C in i N 1 i 2 i 1 i N,i N 1,,i 2,i 1 {,2} C in i N 1 i 2 i 1 3 N N C in i N 1 i 1 ( i 1, i 2, {, 2}) i n 3 n, i n {, 2}, n=1 C = lim C N = C N (8) N N=1 Cantor C 9 ( ) N C N ɛ > 3 N > log ɛ log 3 log 2 Cantor C ɛ C N 1.1 [, 1] Cantor C x 1 2 x n x = 2 n, x n {, 1}. n=1 n=1 1 2 n = = 1 2 x 2 m x m = 1 x n =, n m+1, x m = x n = 1, n m + 1, x 2 x = 1 N x n = n > N 2 {x n } x y 1 3 y = y n 3 n, y n {, 1, 2}. n=1 9 {c n } C c c C N {c n } C N c C N 1 Cantor C 11

12 11 y 1 = y 1 3, y 1 = y 1 x n [, 1] x = 2 n x = n=1 n=1 2x n [, 1] 3n [, 1] Cantor C 1 1 = [a, b], a < b, 1 f(t) = ct + d 3 2 ct 1 2 ca + d, a t 2a 3 + b 3 g(t) = 1 2 c(a + b) + d, 2a 3 + b 3 x a 3 + 2b ct 1 2 cb + d, a 3 + 2b 3 t b f(t) 1 g(t) g(t) C, C 1, C 2 C f(t) C 1 f(t) C 2 f(t) c(t a), a t 2a 3 + b 3 f(t) g(t) = 1 2 c(a + b 2t), 2a 3 + b 3 t a 3 + 2b c(t b), a 3 + 2b 3 t b 1.6 b a f(t) g(t) 1 c (b a), a t b, 6 f(t) dt = 1.7 b a b ϕ(t) a ( ) a + b g(t) dt = (b a) c + d. 2 ϕ (t) g(t) dt = (ca + d) ϕ(a) + (cb + d) ϕ(b) 3 2a/3+b/3 2 c ϕ(t)dt 3 b 2 c a a/3+2b/3 ϕ(t)dt 11 X 1 3 n=1 n = = q m = 1, 2 q n =, n m + 1, 12

13 f (t) = t, t = [, 1], C in i N 1 i 2i 1 f N (t) = C in i N 1 i 2 i 1 f (t), t C in i N 1 i 2 i 1 (9) f N (t), t, t, t 1 3 f 1 (t) = 1 2, 1 3 < t t 1 2, 2 3 < t 1 ϕ(t) ( 1 f 1 (t) ϕ (t) dt = ϕ(1) 3 1/ ) 1 + ϕ(t) dt 2/3 f 2 (t) ϕ(t) ( f 2 (t) ϕ (t) dt = ϕ(1) 9 1/9 3/9 7/9 ) ϕ(t)dt 4 2/9 6/9 8/ N = 1, 2,, 1.1 f C (t) = N=1 f N (t) f N+1 (t) 1 1, t N+1 (f N (t) f N 1 (t)) + f (t) = lim N f N (t), t 1, (1) t 1 f C (t) f N (t) n=n f n+1 (t) f n (t) n+1 = n=n t N f N (t) f C (t) f C (t) Cantor devil s staircase. 2 N 13

14 1.2 f C (t) [, 1] Cantor C t C (t) = 1 f C f C (t) dt = f N (t) ϕ (t) dt = ϕ(1) 3N 2 N i 1,,i N {,2} C in i N 1 i 1 ϕ(t) dt C in i N 1 i 1 = [γ in i N 1 i 1, γ in i N 1 i 1 + 1/3 N ] ɛ N 3 N 2 N i 1,,i N {,2} C in i N 1 i 1 ϕ(t) dt = = 1 2 N ϕ(γ in i N 1 i 1 ) + ɛ N i 1,,i N {,2} ɛ N 1 3 N max t 1 ϕ (t) 1 2 N ϕ(γ in i N 1 i 1 ) max ϕ(t) t 1 i 1,,i N {,2} ϕ(t) 3 N γ C (ϕ) = lim N 2 N ϕ(t) dt i 1,,i N {,2} C in i N 1 i 1 = lim N 1 2 N ϕ(γ in i N 1 i 1 ) i 1,,i N {,2} γ C (ϕ) max t 1 ϕ(t) 1 f C (t) ϕ (t) dt = ϕ(1) γ C (ϕ) (11) 15 γ C (t) = 1 2 γ C(1) = (11) 14

15 cb + d a+b 2 c + d 3 2 ct 1 2 ca + ca + d d 3 2 ct 1 2 cb + d ct + d a 2 3 a b 1 3 a b b t 3: trisect([[c, d], [a, b]]) 1.12 p n (t) = t n, n = 1, 2,, γ C (p n ) = 1 n 1 f C (t) p n 1 (t) dt Cantor Maple 16 trisect line 3 trisect 17 F L = [[[c, d], [a, b]] [a, b] 1 f(t) = ct + d f(t) [[ 3 2 c, 1 2 ca + d], [a, 2a 3 + b 3 ]], [[, 1 2 c(a + b) + d], [2a 3 + b 3, a 3 + 2b 3 ]], [[ 3 2 c, 1 2 cb + d], [a 3 + 2b 3, b]] trisect trisect a < b f(t) line F L = [[c, d], [a, b]] t, t < a h : t ct + d, a t b, t > a 4 16 Maple Waterloo, nc. 17 Maple [a, b] Maple 15

16 c t + d a t b 4: line([[c, d], [a, b]], t) trisect line 18 trisect := proc(fl) local a, b, c, d, f, t, n, m, F, L, FF, LL; a := FL 21 ; b := FL 22 ; if b a then RETURN( bad data in abscissae ) end if ; for n to 4 do L n := 1/3 (4 n) a + 1/3 (n 1) b end do ; c := FL 11 ; d := FL 12 ; f := t c t + d ; F 1 := (1/2 f(a) 1/2 f(b))/(a L 2 ) ; F 2 := f(a) F 1 a ; F 3 := ; F 4 := 1/2 f(a) + 1/2 f(b) ; F 5 := (1/2 f(b) 1/2 f(a))/(b L 3 ) ; F 6 := f(b) F 5 b ; for m to 3 do FF m := [F 2 m 1, F 2 m ] end do ; for m to 3 do LL m := [L m, L m+1 ] end do ; seq([ff m, LL m ], m = 1..3) end proc 18 cantor fcn deriv Cantor 16

17 line := proc(fl, t) local a, b, c, d; a := FL 21 ; b := FL 22 ; c := FL 11 ; d := FL 12 ; t piecewise(t < a,, a t and t b, c t + d, b < t, ) end proc fcn (9) N t trisect [[1, ], [, 1]] N 3 N line 1 3 N f N (t) 1.13 f N (t) 1 2 N 1 fcn f 3 (t) f 4 (t) f 7 (t) 8 7 fcn := proc(n, t) local output, num, fcn, m, n; option remember; output 1 := trisect([[1, ], [, 1]]) ; num 1 := 3 ; for n from 2 to N do output n := seq(trisect(output n 1 m ), m = 1..num n 1) ; num n := nops([output n ]) end do; fn := seq(line(output N m, t), m = 1..num N ) ; t sum(fn m (t), m = 1..num N ) end proc 1 f N (t) φ (t) dt deriv 19 deriv(n, φ) 1 f N(t) φ (t) dt deriv(n, φ) = φ(1) 3N 2 N [,1]\C N 19 cantor φ(t) dt 17

18 t 5: f 3 (t) t 6: f 4 (t) 18

19 t 7: f 7 (t).5.4 y t 8: f 7 (t) 19

20 deriv := proc(n, φ) local fn, t, a, b, c, R, n, m, l, M; fn := cantor : fcn(n, t) ; M := nops(fn (t)) ; c := [seq([op(3, op(n, fn (t))), op(4, op(n, fn (t)))], n = 1..M)] ; a := [seq(subs(t =, op(1, op(1, c n1 ))), n = 1..M)] ; b := [seq( subs(t =, op(1, op(2, c n1 ))), n = 1..M)] ; for m to M do R m := student intparts (nt(c m2 diff(φ(t), t), t = a m..b m ), c m2 ) end do; simplify(sum(r l, l = 1..M)) end proc deriv op(n, X) X nops(x) n fn (t) M s 1 +s 2 + +s M s n = op(n, fn (t)) s n, t < a n s n = c n t + d n, t a n and t b n, b n < t 6 t < a n,, t a n and t b n, c n t + d n, b n < t, [op(3, op(n, fn (t))), op(4, op(n, fn (t)))] [ t a n and t b n, c n t + d n ] 2 student intparts Maple student deriv deriv(2, φ) = 9 4 1/9 φ(t) dt 9 4 1/3 2/9 φ(t) dt 9 4 7/9 2/3 φ(t) dt /9 φ(t) dt + φ(1) Ω Ω S Ω Ω 2 op(1, t a n and t b n ) op(2, t a n and t b n ) t a n, t b n 2

21 21 Γ = { D µ ; µ M } S µ M D µ ( D µ S ) (12) S Γ D µ S Ω S Cantor C (C, = [, 1]) {C in i N 1 i 2i 1 ; i 1,, i N {, 2}} Cantor {C in i 1 } = [, 1] i 1,, i N {,2} C in i N 1 i 2 i 1 = 2N 3 N, N, s log 2 3 > 1 C in i N 1 i 2i 1 s 1, s log 2 3 = 1 i 1,, i N {,2}, < s log 2 3 < (Besicovitch 1 ) 25 C ( [a, b], < a < b < + ) c = [ ρ c + c, c + ρ c ], c C, 21 µ D µ (family) µ M 22 system 23 C in 1 i 1 C 2iN 1 i 1 3 N C 24 log 2 3 = , log 3 2 = C centers Cantor 21

22 = { c } ρ c > c c C c C c c = 2ρ c R = sup ρ c < + c C 26 c, c 1, { cj } C R > 3 4 R c C ρ c C 1 = { c C ; c c } C 1 R 1 = sup c C1 ρ c ρ c1 > 3 4 R 1 c 1 C 1 R 1 ρ c1 j 1 C j = C \ ci, i= R j = sup c C j ρ c, c j C j, ρ cj > 3 4 R j, j = 2, 3,, 27 C j = c j, R j J = j 1 C j J = B = { cj ; j =,, J} B C J C cj (13) j= J < J = k > j ρ ck 4 3 ρ c j (14) 28 [c i 1 3 ρ c i, c i ρ c i ] i = 1,, J (15) 26 sup least upper bound ρ c R < R R < ρ c ρ c k > j c k C k C j ρ cj > 3 4 R j 3 4 ρ c k 22

23 29 J [c i 1 3 ρ c i, c i ρ c i ] [a R, b + R ] i=1 J i=1 2 3 ρ c i b a + 2R < + J =, i, ρ ci (13) J = c C j ρ cj < 3 4 ρ c c j 1 i= c i c C j ρ cj > 3 4 R j 3 4 ρ c 2.1 J = {c i } C J = {c i } c C ρ c = ρ c >, c C, (12) S Γ = { D ν ; ν N} N N Γ = { D ν ; ν N } S S ν N D ν Γ 2.2 C B B B 29 i < j c j ci c i c j > ρ ci x [c i 1 3 ρ c i, c i ρ c i ] [c j 1 3 ρ c j, c j ρ c j ] c i c j c i x + x c j 1 3 (ρ c i + ρ cj ) 1 3 ( )ρ c i < ρ ci 23

24 2.2 θ > 1 cj ck, 1 j < k < J, ρ cj θ ρ ck j θ > 1 cj ck, 1 j < k < J, ρ cj > θ ρ ck j J c14 cj = j = 1,, J 2.2 B (j) B B = 13 j=1 (j) B δ : B {1, 2,, 13} = { c k ; δ( ck ) = j} δ (j) B δ(i) = i, i = 1, 2,, 13, δ(14) 2.3 j k 13 i k δ(i) 1 j 13 ck+1 δ(i)=j, 1 i k ci (16) 3 j k c j c k ρ cj +ρ ck (14) θ = θ [ 1 4 ρc k + c j, c j ρc k ] [ 1 3 ρc j + c j, c j ρc j ] [ θ ρ ck + c k, c k + θ ρ ck ] j N k (15) 1 2 N kρ ck 2θ ρ ck, N k 4θ = θ 2 k 31 j 2 2 i, j ci ck, cj ck, 1 i, j < k ρ ci > θρ ck, ρ cj > θρ ck i, j < k c k ci cj c k = c i cj i < j c j ci ci cj c i, c j ρ ck > ρ cj ρ ck > ρ ci c k c j ci ci cj = c i c j > ρ ci, ρ cj > θρ ck θ + 2 = 12, θ =

25 ci, δ(i) = j, 2.4 (16) ck+1 ci i 1 j 13 j (16) 2.2, 2.3 (16) j δ(k + 1) 2.2 Besicovitch S Γ = {D ν } Γ = {D µ} D ν D µ D µ D ν Γ Γ 2.4 C [a, b], < a < b < +, C O = {O λ } {[ ρ s + s, s + ρ s ]; s S} S O O S S O λ s S O λ O λ ( 2ρ s + s, s + 2ρ s ) O λ ρ s > [ ρ s + s, s + ρ s ] O λ Heine-Borel 2.1 S [a, b], < a < b < +, S O = {O λ, λ Λ} [ ] M = {s n, n = 1, 2, } S S a, b a {s n } [a, a ] {s n } [a, b] M 1 [a, a ] M 1 M 2 S M M 1 M 2 M n s n M n, n =, 1, 2,, {s n} {s n }(= M ) s n s n+1 b a, n =, 1, 2, 2n s = n= (s n+1 s n ) + s = lim n s n Miller, G. L. Separators for sphere-packings and nearest neighbor graphs, Journal of the ACM, 44, 1 29 (1997) 25

26 34 S s S O {[ ρ s + s, s + ρ s ]; s S} B = { si, i = 1,, J} 2.1 J < + O λi si O λi { O λi ; i = 1,, J} O 2.2 S S R J = {J ν ; ν N} S J N N J J = ν N J ν ( J ν = b ν a ν, J ν = [a ν, b ν ]) ɛ > ɛ J ɛ S 2.5 S { s n ; n =, 1, 2, } N = 1, 2, J n,n = [ 2 N n + s n, s n + 2 N n ] S J N = {J n,n } J N J N = 2 N n = 2 N 1 n= N 2.5 S R 2.6 Q + = { q > ; q } 2.5 Q + {q n ; n =, 1, 2,, } q Q + q = i + 1 j + 1 (i, j =, 1, ) q 35 9 m n (m, n) (i + 1)(j + 1) = (i + 1)(j + 1) i, j q = i +1 j +1 26

27 Q + Cantor 2.7 Cantor C 2.1 Cantor 1.1 [, 1] Cantor [, 1] x 1- [, 1] x m = n=1 ξ mn 1 n, ξ mn {, 1,, 9}, m =, 1, 2,, [, 1] z = n=1 ζ n 1 n, ζ n {, 1,, 9} \ {ξ nn }, [, 1] z x m m- (ξ mn ) [, 1] 2.6 S n, n =, 1, 2,, S = n= S n 36 (1, 1) (2, 1) (3, 1) (1, 2) (2, 2) (3, 2) (1, 3) (2, 3) (1, 4) (4, 1) 9: 36 ɛ > S n J n,ɛ = {J nm} J n,ɛ < 2 n 1 ɛ 27

28 2.3 Besicovitch 2.2 Besicovitch Besicovitch c = (a, b) R 2 ρ > B ρ (c) = { (x, y) ; (x a) 2 + (y b) 2 ρ 2 } 2.2 C R 2 c C ρ c > B = { B ρc (c) ; c C } C R = sup ρ c < + c C c k C, k =, 1, 2,, J, B = { B ρk (c k ) ; k =, 1,, J }, r k = ρ ck, C B B B rj/3(c j ) B ri/3(c i ) =, j i, (17) J = c i B rj /2(c j ), i j, (18) lim r j = (19) j L B L, l = 1,, L, B l B = L l=1 B l B l J ɛ S S = n= J n,ɛ S J ɛ < ɛ J ɛ J nm n= Jn,ɛ ι : N 2 (n + m)(n + m + 1) (n, m) + n N 2 28

29 c B ρc (c) c j j 1 k= B rk (c k ) i > j r i 3 4 r j (17) (13) 29 (18) i > j i < j c i B rj (c j ) c i c j r j < r i c j B ri (c i ) (19) Maple Besicovitch C B B B l 2.8 C {ρ c } Maple ingredients ingredients centres radii centres ( r, r) N n 39 radii (.1, r) N N uniform 4 > ingredients:=module() > export uniform, centres, radii; > uniform:=proc(r::constant..constant) > local intrange, f: > intrange:=map(x->round(x*1^digits),evalf(r)): > f:=rand(intrange): > (evalf@eval(f))/1^digits: > end; 37 J < L < J n = 2 4 M.B.Monagan Maple 8 Advanced Programming Guide. 22. Waterloo Maple, nc., p.9. 29

30 > centres:=proc(n,r,n) # n::dimension > local c,i,j: > for i from 1 to n do > c[i]:=uniform(-r..r): > end do: > [seq([seq(c[i](),i=1..n)],j=1..n)]: > end proc; > radii:=proc(r,n) > local r1,j: > r1:=uniform(.1..r): > [seq(r1(),j=1..n)] > end proc; > end module: 2.9 C r > R covering covering C r base C r covering C R covers > covering:=module() > export base, covers; > base:=proc(c,r) # C:: [seq], r::scalar> > local i, N: > N:=nops(C): > [seq([c[i],r],i=1..n)] > end proc; 3

31 > covers:=proc(c,r) # C:: [seq], R::list > local N, k: > N:=nops(C): > if nops(r)<>n then RETURN( bad data in covering:-covers ) end if: > [seq([c[k],r[k]],k=1..n)]: > end proc; > end module: coversplot besicovitch pickone remains besico pickone cc cc cc remains cc pickone cc besico cc pickone remains cc c = cc g 1 =pickone(c ) c 1 =remains(c ) m = 1, 2, g m =pickone(c m 1 ) c m =remains(c m 1 ) c M = M besico [g 1, g 2,, g M ] > besicovitch:=module() > export pickone, remains, besico; > pickone:=proc(cc) # cc:=covering:-covers: > local i,rr,m,j,c1,c2: > m:=nops(cc): > RR:=max(seq(cc[i][2],i=1..m)): > c1:={}: > for j from 1 to m do > if cc[j][2]=rr then c1:=c1 union {cc[j]} > end if: 31

32 > end do: > c2:=convert(c1,list): > c2[1]: > end; > remains:=proc(cc) > local i, f, r, c, m: > f:=pickone(cc)[1]: > r:=pickone(cc)[2]: > c:={}: > m:=nops(cc): > for i from 1 to m do > if (cc[i][1][1]-f[1])^2+(cc[i][1][2]-f[2])^2>r^2 > then c:=c union {cc[i]} > end if > end do: > convert(c,list) > end proc; > besico:=proc(cc) > local g, c; > g:={}: > c:=cc: > while nops(c)> do > g:=g union {pickone(c)}: > c:=remains(c): > end do: > convert(g, list): > end proc; > end module: besico

33 2. x, y 2. 1 C > C:=ingredients:-centres(2,2.,1): 1 R > R:=ingredients:-radii(.5,1): C C.1 CB > CB:=covering:-base(C,.1): C C R CV > CV:=covering:-covers(C,R): CV Besicovitch BCV > BCV:=besicovitch:-besico(CV): CB CV BCV coversplot 2.3 C coversplot coversplot C iro hutosa 41 > coversplot:=proc(c,iro,hutosa) > local k, N, d: > N:=nops(C): > with(plottools): > for k from 1 to N do > d[k]:=circle(c[k][1],c[k][2], > color=iro,thickness=hutosa): > end do: > plots[display](seq(d[k],k=1..n)): > end proc: circle plottools 41 iro Maple coloris > coloris:=[aquamarine, black, blue, navy, coral, cyan, brown, gold, green, gray, khaki, magenta, maroon, orange, pink, plum, red, sienna, tan, turquoise, violet, wheat, white, yellow]: coral coloris[5] hutosa Maple thickness 33

34 1: 2.2 CB CV dbcv dcb dcv dbcv > dcb:=coversplot(cb,black,2): > dcv:=coversplot(cv,yellow,1): > dbcv:=coversplot(bcv,red,1): dcb dcv dbcv decomposer Besicovitch meets meetsnr meetsmax indeks decomp meets [[a, b], r] E = [E 1,, E N ] E i, i = 1,, n 1, D = E n D 42 dcb dcv dbcv 43 > plots[display](dbcv,dcb,dcv,scaling=constrained,axes=none); 34

35 E i i meets(d, n, E) meetsnr E meets(e n, n, E) E E n n meetsnr(e, n) = nops(meets(e n, n, E)) meetsmax E meetsmax(e) = max{ meetsnr(e, n) ; n = 1,, N } indeks 2.3 δ indeks(e, n) 44 L = meetsmax(e) + 1 indeks : E E n (= (E, n)) m {1,, L} indeks(e, k) = indeks(e, l) k = l E k E l decomp indeks(e k ) = n E E k decomp(e, n) > decomposer:=module() > export meets, meetsnr, meetsmax, indeks, decomp; > meets:=proc(d,n,e) # n<=nops(e) > local j, s: > s:={}: > for j from 1 to n-1 do > if (E[j][1][1]-D[1][1])^2+(E[j][1][2]-D[1][2])^2 > >(E[j][2]+D[2])^2 > then s:=s union {j} > end if: > end do: > {seq(j,j=1..n-1)} minus s: > end proc; > meetsnr:=proc(e,n) # n<=nops(e) > local D, s: 44 δ(e n) = indeks(e, n) 35

36 > D:=E[n]: > s:=meets(d,n,e): > nops(s): > end proc; > meetsmax:=proc(e) > local i, N: > N:=nops(E): > max(seq(meetsnr(e,i),i=1..n)): > end proc; > indeks:=proc(e,n) > local i,j,k,m,mm,n,t,k,l,m: > K:=nops(E): > L:=meetsmax(E)+1: > N:={seq(i,i=1..L)}: > for i in N do m[i]:=i end do: > for k from L+1 to K do > M:=meets(E[k],k,E): > MM:={seq(m[M[i]],i=1..nops(M))}: > T:=N minus MM: > m[k]:=min(seq(t[j],j=1..nops(t))): > end do: > m[n]: > end proc; > decomp:=proc(e,n) > local N,i,j,s: > if n>meetsmax(e)+1 then [] end if: > N:=nops(E): > s:={}: > for i from 1 to N do > if indeks(e,i)=n > then s:=s union {E[i]} > end if: 36

37 > end do: > convert(s,list): > end proc; > end module: Besicovitch BCV meetsmax decomp mm = meetsmax(bcv )+1 45 decomp(bcv, m) m = 1,, mm coversplot dbd[m], m = 1,, mm, 46 dbd[m] BCV dbd[m] S [a, b], < a < b < +, J = {J n } J = n= J n S = inf J (2) J S S S S S S 49 S [a, b] {[a, b]} S S b a 45 decomposer > mm:=decomposer:-meetsmax(bcv)+1: 46 > for m from 1 to mm do dbd[m]:= coversplot(decomposer:-decomp(bcv,m), coloris[m],3) end do: > plots[display](seq(dbd[m],m=1..mm), scaling=constrained,axes=none): 48 dbd[m] dcb > plots[display](dbd[1],dcb,scaling=constrained,axes=none); 49 m (S), m(s), µ (S) 37

38 11: BCV 12: BCV (1) 38

39 13: BCV (2) 14: BCV (3) 39

40 15: BCV (4) 16: BCV (5) 4

41 17: BCV (6) 18: BCV (7) 41

42 19: BCV (8) 3.1 S S = 3.1 S = [c, d] [a, b] S = d c S = [c, d] S d c S ɛ > S = { n } < S + ɛ n = [c n, d n ] J n = (c n 2 n 2 ɛ, d n + 2 n 2 ɛ) [c, d] J 2.1 J S J J S d c (d n c n + 2 n 1 ɛ) S + ɛ J n J 3.1 S = (c, d) [a, b] S = d c 3.2 S n [a, b], n =, 1, 2,, S S 1 S S 1 (21) S n S n (22) n= n= 42

43 (21) S 1 S (22) S n J n = {J nm } J = n= J n S n J = {J k } 5 n= S n J = J k = k= n= m= J nm = J n n= (22) N S n J n J n = m= J nm < S n + 2 N n S n J S n + 2 N+1 n= n= N (22) Cantor C = [, 1] S = \( 2 ) S 1 = ( 2 )\C 2 S n = C n 1 \C n S n S m =, m n, \ C = \ C = n= S n S n = 1 n= S n 3 n 1 2 n S n = 2n, n =, 1, 2,, 3n+1 n= S n = 1 \ C 1 \ C < 1 \ C J = {J n } ρ = 1 J > \ C J = n= J n 5 36 ι J ι(n,m) = J nm 43

44 \ J C x \ J ( r + x, r + x) J = r > J n 2 n 2 ρ \ C [, 1] ρ < 1 (c, d) \ J, c d, (c, d) C < d c C = [a, b] S b a [a, b] \ S S S [a, b] = S ([a, b]\s) b a S + [a, b]\s S S 3.1 S [a, b] ɛ > O ɛ S O ɛ S + ɛ F ɛ S F ɛ S ɛ 3.1 O R R x O x x O (x ɛ, x + ɛ ) O ɛ, ɛ > x 51 S S S + [a, b] \ S = b a S [a, b] [a, b ] S + [a, b ] \ S = b a S S S Lebesgue) S 52 S [a, b] \ S 51 R O 1, O 2 O 1 O 2 O α, α A, O α [ α A 52 S m(s), µ(s) 44

45 3.3 Cantor C 3.2 S 1, S 2 [a, b] S 1 S 2 = S 1 S 2 S 1 S 2 = S 1 + S S 1, S 2 [a, b] S 1 S 2 S 1 S 2 2. S 1, S 2 [a, b] S 1 S 2 S 1 S 2 S 1 \ S S n [a, b], n =, 1, 2, n= S n 2. S n S m =, m n, 3.2 n= S n = S n n= S n [a, b], n =, 1, 2,, S n S n+1, n =, 1, 2,, S = S, S n = S n \ S n 1, n 1 S m S n =, n m n= S n = n= S n 3.3 S n [a, b], n =, 1, 2,, S = n= S n S = lim n S n 3.2 = [a, b] τ = a < τ 1 < < τ N = b {τ, τ 1,, τ N } τ, τ 1,, τ N 53 S 1 S 2 = S 1 S 2 S 1 + S 2 45

46 54 w 1,, w N s(t) = w i, τ i 1 t < τ i, i = 1,, N (23) (23) τ = (τ, τ 1,, τ N ) w = (w 1,, w N ) s(t) = kaidan(t, τ, w) Riemann Riemann N s(t) dt = w i (τ i τ i 1 ). (24) i=1 τ i w i w i+1 f(t) s n (t) = kaidan(t, τ n, w n ), n = 1, 2,, f(t) s n (t) s n (t) f(t) t e = \ { t ; lim n s n(t) = f(t) } lim s n(t) = f(t), t \ e, e = n e s n (t) f(t) s n (t) f(t), a.e f(t) = log(1+t) [, 1] [, 1] τ = (τ, τ 1,, τ n ) ( τ1 1 τn ) w = τ 1 + x dx,, 1 τ 1 + x dx 54 i = [τ i 1, τ i ], i = 1,, N, o (τ i 1, τ i ) i i 1,, S N N i=1 o i = i o j =, i j, = N

47 kaidan(t, τ, w) t log(1 + t) t t [, 1] Riemann 3.6 f 1 (t), f 2 (t) α, β 1 f(t) = α f 1 (t) + β f 2 (t) g(t) = f 1 (t) f 2 (t) f 2 (t) h(t) = f 1 (t)/f 2 (t) S { 1, t S c S (t) =, t S S \ S c \S (t) = 1 c S (t), t, c (t) c (t) J = [c, d] { 1, c t d c J (t) =, t < c t > d o J = (c, d) { 1, c < t < d 3.5 S 1, S 2 c o J (t) =, t c t d c S1 (t), c S2 (t) S 1 S 2 c S1 (t) c S2 (t) 3.6 S 1, S 2 c S1 (t), c S2 (t) S = S 1 S 2 c S (t) = c S1 (t) c S2 (t) S = S 1 S 2 c S (t) = c S1 (t) + c S2 (t) c S (t) 47

48 3.7 kaidan(t, τ, w) S c S (t) S 3.8 f n (t), n = 1, 2,, e t f n (t) f(t) f(t) = lim n f n(t), t e. f(t) =, t e, f(t) e f(t) g(t) [a, b] F (x) h(t) = F (g(t)) [a, b] [ ] s n (t) = kaidan(t, τ n, w n ) τ n w n lim n s n (t) = g(t) t e, e h n (t) = F (s n (t)) 3.8 F (x) t e lim n h n (t) = h(t) 3.8 [ ] F (x) s(t) = kaidan(t, τ, w) f(t) = F (s(t)) τ = (τ, τ 1,, τ N ) [a, b] w = (w 1,, w N ) f(t) [a, b] f(t) 56 F w = (F (w 1 ),, F (w N )) f(t) = kaidan(t, τ, F w) 48

49 s 2: φ(s) 57 φ(s) = s2 e s 1 + s 2 e s 3.1 f(t) = [a, b] α S α = {t ; f(t) > α} n = 1, 2, F n (t) = φ(n(f(t) α)) F n (t) { lim F 1, f(t) > α n(t) = n, f(t) α S α S α 3.1 f(t) = [a, b] β { t ; f(t) β } 3.11 f(t) = [a, b] n = 1, 2, k =, ±1, ±2, S nk = { t ; k 1 < f(t) k n n } 57 φ() = lim s φ(s) = lim s φ(s) = 1 49

50 c nk (t) n f n (t) = + k= k 1 n c nk(t) f n (t) 3.11 n f n (t) f(t) [a, b] f(x) c { x ; a x b, f(x) c } 3.13 [a, b] f(x) f(x) = [a, b] {f n (t)} f(t) ɛ > 58 α R S n (ɛ) = { t ; f n (t) f(t) ɛ } f n (t) > α t S nk, k [nα] + 2, [nα] Gauss nα 59 S nk = S pn,pk p+1 S pn,pk, p = 2, 3,, f n (t) f pn (t) 6 f(x) c { x ; f(x) c } 5

51 S n (ɛ) f n (t) f(t) S n (ɛ) ɛ > n S n (ɛ) f n (t) f(t) 3.7 f n (t) = exp( n sin πt) f(t) ɛ > S n (ɛ) = 2 π arcsin( 1 n log 1 ɛ ) f n (t) f(t) f(t), g(t) h(t) = f(t)+g(t) ɛ > S f (ɛ) = { t ; f(t) ɛ } S g (ɛ), S h (ɛ) 61 S h (ɛ) S f ( ɛ 2 ) + S g ( ɛ 2 ) 3.15 f(t), f n (t) S n (ɛ) S n (ɛ) n {f n (t)} = [a, b] f(t) n f n (t) f(t) f n (t) f(t) f n (t) f(t) {f n (t)} {f n (t)} f n (t) f(t) {f n (t)} ɛ > S nm (ɛ) S nm (ɛ) = { t ; f n (t) f m (t) ɛ } lim S nm(ɛ) = m,n f(t) f n (t) f(t) 61 S h (ɛ) S f ( ɛ 2 ) Sg( ɛ 2 )

52 3.8 = [ 1, 1] { sin 1 t f n (t) =, 2 n t 1, n = 1, 2, 3,, t < 2 n n > m 1 { f n (t) f m (t) = sin 1 t, 2 n t < 2 m, t ɛ > S nm (ɛ) < 2 m+1 2 n+1, n, m f n (t) f(t) = { sin 1 t, < t 1, t = {f n (t)} = [a, b] J n, n = 1, 2,, J n = k=1 nk, nk = [a nk, b nk ], a a nk < b nk b, n = 1, 2,, J 1 J 2 64 e = n=1 J n lim J n = n 63 f(t) g(t) ɛ > {t ; f(t) g(t) > 2ɛ } {t ; f(t) f n (t) > ɛ } {t ; f n (t) g(t) > ɛ } 64 a nk, b nk J n nk nk 52

53 J n J n = ( \ J 1 ) (J 1 \ J 2 ) (J n 1 \ J n ) J n n = \ J 1 + J 1 \ J J n 1 \ J n + J n J k \ J k+1 < + k=1 J n = (J n \ J n+1 ) (J n+1 \ J n+2 ) e J n = J k \ J k+1, n, k=n 3.17 J = k=1 k J = k=1 k 1 k, k = k \ j=1 j, k k l =, k l J k J = k= k, k l =, k l, 3.17 J ɛ > J ɛ J 66 J \ J ɛ ɛ k P P k=1 k = J < + lim N k N k P S = N k N+1 k ɛ Jɛ = N k=1 k 53

54 η > J n,η J n 2 n η J n = J n,η e n,η, e n,η = J n \ J n,η, e n,η < η 2 n. J 1,η = J 1,η, J 2,η = J 2,η J 1,η, J 3,η = J 3,η J 2,η,, J n,η = J n,η J n 1,η, J 1,η J 2,η J n,η J n,η e n=1 lim n J n,η = J 1 = J 1,η e 1,η, e 1,η = e 1,η J 2 = J 2,η e 2,η, e 2,η = (J 2,η \ J 1,η ) e 2,η e 1,η e 2,η J 2,η J 2 J 1 J n = J n,η e n,η, e n,η e n 1,η e n,η (25) J n 3.2 (25) 67 n e n,η e j,η, j=1 e n,η < η 2 J n = J n,η e n,η J n,η + e n,η J n,η < 1 2 η n J n < η η > = [a, b] {s n (t) = kaidan(t, τ n, w n )} f(t) s n (t) { s n (t), t τk n s n (t) =, t = τk n 67 n e n,η = (J n,η \ J n 1,η ) en,η 54

55 s n (t) f(t) ɛ > n = 1, 2, J n = { t [a, b] ; s p (t) s q (t) ɛ } p, q n J n 3.3 e = n=1 J n t e p, q s p (t) s q (t) ɛ s n (t) n f(t) e 3.21 J n = p, q n { t [a, b] s p (t) s q (t) ɛ } s n (t) f(t) f(t) e = { t ; s n (t) f(t) } e t e t J n p n s n (t) s p (t) < ɛ p s n (t) f(t) ɛ < 2ɛ { t ; s n (t) f(t) 2ɛ } J n e f n (t) s nk (t) 3.4 ɛ n >, η n > k κ n (ɛ n, η n k κ n e nk = { t ; f n (t) s nk (t) ɛ n } = e nk < η n s n (t) = s nκn (t) η n n=1 η n < + e = e nκn (26) i=1 n=i 55

56 3.22 (26) e 68 t e n f n (t) s n (t) ɛ n 3.8 f n (t) f(t) t e e e t e e lim s n(t) = f(t) n f(t) s n (t) s n (t) f(t) ɛ >, η > n n ν) { t ; f(t) s n (t) 1 2 ɛ } < 1 2 η n ν ɛ n < 1 2 ɛ, η n < 1 2 η n ν { t f(t) f n (t) ɛ } < η 69 f n (t) f(t) Lebesgue Banach Hilbert 3.11 f n (t) f(t) ɛ >, η > ν(ɛ, η) N n ν(ɛ, η) e n (ɛ) = { t ; f n (t) f(t) ɛ } = e n (ɛ) < η n k = ν( 1, 1 ) n 2 k 2 k 1 < n 2 < e = e k = e nk (2 n k ), i=1 k=i e k e t e f nk (t) f(t) 68 i e [ n=i e nκn X n=i η n, i, 56

57 3.12 f(t) ɛ >, η > µ(ɛ, η) N m, n µ(ɛ, η) S nm (ɛ) = { t ; f n (t) f m (t) ɛ } = S nm (ɛ) < η m k = µ(2 k, 2 k ) { e k = S mk m k+1 (2 k ) = t ; f mk (t) f mk+1 (t) 12 } k e k < 2 k e = j=1 k=j e k e { ( k=1 fnk+1 f(t) = (t) f n k (t) ) + f n1 (t), t e, t e t e j t ( fnk+1 (t) f n k (t) ) k=j k=j 1 2 k < + k=j e k f nk (t) f(t) f nk (t) f(t) ɛ >, η > λ(ɛ, η) N n k λ(ɛ, η) = ẽ nk (ɛ) < η ẽ n (ɛ) = { t ; f n (t) f(t) ɛ } ( ) ( ) 1 1 ẽ n (ɛ) ẽ nk 2 ɛ S nm 2 ɛ n, n k max { ( 1 µ 2 ɛ, 1 ) ( 1 2 ɛ, λ 2 ɛ, 1 ) } 2 η ẽ n (ɛ) < η f n (t) f(t) 57

58 4 4.1 Lebesgue f(t) = [a, b] M > f(t) M, t, f(t) f(t) dt f(t) s n (t) = kaidan(t, τ n, w n ) s n (t) s n (t) sup t f(t) Riemann f(t) dt = lim s n (t) dt (27) n f(t) 4.1 s n (t) f(t) { s n(t) dt } s n (t) M s n (t) f(t) 3.11 ɛ >, η > n n N(ɛ, η) S n (ɛ) < η, S n (ɛ) = { t ; s n (t) f(t) ɛ }, n, m N(ɛ, η) n,m (ɛ) = { t ; s n (t) s m (t) 2ɛ } S n (ɛ) S m (ɛ) 2η n,m (ɛ) 7 58

59 Riemann n, m N(ɛ, η) s n (t) dt s m (t) dt s n (t) s m (t) dt = s n (t) s m (t) dt + s n (t) s m (t) dt n,m (2ɛ) \ n,m (2ɛ) 4Mη 2ɛ 2ɛ { s n(t) dt } 4.1 f(t), g(t) (a) f(t) = g(t) f(t) dt = g(t) dt (b) { f(t) + g(t) } dt = c f(t) dt = c c (c) (d) f(t) = f(t) dt + g(t) dt, f(t) dt, f(t) dt f(t) dt f(t) dt Lebesgue 4.1 = [a, b] f n (t) f n (t) M (M ) 71 f n (t) f(t) a.e. 71 M g(t) 59

60 f n (t) dt = f(t) dt lim n f n (t) f(t) g m (t) = inf { f m (t), f m+1 (t), }, m = 1, 2,, g 1 (t) g 2 (t) g m (t) g m (t) f(t) { } g m (t) dt inf f m (t) dt, f m+1 (t) dt,, m = 1, 2,, j m {j m } f n (t) f n (t) dt n f(t) = lim n f n (t) f(t) dt = lim f n (t) dt n 4.3 f n (t) n=1 f n(t) h(t) n=1 f n (t) dt = f n (t) dt n= f n (t) n=1 f n(t) M < + 6

61 4.2 S c S (t) dt = S 4.1 f(t) f n (t) = 1 k nm k 1 n c nk(t), c nk (t) = c Snk (t), S nk = { t ; k 1 < f(t) k n n }, f n (t) f(t) f n (t) f(t) f(t) dt = lim f n (t) dt n f n (t) dt = 1 k nm k 1 n S nk f n (t), n = 1, 2,, = [a, b] f n (t) M < + g k (t) = inf n k f n(t), k = 1, 2, g k (t) M g(t) = lim k g k(t), t, g(t) {f n (t)} lim inf n f n(t) 4.1 g(t) dt = lim g k (t) dt k j k = inf n k f n (t) dt, k = 1, 2,, 61

62 j = lim k j k j {j k } lim inf n j n g k (t) dt j k = inf f n (t) dt n k k lim inf f n(t) dt lim inf f n (t) dt (28) n n 4.2 f n (t) = [a, b] (28) 72 f n (t) n f(t) f(t) = lim inf n f n(t) f n (t) M M ± f n (t) 2M M ± f(t) = lim inf n (M ± f n(t)) f(t) = lim n f n (t) (M ± f(t)) dt lim inf (M ± f n (t)) dt n M ± f(t) dt M + lim inf n (±f n(t)) dt lim inf ( f n (t)) dt f(t) dt lim inf f n (t) dt n n {r n } r k = inf n k r n lim inf n r n r k = sup n k r n lim sup n r n lim inf ( r n) = lim sup r n n n 72 Fatou 62

63 lim sup f n (t) dt f(t) dt lim inf f n (t) dt n n lim inf n r n lim sup n r n lim r n = ρ lim inf n n r n = lim sup n r n = ρ (, + ) (, + ) + e t f(t) dt s n (t) f(t) s n(t) dt f(t) f(t) dt = lim s n (t) dt n s n (t) f(t) M < +, t, 4.1 s n(t) dt 63

64 4.3 f n (t) = [a, b] f n(t) dt f n (t) f(t) f(t) dt = lim f n (t) dt n 73 f n (t) M < +, t, f(t) 4.1 f n (t) M < + M, c = sup n f n (t) dt N = 1, 2, e N,n = { t ; f n (t) > N } f n (t) e N,n e N,n+1 f n (t) n c f n (t) dt f n (t) dt N e N,n e N,n e N = n=1 e N,n e N c N e = N=1 e N t e t e N N t e N,n f n (t) N f(t) = lim n f n (t) f(t) N f n (t) f(t) f n (t) f n (t) M < f(t) 73 Beppo Levi f ± (t) = max {±f(t), } (29) 64

65 f ± (t) f(t) = f + (t) f (t) (3) f(t) = f + (t) + f (t) (31) 4.6 f(t) f(t) f(t) f(t) 4.7 f(t) f ± (t) f(t) f ± (t) f(t) dt = f + (t) dt 74 f (t) dt (32) 4.8 f(t) f(t) f(t) f(t) f(t) dt < f(t) f(t) dt f(t) dt (33) 4.4 R (, + ) f(t) R f(t) (29) (3) f(t) = f + (t) f (t) f(t) 74 65

66 f n (t) f(t) f n (t) dt R f(t) dt R f n (t) {, t > n f n (t) = f(t), t n f n (t) n n f n (t) f(t) f n (t) f n (t) f n (t) f(t) f(t) f(t) (34) f n (t) (34) f(t) dt = lim f n (t) dt R n R f(t) (29) f ± (t) f(t) = f + (t) + f (t) R a R t t R = t a R R f(t) f a (t) = f(t a) (35) f a (t) f(t) a f a (t) f(t) 4.3 f(t) f a (t), a R, f(t) dt = f a (t) dt R R 66

67 f(t) f(t) = kaidan(t, (τ 1, τ 2 ), (w 1 ), c) S 1 Q = {(x, y); a 1 x a 2, b 1 y b 2 } = [a 1, a 2 ] [b 1, b 2 ] Q Q = (a 2 a 1 ) (b 2 b 1 ) = [, 1] [, 1] 2 = 1 S [, 1] [, 1] 75 Q = { Q i ; Q i } S Q i Q Q i Q S 1 Q = Q i Q i Q S S = inf Q Q S 75 S S S [ 1, 1] [1, 11] 67

68 5.2 1 {(x, x); x 1} S 2 = [, 1] [, 1] S + 2 \ S = 2 = 1 S S S S 1 c 1, c 2,, c N Q i, i = 1, 2,, N Q i Q j, i j, Q = [a 1, a 2 ] [b 1, b 2 ] Q = (a 1, a 2 ) (b 1, b 2 ) Q i Q j Q i Q j = s(x, y) { c i, (x, y) s(x, y) = Q i, s(x, y) dxdy = N c i Q i Riemann i= Fubini S 2 1 a, b Y a (S) = { y ; (a, y) S}, X b (S) = { x ; (x, b) S} 1 S 2 Y a (S) X b (S) 1 68

69 y 5 y 4 b y 3 y 2 y 1 x 1 x 2 a x 3 x 1 21: S 5.1 S 2 1 N a N Y a (S) 1 S ɛ > S Q = Q(ɛ) = {Q i } Q(ɛ) < ɛ Q a (ɛ) = {Y a (Q i ); Q i Q(ɛ)} X a (Q i ) 1 Q a (ɛ) Y a (S) 1 1 { ( n ) ξ n = x ; Y x Q i > } ɛ ξ n ɛ n { ( ) x ; Y x Q i > ɛ} ɛ i=1 S Q(4 n ) = {Q kn } P n = i=1 k=1 Q kn { x ; Y x (P n ) > 1 } 2 n 1 2 n { R m = x ; Y x (P n ) > 1 } 2 n n=m 69

70 N = m=1 R m N 1 a N a R m m Y a (P n ) 1 2 n, n m, Y a (S) f(x, y) 1 M b M x f(x, b) f(x, y) 2 ϕ n (x, y) ϕ n (x, y) f(x, y), (x, y) S S M b M X b (S) = {x; (x, b) S} ϕ n (x, b) x ϕ n (x, b) f(x, b), x X b (S), f(x, b) x Fubini 5.1 f(x, y) 2 1 N x N F (x) = f(x, y) dy f(x, y) dxdy = F (x) dx f(x, y) f(x, y) M ϕ n (x, y) ϕ n (x, y) f(x, y), (x, y) 2 \ S, S =, 2 7

71 ϕ n (x, y) M ϕ n (x, y) dxdy f(x, y) dxdy 2 dx ϕ n (x, y) dy = F n (x) dx 2 S N x N Y x (S) = {y; (x, y) S} 1 x N ϕ n (x, y) f(x, y), y Y x (S), Y x (S) =, y x N F n (x) = ϕ n (x, y) dy f(x, y) dy = F (x) F n (x) = ϕ n (x, y) dy M F n (x) dx F (x) dx 5.1 f(x, y) f(x, y) = x2 y 2 (x 2 + y 2, (x, y) (, ), ) 2 Q = [, 1] [, 1] f(x, y) n = 1, 2,, { f(x, y), n 2 (x 2 + y 2 ) 1 f n (x, y) =, n 2 (x 2 + y 2 ) < 1 71

72 f n (x, y) f(x, y) f n (x, y) dxdy log n Q Q 1 f n (x, y) dxdy 2 rdr 1/n π/4 dθ r2 cos 2θ r 4 = log n f(x, y) Fubini 1 { 1 } f(x, y) dy dx = π4 1 { 1, } f(x, y) dx dy = π f(x, y) = f(y, x) 1 x > 1 x 2 y 2 (x 2 + y 2 ) 2 dy = 1 x 1/x ( z ) dz = z x f(y) t > + F t (x, y) = 1 π f(y) dy < + t (x y) 2 + t 2 f(y) R 2 + F t (x, y) dxdy f(y) dy R 2 + F t (x, y) dxdy = f(y) dy R 2 = + + f(y) dy < + 1 t π (x y) 2 + t 2 dx F t (x, y) u(x, t) = 1 π + t (x y) 2 f(y) dy (36) + t2 < x < + 72

73 5.1 (36) u(x, t) x 76 u(x, t) x 5.2 t > u(x, t) (36) 77 lim t + u(x, t) f(x) dx = 5.1 f(x), g(x) R + f(x) dx < +, + g(x) dx < + x h(x) = + f(x y) g(y) dy (37) h(x) + h(x) dx + f(x) dx + g(x) dx (38) h(x) f g (x) 4.3 f(x y) g(y) dxdy = R 2 = f(x) dx 76 Lebesgue x n a 1 π f(x y) dx g(y) dy t (x n y) 2 + t 2 f(y) 1 πt f(y) f(y) 1 lim x n a π t (x n y) 2 + t 2 f(y) = 1 π g(y) dy 77 Z u(x, t) f(x) = {f(x ty) f(x)} dy π 1 + y2 Z + Z + u(x, t) f(x) dx 4.3 R t (a y) 2 + t 2 f(y) 1 π Z π 1 + y 2 f(x ty) f(x) dx Z y 2 f(x ty) f(x) dx dy Z π 1 + y 2 f(x) dx Lebesgue

74 f(x y) g(y) R h(x) (38) x f(x) =, g(x) = x (37) h(x) = f(x) [, a] a > h(x) 5.4 f(x) g(x) [a, b] [c, d] a < b, c < d f(x), g(x) h(x) 5.5 f(x), g(x) 5.1 x f(x) =, g(x) = h(x) f(x), g(x) 5.1) + e x h(x) dx = + e x f(x) dx + e x g(x) dx f(t) [, 1] c f (x) = 1 f(t) cos 2πxt dt (39) x n Fourier n c f (x), < x < +, δ cos 2π(x + δ)t cos 2π xt δ 2πt 2π, t 1, lim δ cos 2π(x + δ)t cos 2πxt δ = 2πt sin 2πxt t t f(t), sin 2πxt [, 1] x f δ (t) = f(t) cos 2π(x + δ)t cos 2πxt δ 74

75 f δ (t) 2π f(t), lim δ f δ (t) = 2πt f(t) Lebesgue 1 1 lim f δ (t) dt = 2π tf(t) sin 2πxt dt δ (39) c f (x) x x d 1 dx c f (x) = ( ) 1 f(t) cos2πxt dt = 2π tf(t) sin 2πxt dt x s f (x) = 1 f(t) sin 2πxt dt, < x < +, s f (x) x c(x) = 1 cos 2πxt dt, s(x) = 1 sin 2πxt dt p(t) = N k= a k t k a k Fourier c(x) s(x) e ax g(x) a > (, + ) + e ax g(x) dx < + s a g(x) Laplace G(s) = + e sx g(x) dx (4) 78 c(x), s(x) x = x R 1 t2 cos nt dt = 1/(4π 2 ) c (n) n 75

76 s > a δ s + δ > a e (s+δ)x e sx g(x) δ x e ax g(x) 1 e a g(x) lim δ e (s+δ)x e sx g(x) = x e s x g(x) δ Lebesgue s G(s) s > a d + ds G(s) = e sx x g(x) dx y (a, b) h(t, y) t t h(t, y) y u(t) h(t, y) y u(t) H(y) = h(t, y) dt H (y) = h(t, y) dt y u(t) u(x, t) (36) F t (x, y) < x < +, t > y P (t, x y) = 1 π t (x y) 2 + t 2 (41) 76

77 79 F t (x, y) = P (t, x y) f(y) f(y) P (t, x y) = π ( 1 + t ) (x y)2 t 2 P (t, x y) 2 x y P (t, x y) = 2π P (t, x y) 2 x t F t (x, y) u(x, t) x, t u(x, t) C u(x, t) t, x 2 2 u(x, t) + u(x, t) = x2 t2 ϕ(x) C ϕ(x) C { exp( 1 t ρ(t) = ), < t (42), t ρ(t) C 81 ρ(1 x 2 ) x 1 x > 1 C µ = + ρ(1 x 2 ) dx = 1 1 ρ(1 x 2 ) dx > ω(x) = 1 µ ρ(1 x2 ) (43) { x ; 1 x 1 } C ω(x) ω( x) = ω(x) 22 + ω(x) dx = 1 1 ω(x) dx = 1 (44) 79 P (t, x) Poisson 8 f(x) { x ; f(x) } supp (f) supp (f) y f(y ) y y 81 ρ(t), t >, t 77

78 y x 22: ω(x) 5.9 f(x) ω f (x) = + ω(x y) f(y) dy x 5.1 ω f (x) 5.2 x C - d n dx n ω f (x) = d n ω(x y) f(y) dy dxn ϕ(x) C ϕ(x) f(x) ϕ f (x) C - (43) ω(x) (44) + ω k (x) = 1 k ω(x ), k >, (45) k ω k (x) dx = 1 ω k (x) f(x) ω k f (x) 5.8 C 78

79 5.3 f(x) + lim k ω k f (x) f(x) dx = (46) (44) ω k f (x) f(x) = + ω k (x y){f(y) f(x)} dy + ω(y) { f(x ky) f(x) } dy ( + ) ω k f (x) f(x) dx ω(y) f(x ky) f(x) dx dy y 2 ω(y) + f(x) dx y k Lebesgue (46) C (46) f(x) ω k (x) f(x) f(x) ω k (x) 5.4 f(x) [a, b] ω k f(x) [a k, b + k] supp (f) [a, b] ω k f(x) = b a ω k (x y) f(y) dy x b x y x a, a y b, x < a k x > b + k x ω k f 79

80 6 Lebesgue f(t) L 1 () 6.1 f(t), g(t) L() 1 h(t) = a f(t) + b g(t) a, b h(t) L 1 () h(t) dt = a f(t) dt + b g(t) dt h h = a f + b g (a f + b g)(t) dt = a f(t) dt + b g(t) dt f(t), g(t) L 1 () f(t) f(t) f 1 = f(t) dt (47) f 1 f L 1 () f(t) = f 1 = (48) 6.2 f 1 L 1 () f 1 (49) a f 1 = a f 1 (5) f + g 1 f 1 + g 1 (51) 82 [a, b] (a, + ) (, 8

81 (49) f 1 = f = 1 (48) f 1 = f(t) 6.1 X ν : X R + x, y X a ν(x) = = x = (52) ν(a x) = a ν(x) (53) ν(x + y) ν(x) + ν(y) (54) L 1 () 1 (52) L 1 () 1 f 1 = f L 1 () N 1 () 6.3 f N 1 () f(t) = f(t) = f, g N 1 () 1 a f + b g N 1 () (48) (5) (51) 6.1 f, g L 1 () f g N 1 () 83 f 1 = g 1 f = (f g) + g (51) f 1 f g 1 + g 1 = g 1 g = (f g) + f g 1 f 1 L 1 () f, g,... f, g d(f, g) f, g, h L 1 () d(f, g) d(f, g) = = f = g (55) d(f, g) = d(g, f) (56) d(f, g) d(f, h) + d(h, g) (57) 83 1 g = ( 1) g, f g = f + ( 1) g 81

82 84 d(f, g) = f g 1 N 1 () N 1 () N 1 () L 1 () f g f g N 1 () (58) 85 f g f = g d(f, g) = f g 1 L 1 () L 1 ()/ f f (59) f g = g f (6) f g, g h = f h (61) (59) (6) (61) 6.1 = C 1 (f) = {g ; g f} f f C 1 (f) C 1 (f) f f f 6.2 f L 1 () C 1 (f) = { f 1 L 1 () ; f 1 f} f, g L 1 () C 1 (f) = C 1 (g) C 1 (f) C 1 (g) = C 1 (f) = C 1 (g) f g 6.3 f, f 1, g, g 1 L 1 () f f 1, g g 1 a f + b g a f 1 + b g 1 a, b = (a.e.) 82

83 6.4 f L 1 () f 1 = f 1 (49) f 1 = f f f 6.4 f g f 1 = g L 1 ()/ = {C 1 (f) ; f L 1 ()} (62) C 1 () = N 1 () 6.3 C 1 (f) C 1 (g) ac 1 (f) + bc 1 (g) C 1 (af + bg) L 1 ()/ N 1 () C 1 (f) = f L 1 ()/ 1 87 L 1 ()/ L 1 () Lebesgue L 1 ()/ 1 L 1 ()/ L 1 () f(t), f 1 (t) L 1 ()/ L 1 () L 1 () L 1 () = L 1 ()/ L 1 () L 1 () L 1 () L 1 () d(f, g) = f g 1 f n L 1 (), n = 1, 2, 3, f L 1 () f n f 1 f n f

84 {f n } f lim f n f m 1 = n, m {f n } {f n } L 1 () Cauchy {f n } L 1 () ɛ > N(ɛ) f n f m 1 < ɛ, n, m N(ɛ) (63) Cauchy N(ɛ) ɛ > ɛ = N(ɛ) < N(ɛ ) lim ɛ N(ɛ) = 6.1 L 1 () Cauchy Cauchy {f n } L 1 () f n f 1, n f L 1 () [ ] (63) {f n } δ > S nm (δ) = { t ; f n (t) f m (t) δ } lim S nm(δ) = n, m δ S nm (δ) f n (t) f m (t) dt = f n f m f(t) {f n } f 3.11 f n (t) f(t) (63) k = 1, 2, n k = N(2 k ) {f nk } f nk f f nk (t) f(t) n k f nk+1 f nk 1 < 2 k, k = 1, 2, g(t) = f nk+1 (t) f nk (t) + f n1 (t) k=1 84

85 g(x) g(t) dt f n1 (t) dt + 1 f(t) = ( fnk+1 (t) f n k (t) ) + f n1 (t) k=1 f(t) g(t) f(t) f nl (t) = ( fnk+1 (t) f n k (t) ) k=l f(t) f f nl 1 f nk+1 f nk 1 < 2 l k=l n N(2 l ) f f n 1 f f nl 1 + f nl f n 1 < 2 l f n f 6.1 lim n f n 1 = f a(x) L 1 (R) 5.1 f 1 = a, f 2 = a f 1,, f n+1 = a f n, a 1 < 1 {f n } L 1 (R) Cauchy 6.1 L 1 () L 1 () 1 Lebesgue L 1 () 6.2 Banach 6.1 ν( ) X {x n } lim n,m ν(x n x m ) = Cauchy 85

86 Cauchy {x n } x X lim n ν(x n x) = Banach 6.1 L 1 () Banach Lebesgue L 1 () f 1 = f(x) dx (f L 1 ()) 88 Banach Banach 6.2 X ν( ) Banach 6.1 X X T c > x, y X ν( T x T y ) c ν( x y) (64) < c < 1 T c T T u = u u X u, v X T T u = u T v = v (64) ν( u v ) c ν( u v ), < c < 1, ν( u v ) = u = v X x x 1 = T x x n+1 = T x n, n = 1, 2, {x n } Cauchy n 1 x n+1 x n = T x n T x n 1 (64) ν( x n+1 x n ) c ν( x n x n 1 ) ν( x n+1 x n ) c n ν( x 1 x ) n > m ν( x n x m ) n 1 k=m c k ν( x 1 x ) < cm 1 c ν( x 1 x ) 88 C 1 (f) f f 86

87 {x n } Cauchy w X lim n ν(x n w ) = T x n = x n+1 ν( T w w ) ν( T w T x n ) + ν( x n+1 w ) c ν( w x n ) + ν( x n+1 w ) T w = w w w = u 6.1 L 1 ([, 1]) T : L 1 ([, 1]) f(x) x f(t) dt L 1 ([, 1]) f, g L 1 ([, 1]) 1 T f(x) T g(x) = 1 2 T f(x) T g(x) dx = {f(t) g(t)} dt ( 1 ) f(t) g(t) dt dx f(t) g(t) dt T 2 T u(x) = x h(x) L 1 ([, 1]) T : L 1 ([, 1]) f(x) h(x) f(t) dt L 1 ([, 1]) 6.8 T : L 1 ((, + )) f(x) e 2x + x e 2x+2y f(y) dy L 1 ((, + )) f(t) f(t) 2 9 L 2 () f(t)

88 L 2 () f 2 = f(t) 2 dt (65) 6.5 f(t), g(t) L 2 () f(t) g(t) f(t) g(t) dt f 2 g 2 (66) r > f(t) g(t) 1 2r f(t) 2 + r 2 g(t) 2 f(t) g(t) dt 1 f(t) 2 dt + r g(t) 2 dt 2r 2 r > (65) (66) f(t), g(t) L 2 () a f(t) + b g(t) 2 a 2 f(t) a b f(t) g(t) + b 2 g(t) L 2 () 6.5 f(t), g(t) L 2 () f, g = f(t) g(t) dt (67) 6.7 f, g, h L 2 () a, b R f, f = f 2 2 (68) f, g = g, f (69) a f + b g, h = a f, h + b g, h (7), L 2 () 88

89 6.2 f 2 = f, f, f L 2 () (71) 6.3 X x, y σ(x, y) σ(x, x) σ(x, x) = = x = σ(x, y) = σ(y, x), σ(a x + b y, z) = a σ(x, z) + b σ(y, z) x, y, z X a, b R σ(, ) X L 2 (), 6.9 f, g L 2 () (66) f, g f 2 g 2 (68)(69)(7) f, g, h L 2 () a, b R f 2 (72) a f 2 = a f 2 (73) f + g 2 f 2 + g 2 (74) (71) (68)(69) (72)(73) (74) (68) (7) f + g 2 2 = f + g, f + g = f, f + 2 f, g + g, g (66) 2 f 2 g 2 ( f 2 + g 2 ) 2 ( f 2 + g 2 ) 2 (74) L 1 () 1 2 N 2 () = { f L 2 () ; f 2 = } 89

90 f, g L 2 () f g f g N 2 () (75) 6.3 N 1 () N 2 () (75) (58) (59)(6)(61) 6.1 N 1 () = N 2 () f, f 1, g, g 1 L 2 () f f 1 g g 1 f, g = f 1, g 1 (76) f(t) g(t) f 1 (t) g 1 (t) = (f(t) f 1 (t)) g(t) + f 1 (t) (g(t) g 1 (t)) (71) 6.3 f, f 1 L 1 () f f 1 f 2 = f 1 2 L 1 ()/ L 2 () C 2 (f) = { f 1 L 2 () ; f f 1 } L 2 () = L 2 ()/, (76) L 1 () L 1 () 6.2 L 2 () L 2 () L 2 () 2 L 1 () d 2 (f, g) = f g 2 f, g L 2 () N () = { f(t) ; f(t) =, a.e. } N () L 1 () L 2 () N 1 () = N () = N 2 () 9

91 6.2 f n L 2 () L 2 () Cauchy (63) 1 2 f L 2 () lim n f n f 2 = 6.2 L ( ) 2 L 2 () (71) L 2 () Hilbert {a k ; k = 1, 2, } {b k ; k = 1, 2, } a k 2 < +, k=1 b k 2 < + (77) k=1 a R = [, 1] L 2 () f(t) = a + {a k cos 2kπt + b k sin 2kπt} (78) k=1 (78) f n (t) = a + n {a k cos 2kπt + b k sin 2kπt}, n = 1, 2, k=1 n L 2 () f n (t) m > n f m (t) f n (t) = f m f n, f m f n = m k=n+1 m k,l=n {a k cos 2kπt + b k sin 2kπt} a k a l cos 2kπt, cos 2lπt m k,l=n+1 m k,l=n+1 cos 2kπt, cos 2lπt = 1 a k b l cos 2kπt, sin 2lπt b k b l sin 2kπt, sin 2lπt cos 2kπt cos 2lπt dt =, k l 1 2, k = l 91

92 sin 2kπt, sin 2lπt = cos 2kπt, sin 2lπt = f n f m 2 2 = 1 2 m k=n+1, k l 1 2, k = l ( a k 2 + b k 2 ) (77) n, m {f n } Cauchy f(t) L 2 () f n f 2 2 = 1 2 k=n+1 ( a k 2 + b k 2 ) 6.11 f n (t), f(t) 6.2 g(t) L 2 () 92 lim f n, g = f, g n = [, 1] 1 f, 1 = a n = 1, 2, f, cos 2nπt, f, sin 2nπt L 2 () Fourier Fourier 6.2 L 2 () f n (t) 6.1 f n (t) f(t) f n (t) f(t) 92 f f n, g f f n 2 g

93 f(t) L 2 () f f n 2, n, {f n } L 2 () Cauchy n 1 < n 2 < n k f nk f nk+1 2 < 2 k f nk (t) f(t) g(t) = f n1 (t) + f nk+1 (t) f nk (t) L 2 () k=1 m = 1, 2,, g m (t) = f n1 (t) + m f nk+1 (t) f nk (t) k=1 t g m (t) g(t)( + ) g m (t) L 2 () g m (t) 2 L 1 g m 2 f n1 2 + f nk+1 f nk 2 < f n k=1 g(t) L 2 () m 1 f nm (t) = f n1 (t) + {f nk+1 (t) f nk (t)} k=1 f nm (t) g(t) Lebesgue f(t) L 2 () f(t) = f n1 (t) + {f nk+1 (t) f nk (t)} k=1 f f nm 2 k=m f nk+1 f nk 2 < 2 m+1 lim n f f n 2 = L 2 (R) ω k (x) (45) f L 2 (R) ω k f(x) ω k f(x) L 2 (R), k >, (79) 93

94 ω k f(x) C lim ω k f f 2 = (8) k (79) ω k f(x) + ωk (x y) ω k (x y) f(y) dy + ω k (x y) dy + ω k (x y) f(y) 2 dy x (79) ω k f 2 2 f 2 2 ω k f(x) 5.9 (8) Hilbert 6.3 X X 6.8 X ν(x) = σ(x, x), x X, (81) 6.14 (81) X X ν( ) Hilbert Hilbert Banach x, y X σ(x, y) = 94 σ x σ y Pythagoras x σ y = ν(x) 2 + ν(y) 2 = ν(x + y) 2 94 σ σ 94

95 ν(x y) 2 = ν(x) 2 + ν(y) 2 2 σ(x, y) ν(x y) 2 + ν(x + y) 2 = 2 ( ν(x) 2 + ν(y) 2 ) (82) σ(x, y) = 1 4 ( ν(x + y)2 ν(x y) 2 ) (83) 6.3 ν(x) (82) (83) σ(x, y) (81) 6.15 (83) σ σ(x, y) ( ν(x) + ν(y)) ν(x) 95 a, b σ( (a b)x, y) a b ( a b ν(x) + ν(y) ) ν(x) [ 6.3 ] σ(x, x) = ν(x) 2 σ(x, y) = σ(y, x) σ(, y) = (82)(83) ( ) x + z σ(x, y) + σ(z, y) = 2 σ, y, x, y, z X, (84) 2 z = x σ( x, y) = σ(x, y) x 2x z = σ(2x, y) = 2 σ(x, y) (84) σ(x, y) + σ(z, y) = σ(x + z, y) m, n n n σ(x, y) = σ( x, y) 2m 2m a m 2 n, m n 6.15 σ( a x, y) = a σ( x, y) σ 6.6 Hilbert 95 ν (82) 95

96 6.5 f(t) f(t) M > { t ; f(t) > M } L () f L () f = inf { M > ; { t ; f(t) > M} } (85) 6.1 f L () f f = f(t) = (85) f = f(t) > t f(t) = 6.4 N () = { f L () ; f = } N () = N () f(t) L () a R a f(t) L () a f = a f (86) a = a a f(t) > M f(t) > M a (86) 6.12 f(t), g(t) L () (f + g)(t) = f(t) + g(t) f + g f + g (87)

97 L >, M > f(t) L g(t) M f(t) + g(t) L + M { t ; f(t) + g(t) > L + M } { t ; f(t) > L } { t ; g(t) > M } L > f M > g (87) L () f, g a f + b g L () L () L () L 1 () L 2 () f, g L () f g f g N () L () C (f) = { f 1 L () ; f 1 f } L () = L ()/ L 1 () L () L () L () L () L () L () d (f, g) = f g L () L () Banach 6.3 f n (t) L () n = 1, 2, Cauchy f(t) L () lim n f n f = f n (t) L () Cauchy 3.12 f(t) f(t) 6.16 n f n f L 1 () L 2 () L () 6.4 L 1 () L () L 2 () 97

98 f(t) L 1 () L () M > {t f(t) > M} f(t) 2 dt M f(t) dt (88) f 2 2 f 1 f 6.17 f L 1 () L () C 1 () C () C 2 () 6.5 [, 1] L () L 2 () L 1 () 1 L 1 () L () f L 2 () f(t) dt = f, 1 f f 1 f 2, f L 2 () (89) f L () M > {t; f(t) > M} f(t) 2 dt M 2 1 dt = M 2 f 2 f, f L () (9) 6.18 f L () C (f) C 2 () C 1 () Lebesgue 7.2 Borel

99 , 13, 44, 37, 7, 46, 21 Gauss, 5, 62, 37, 65, 44, 46, 7 Cantor, 13 Cantor, 11, 51, 85, 26, 84, 3, 81, 58, 26, 3, 48, 73 Cauchy, 84, 25 3, 11, 87, 87, 86, 86, 62, 22, 4, 15, 65, 8, 22, 89, 21, 7, 44, 51, 77, 82, 77, 94, 82, 82, 47, 89, 89, 44, 6, 79 2, 11, 81, 81 Heine-Borel, 25 Banach, 86 Pythagoras, 94, 21, 26, 3 99

100 Hilbert, 91 Fatou, 62 Fourier, 2 Fourier, 92, 86 Fubini, 7, 23, 5, 5, 3, 3, 66, 44, 21 Besicovitch, 21, 28 Beppo Levi, 64, 46 Poisson, 77, 96 Maple, 15, 95, 63 Riemann, 4 Riemann, 4 Lipschitz, 4 Lebesgue, 83 Lebesgue, 5 Lebesgue, 7 Lebesgue, 59, 5, 26, 5 1