8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) 10.1 10.2 Carathéodory 10.3 Fubini 1 Introduction [1],, [2],, [3],, [4],, [5],, [6],, [7],, [8],, [1, 2, 3] 1980

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% 100% 1 Introduction 2 (100%) 2.1 2.2 2.3 3 (100%) 3.1 3.2 σ- 4 (100%) 4.1 4.2 5 (100%) 5.1 5.2 5.3 6 (100%) 7 (40%) 8 Fubini (90%) 2006.11.20 1

8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) 10.1 10.2 Carathéodory 10.3 Fubini 1 Introduction [1],, [2],, [3],, [4],, [5],, [6],, [7],, [8],, [1, 2, 3] 1980 [4] 1980 [5, 6, 7, 8] 2000 [1, 4] [7, 4] [7] Stieltjes [6] R n [3] 1 [5] 2 2.1 2 ( ) E = {(x, y) x [a, b], y [c, d]} f(x, y) E f(x, y)dxdy E 2

2.1 E : a = x 0 < x 1 < < x n = b, c = y 0 < y 1 < < y m = d S(f, ) = s(f, ) = sup {f(x, y) x i 1 x x i, y j 1 y y j } (x i x i 1 )(y j y j 1 ) 1 i n,1 j m 1 i n,1 j m inf {f(x, y) x i 1 x x i, y j 1 y y j } (x i x i 1 )(y j y j 1 ). S(f) = inf {S(f, ) } s(f) = sup {s(f, ) } S(f), s(f) Darboux 2.2 = max{x i x i 1, y j y j 1 1 i n, 1 j m} lim S(f, ) = S(f), lim s(f, ) = s(f) 0 0 2.3 S(f) = s(f) f(x, y) E E f(x, y)dxdy 2.4 (1) f(x, y) f(x, y) (2)f(x, y) Darboux ξ i,j [x i 1, x i ] [y j 1, y j ] f(x, y)dxdy = lim f(ξ i,j )(x i x i 1 )(y j y j 1 ) E 0 1 i n,1 j m f(x, y) Darboux 2.2 A R 2 1 (x, y) A 1 A (x, y) = (2.1) 0 (x, y) A c 1 A A 3

2.5 (A ) A E 1 A E A = 1 A (x, y)dxdy. (2.2) E E 1 A A E 1 A E 1 A (x, y)dxdy = E 1 A (x, y)dxdy E A A E 2.6 S(1 A ) m J (A), s(1 A ) m J (A) A Jordan Jordan A A Jordan m J (A) S(1 A ), s(1 A ) A E E Jordan Joran A Jordan 2.7 (1) c(t) = (x(t), y(t)) C 1 (c(0) = c(1) t t c(t) c(t )) A (2) A E = [0, 1] 2 x y m J (A) = 0, m J (A) = 1 Jordan 2.8 A, A i (1) m J (A) m J (A). (2) (Jordan )A 1, A 2 Jordan A 1 A 2, A 1 A 2 Jordan m J (A 1 A 2 ) = m J (A 1 ) + m J (A 2 ) m J (A 1 A 2 ). (3) {A i } n i=1 Jordan n i=1 A i Jordan m J ( n i=1a i ) n m J (A i ). i=1 (4) A E (E ) m J (A) = E m J (A c E). A Jordan A c E Jordan 4

2.9 A R 2 1 A A A A A = { P R 2 ε > 0 B ε (P ) A, B ε (P ) A c }. (2.3) P = (p, q) B ε (P ) = {(x, y) (x p) 2 + (y q) 2 < ε}. 2.10 A R 2 A Jordan 0 2.11 R I = [a, b] (i) I 1 3 (ii) (i) I 1, I 2 I 1 ( ) 1 2 3 I2 ( 1 2 3) (iii) (ii) I 1,1, I 1,2, I 2,1, I 2,2 ( 1 3) 3 C Cantor m J (C) = 0 n r n (0 < r < 1 3 ) C r C r Jordan 2.3 2.7 (2) A E 0 Jordan 0 Jordan 2 2.12 A R n A m L (A) m L (A) { = inf I i I i R n ( } n i=1 [a i, b i ] ) A i=1 I i i=1 (2.4) (1)-(4) (5)-(7) 2.13 (1) A B m L (A) m L (B). (2) A i i=1 A i (3) A m L (A) = 0. (4) A m L (A) m J (A). 5

(5) A A 0 Jordan 0 (6) E R n m L (E) E (7) E R 2 A E m L (A) + m L (A c E) m L (E). (6) A, B m L (A B) + m L (A c B) m L (B). (2.5) 10.2 2.14 A R n R n B m L (A B) + m L (A c B) = m L (B) (2.6) A m L (A) A m L (A) R n B L (R n ) 2.15 A E A E B = E (2.6) m L (A) = E m L (A c E) (2.7) 2.8 (4) A Jordan Jordan 2.8 (4) 2.13 (4), (6) 2.16 R n A Jordan A m J (A) = m L (A). 2.17 (1) R n, B L (R n ). (2) A B L (R n ) A c B L (R n ). (3) A i B L (R n ) (i N) i=1 B L(R n ). (4) () A i B L (R n ) (i N) i j A i A j m L ( i=1a i ) = m L (A i ). (2.8) i=1 2.17 (3), (4) 2.16 10.1 6

3 3.1 3.1 2 ( ) 2 3.2 () F 2 ( F ) (, F) (1), F. (2) A F A c F. (3) A n F (n = 1, 2,...) n=1 A n F. (1), (2), (3) σ-algebra, σ-field,σ-, σ- 3.3 (, 2 ) 3.4 (, F) (3) A n F (n = 1, 2,...) N n=1 A n F (N N). 3.2 (2) A n = (n = N + 1, N + 2,...) 3.5 3.2 (1), (2) (3) σ- 3.6 () (, F) m F m m (, F, m) (1) A F 0 m(a) + m( ) = 0. (2) (, ) A n F A n A m = (n m) m( n=1a n ) = m(a n ). (3.1) (i) m( n ) < (n N) = n=1 n σ- (ii)m() < (iii) m() = 1 (m ) 3.7 (1) A 2, (, A, m) n=1 (i) A (ii) A A 0 m(a) + m( ) = 0. 7

(iii) A i A (1 i n, n N), A i A j = m( n i=1 A i) = n i=1 m(a i). Jordan 3.8 (1) F = 2 A A m(a) = + A. (3.2) (2) = R n, F B L (R n ),,m(a) A (R n, B L (R n ), m L ). = {w : [0, ) R n w(0) = 0 w(t) t } (3.3) = [0, 1] N (3.4) () σ- 3.2 3.9 (, F, m) (1) A i F (1 i n) n i=1 A i. A i A j = m( n i=1 A i) = n i=1 m(a i). (2) A, B F A B m(a) m(b). m(b) < m(a) < m(b \ A) = m(b) m(a). (3) A n A n+1, A n F (n = 1, 2,...) lim m(a n) = m ( n=1a n ). (3.5) (4) A n F (n = 1, 2,...) m ( n=1 A n) n=1 m(a n). (5) A n A n+1, A n F (n = 1, 2,...) n 0 m(a n0 ) < lim m(a n) = m( n=1a n ). (3.6) (6) A B := (A\B) (B\A) m(a) <, m(b) < m(a) m(b) m (A B). (1) 3.6 A n+1 = A n+2 = = (2) B = A (B \ A) A (B \ A) =. B \ A = B A c (1) m(b) = m(a) + m(b \ A) m(a). m(b) < m(a) < m(b \ A) = m(b) m(a). (3) B n = A n \ A n 1 (n 1, A 0 = ) 1. A N = N n=1 A n = N n=1 B n (N N + ) 2. n > m B n B m A c n 1 A m =. 8

(1) (4) m ( n=1a n ) = m ( n=1b n ) = lim N N n=1 m(b n ) = lim N m ( N n=1b n ) = lim N m(a N). (3.7) m ( N ) N n=1a n m(a n ) (3.8) (i) N = 2 A 1 A 2 = A 1 (A 2 \ A 1 ) m(a 1 A 2 ) = m(a 1 ) + m(a 2 \ A 1 ) m(a 1 ) + m(a 2 ). (ii) N OK N = 2 ( ) m N+1 n=1 A n = m ( ( N ) n=1a n ) A N+1 n=1 m ( N n=1a n ) + m(an+1 ) N+1 n=1 m(a n ). (3.9) (3.8) N (3) (5) A n \ A n+1 = B n, n=1 A n = C n > m B n B m A n A c m+1 =. A n0 \ C = n=n 0 B n. (3.10) (2) m(c) <, m(a n ) < n n 0 N m(a n0 ) m(c) = m(b n ) = lim (m(a n ) m(a n+1 )) N n=n 0 n=n 0 = lim (m(a n 0 ) m(a N+1 )). (3.11) N m(c) = lim N m(a N ). (3.10) (i) x A n0 \ C n n 0 x A n x / A n+1. x A n \ A n+1 = B n. (ii) x n=n0 B n n n 0 x B n = A n \ A n+1. x A n0 x / C. (6) m(a) = m(a B)+m(A B c ), m(b) = m(a B)+m(B A c ). m(a) m(b) = m(a B c ) m(b A c ) m(a B c ) + m(b A c ) = m(a B). 3.10 lim sup A n = n=1 { i=na i } (3.12) lim inf n = n=1 { i=na i } (3.13) 9

lim inf A n = lim sup A n lim A n (1) lim inf A n lim sup A n. (2) m(lim inf A n ) lim inf m(a n ). (3) n 0 N m ( ) n=n 0 A n < lim sup m(a n ) m ( ) lim sup A n. (3.14) (4) n 0 N m ( ) n=n 0 A n < lim A n ( ) lim m(a n) = m lim A n. (3.15) (1) x lim inf A n n 0 x i=n0 A i. n x i=n A i. x lim sup A n. (2) B n = i=n A i lim inf A n = n=1 B n B 1 B 2 B 3 3.9 (3) ( ) m lim inf A n = m ( n=1b n ) = lim n) = lim inf n) lim inf n) (3.16) (3) C n = i=n A i lim sup A n = n=1 C n C 1 C 2 C 3, m(c n0 ) < 3.9 (5) (4) (2),(3) ( ) m lim inf A n lim inf m (lim sup A n ) = m ( n=1c n ) m(a n) lim sup = lim n) = lim sup m(c n ) lim sup m(a n ). (3.17) ( ) m(a n ) m lim sup A n. (3.18) 3.11 lim sup A n = { x n(1, x) < n(2, x) < < n(k, x) < x k=1 A } n(k,x) (3.19) { } lim inf A n = x n(x) x n=n(x) A n (3.20) 10

3.12 {f n (x)} n=1, f(x) [0, + ] n x f n(x) f n+1 (x) lim f n (x) = f(x) a, b 0 a < b + lim A n = A A n = {x a < f n (x) b} (3.21) A = {x a < f(x) b} (3.22) 3.10 (2) 3.12 ( 5.13) 3.2 σ- σ- 2 (1) (3)? 3.13 C 1, C 2 2 C 1 C 2 C 1 C 2 (C 2 C 1 ) () 3.14 C 2 σ(c) = λ Λ F λ {F λ λ Λ} C F λ σ- F λ σ(c) C σ- 2 σ- C F λ C σ(c) (i) σ(c) σ- (ii) F σ- C F σ(c) F (i) 3.2 (1)-(3) (1) λ, F λ, σ(c). (2) A σ(c) λ Λ A F λ. F λ σ- A c F λ λ Λ. A c σ(c). (3) A n σ(c) (n = 1, 2,...) A n F λ n A n F λ. λ n A n σ(c). σ(c) σ- (ii) C F F {F λ } σ(c) F. 3.15 σ(c) C σ- σ- R n Borel( ) 11

3.16 = R n C = {O R n O R n } σ(c) B(R n ) R n B(R n ) (?) B(R n ) B L (R n ) B L (R n ) 2 S σ- Borel B(S) S (3.3), (3.4) 3.17 = R n C i (i = 1,..., 5) σ(c i ) B(R n ) (1) C 1 = {R n } (2) C 2 = { n i=1 [a i, b i ] < a i < b i < } (3) C 3 = { n i=1 [a i, b i ) < a i < b i < } (4) C 4 = { n i=1 (a i, b i ] < a i < b i < } (5) C 5 = {B r (a) r a R n a }. B r (a) = {x R n d(x, a) < r}. 4 ( ) (1) 4.1, Y f : Y A f(a) := {f(x) x A} A f B Y f 1 (B) := {x f(x) B} B f (1) B n Y (n = 1, 2,...) f 1 ( n=1b n ) = n=1f 1 (B n ) (4.1) f 1 ( n=1b n ) = n=1f 1 (B n ). (4.2) (2) A n (n = 1, 2,...) f ( n=1 A n) = n=1 f(a n), f ( n=1 A n) n=1 f(a n) (3) B Y f 1 (B c ) = ( f 1 (B) ) c. 4.1, + 4.2 () (1) (, F) f : [, + ] F- (F-measurable function) ( F- ) a R f 1 ((a, + ]) := {x f(x) > a} F. (2) = R n F = B(R n ), B L (R n ), R ( +, ) 12

4.3 (1) F (2) f : R n R (?) (3) A F A f : A [, + ] a R {x A f(x) > a} F F- 4.4 f : R n R f f : R n R f 1 ((a, + ]) B(R n )- 4.5 f : [, + ] f 1 (R), f 1 ({+ }), f 1 ({ }) F f 1 ({+ }) = f 1 ( n=1(n, + ]) = n=1f 1 ((n, + ]) f 1 ({ }) = f 1 ( n=1[, n]) = n=1f 1 ([, n]) = n=1 ( f 1 (( n, + ]) ) c. f 1 ((a, + ]) F (2), (3) f 1 ({+ }), f 1 ({ }) F. f 1 (R) = \ ( f 1 ({± }) ) F. 4 4.6-4.9 4.6 (, F) f : [, + ] 4 (1), (2), (3), (4) (1) f(x) (2) a R {x f(x) a} F. (3) a R {x f(x) a} F. (4) a R {x f(x) < a} F. (1), (2), (3) (4) (1) (2): f 1 ([, a]) = f 1 ((a, + ]) c F. (2) (3): f 1 ([a, + ]) = f 1 ([, a)) c = f 1 ( n=1[, a 1 n ] ) c = (3) (1): ( n=1f 1 ( [, a 1 n ] )) c F. f 1 ((a, + ]) = f 1 ( n=1[a + 1 n, + ] ) = n=1f 1 ( [a + 1 n, + ] ) F. 4.7 (, F) f : [, + ] (1), (2), (3) (1) f(x) (2) f 1 ({+ }), f 1 ({ }) F R C 13

(i) σ(c) = B(R). (ii) A C f 1 (A) F. (3) f 1 ({+ }) F A B(R) f 1 (A) F. (1) (2) (3) (1) (1) (2) : f 1 ({+ }), f 1 ({ }) F C = {(a, b] < a < b < + } 3.17 (4) σ(c) = B(R)., f 1 ((a, b]) = f 1 ((a, + ]) ( f 1 ((b, + ]) ) c F (ii) C (2) (2) (3) H = { A A R f 1 (A) F } B(R) H H (a) C H, (b) H σ-. B(R) = σ(c) H (a) (ii) (b) (1) R, H: f 1 (R) =, f 1 ( ) = OK (2) A H A c H: f 1 (A c ) = (f 1 (A)) c F OK (3) A n H (n = 1, 2,...) n=1 A n H: f 1 ( n=1 A n) = n=1 f 1 (A n ) F OK H σ- (3) (1) a R f 1 ((a, + ]) = f 1 ((a, )) f 1 ({+ }) F. 4.8 f(x), g(x) +, ϕ : R 2 R h(x) = ϕ(f(x), g(x)) a R ϕ S a = {(x, y) ϕ(x, y) > a} S a S a H = {(α, β) (γ, δ) α, β, γ, δ (α, β) (γ, δ) S a. } H = {I i J i i = 1, 2,...} I i = (α i, β i ), J i = (γ i, δ i ). {x h(x) > a} = {x ϕ(f(x), g(x)) > a} = {x (f(x), g(x)) S a } = i=1{x (f(x), g(x)) I i J i } = i=1 ( f 1 (I i ) f 1 (J i ) ) F. 4.8 14

4.9 f n : [, + ] (n = 1, 2,...) (1) sup n N f n (x), inf n N f n (x) (2) lim sup f n (x), lim inf f n (x) x lim f n (x) (+, ) (1) {x sup f n (x) a} = n=1{x f n (x) a} F, n N {x inf n(x) a} n N = n=1{x f n (x) a} F. 4.6 (2) lim sup f n (x) = inf n { supm n f m (x) }, lim inf f n (x) = sup n {inf m n f m (x)} (1) 4.10 f, g : [, + ] A = {x f(x) g(x)} 4.11 f n (x) (n = 1, 2,...) R (, F) 0 F 0 = {x lim f n (x) } 4.12 f(x) [0, 1] f(x) 3.16 4.13 (1) f : R F f := {f 1 (A) A B(R)} F f σ- F f F (2) σ- G (i) G F. (ii) f G- ( 3.13 ) F f σ- F f σ(f) f σ- 4.2 4.14 ( i, F i ) (i = 1, 2) f : 1 2 F 1 /F 2 - : A F 2 f 1 (A) F 1 4.15 (, F) S S σ- ( S ) B(S) (S, B(S)) f : S F/B(S)- 15

4.16 2 = R, F 2 = B(R) F 1 /F 2-4.2 F 1-4.17 f(x) = (f 1 (x),..., f n (x)) : R n (1), (2) (1) f F/B(R n )- (2) i f i : R 4.2 F- 5 5.1 5.1 (, F) f : R (simplefunction) (1) f (2) f(x) 5.2 A I A (x) 1 if x A, I A (x) = (5.1) 0 if x A c A (indicator function) 5.3 f(x) {a 1,..., a n }, E i = f 1 ({a i }) (1) E i F. i j E i E j = n i=1 E i =. (2) f(x) = n i=1 a ii Ei (x). (2) a i = 0 a i I Ei (x) = 0 (2) f(x) 5.4 f(x) α i (i = 1,..., n) E i F f(x) = n i=1 α ii Ei (x). 5.5 f f(x) = n i=1 a ii Ei (x) 0 = 0 f(x)dm(x) = n a i m(e i ). (5.2) i=1 16

5.6 (1) α, β 0 (αf(x) + βg(x))dm(x) = α f(x)dm(x) + β g(x)dm(x). (5.3) (2) f(x) = l i=1 α ii Ai (x) (α i 0 1 i n, ) fdm = l i=1 α im(a i ). (3) 0 f(x) g(x) fdm gdm. (5.4) (1) f(x) = n i=1 a ii Ei (x), g(x) = m j=1 b ji Fj (x) αf(x) = n i=1 αa ii Ei (x) αfdm = α fdm. α = β = 1 E i F j i, j a i + b j {c l } N l=1 {c l} S l = {(i, j) a i + b j = c l, E i F j } l = (i,j) Sl E i F j 1. l k S l S k = 2. N l=1 l = 3. l k l k = 4. N l=1 c li l (x) f + g (f + g)dm = N l=1 c lm( l ). 1,2 (f + g)dm = = = = = N c l m( l ) l=1 N c l m ( ) (i,j) Sl E i F j l=1 N l=1 (a i + b j )m (E i F j ) (i,j) S l (a i + b j )m(e i F j ) (5.5) 1 i n,1 j m n a i m(e i ) + i=1 m b j m(f j ) = j=1 fdm + gdm (5.6) (5.5) (5.6) 1 i n,1 j m a im(e i F j ) = n i=1 a im(e i ) (2) α ii Ai dm = α i m(a i ) (1) 17

(3) (1) f, g f(x) = (i,j) N l=1 S l a ii Ei F j (x), g(x) = (i,j) N l=1 S l b ji Ei F j (x) x f(x) g(x) (i, j) l S l a i b j. fdm = a i m (E i F j ) (i,j) N l=1 S l b j m (E i F j ) = (i,j) N l=1 S l gdm. (5.7) 5.2 5.7 [0, ] ϕ N (t) N N. 0 0 t 1 2 N ϕ N (t) = k k < t k+1, 0 < k 2 N N 1 2 N 2 N 2 N N t > N (5.8) 5.8 f(x), g(x) [0, + ] (1) ϕ N (f(x)) = 2 N N k=0 k 2 N I E N,k (x), (5.9) E N,k = f 1 ( ( k 2 N, k+1 2 N ] ) (0 k 2 N N 1), E N,2 N N = f 1 ((N, ]). ϕ N (f(x)) (2) N N, x ϕ N (f(x)) ϕ N+1 (f(x)). (3) x lim N ϕ N (f(x)) = f(x). (4) f(x) g(x) ϕ N (f(x)) ϕ N (g(x)). Lemma ϕ N (f(x)) f(x) ϕ N (f(x)) 5.8 (2) 5.6 (3) 0 ϕ N(f(x))dm(x) ϕ N+1(f(x))dm(x) I(f) f(x) f(x) f N (x) = 2 N N k=0 E N,k := f 1 ( [ k 2 N, k + 1 2 N ) k 2 N I E N,k (x) (5.10) ) (0 k 2 N N 1) (5.11) E N,2 N N := f 1 ([N, ]). (5.12) 18

5.9 [0, + ] f(x) I N (f) = ϕ N (f(x))dm(x) (5.13) I(f) = lim N I N(f). (5.14) 5.10 f(x) (, F, m) I(f) = f(x)dm(x). f(x) = n i=1 a ii Ei (x) ϕ N (f(x)) = n i=1 ϕ N(a i )I Ei (x). I N (f) = ϕ N(f(x))dm(x) = n i=1 ϕ N(a i )m(e i ). lim N ϕ N (a i ) = a i I(f) = lim N I N (f) = n i=1 a im(e i ) = f(x)dm(x). f I(f) I(f) [0, + ] 5.11 [0, + ] f(x) f(x)dm(x) := I(f). (5.15) [0, + ] ( ) 5.12 0 f(x) g(x) ( x ) f(x)dm(x) g(x)dm(x). f(x) g(x) ϕ N (f(x)) ϕ N (g(x)) ϕ N(f(x))dm(x) ϕ N(g(x))dm(x). N 5.13 ( (Monotone convergence theorem,mct )) f(x) [0, + ] [0, + ] {f n (x)} n=1 (1) f 1 (x) f 2 (x)... f n (x)... (2) lim f n (x) = f(x). lim f n(x)dm(x) = f(x)dm(x) 5.12 f n(x)dm(x) lim f n(x)dm(x) f(x)dm(x) I N (f n ), I N (f) I N (f n ) = I N (f) = 2 N N k=1 2 N N k=1 k ( 2 N m E (n) N,k ) (5.16) k 2 N m (E N,k). (5.17) 19

E (n) E (n) N,k = { x k 2 N < f n(x) k + 1 } 2 N (1 k 2 N N 1) (5.18) N,2 N N = {x f n(x) > N} (5.19) { k E N,k = x 2 N < f(x) k + 1 } 2 N (1 k 2 N N 1) (5.20) E N,2 N N = {x f(x) > N}. (5.21) f n (x) lim f n (x) = f(x) lim E (n) N,k = E N,k. 3.10 (2) m(e N,k ) lim inf m(e (n) N,k ). lim f n (x)dm(x) lim inf N(f n ) I N (f). (5.22) lim f n(x)dm(x) lim N I N (f) = f(x)dm(x). 5.11 f(x) MCT 5.14 f [0, + ] { } f(x)dm(x) = sup g(x)dm(x) g x 0 g(x) f(x). (5.23) (5.23) [0, + ] f 5.15 3.9 (3) {A n }, A f n (x) = I An (x), f(x) = I A (x) MCT lim m(a n ) = m(a) MCT 3.9 (3) 5.3 f(x) 5.16 (1) f(x) (, F, m) f + (x) := max (f(x), 0), f (x) := max ( f(x), 0) f + (x)dm(x) < f (x)dm(x) < (5.24) f(x) f(x)dm(x) := f + (x)dm(x) f (x)dm(x) (5.25) 20

(2) A F f(x) A f(x) A (A, F A, m A ) A f(x)dm(x) F A = {B F B A}, m A (B) = m(b) 5.17 f +(x)dm(x) = f (x)dm(x) < f(x)dm(x) = +. f +(x)dm(x) < f (x)dm(x) = f(x)dm(x) =. 5.18 (1), (2) (1) f (5.24) (2) f(x) dm(x) <. ϕ N ( f(x) ) = ϕ N (f + (x)) + ϕ N (f (x)) 5.19 5.18 f () L 1 (, F, m) f(x) dm(x) f L 1 (,F,m), f L 1 f L 1-5.20 (1) x f(x) 0, g(x) 0 α 0, β 0 (αf(x) + βg(x))dm(x) = α f(x)dm(x) + β g(x)dm(x). (5.26) (2) x f(x) 0, g(x) 0 f, g L 1 h(x) := f(x) g(x) L 1 h(x)dm(x) = f(x)dm(x) g(x)dm(x). (5.27) (3) f, g L 1, α, β R αf(x) + βg(x) L 1 (αf(x) + βg(x))dm(x) = α f(x)dm(x) + β g(x)dm(x). (5.28) (1) f n (x), g n (x) f(x), g(x) αf n (x) + βg n (x) αf(x) + βg(x) (αf + βg)dm = lim (αf n + βg n )dm { } = lim α f n dm + β g n dm = α fdm + β gdm. (5.29) (2) h(x) = h + (x) h (x) = f(x) g(x) h + (x) + g(x) = f(x) + h (x). h + (x) f(x) h (x) g(x). h +, h L 1. f g L 1. (1) h + dm + gdm = fdm + h dm. (5.30) 21

hdm = h + dm h dm = fdm gdm. (5.31) fdm. (3) α, β 0 (3) f L 1 ( f)dm = αf(x) + βg(x) = (αf + (x) + βg + (x)) (αf (x) + βg (x)). αf + + βg +, αf + βg L 1 (2) αf + βg L 1 (αf + βg)dm = (αf + + βg + )dm (αf + βg )dm ( ) ( ) = α f + f dm + β f + dm f dm = α fdm + β gdm. (5.32) 5.21 (1) x f(x) g(x) g L 1 fdm f(x) dm(x) g(x)dm(x). (5.33) f, f L 1. (2) f, g L 1 x f(x) g(x) fdm gdm. (1) 5.12, 5.18 fdm = f + dm f dm f dm gdm. (2) 0 f(x) g(x) 5.12 5.20 (3) gdm fdm = (g f)dm 0 (4) 5.22 f(x), g(x) m ({x f(x) g(x)}) = 0 f(x) g(x) x a.e. x, m a.s. x, a.s. x, a.a. x f(x) = g(x) m a.e. x. (5.34) 5.23 f(x), g(x) f(x) = g(x) a.e. x fdm = gdm. (5.35) f f(x) = n i=1 α ii Ei (x) α 1,..., α n g(x) β 1,..., β m g(x) = n i=1 α ii Fi (x) + m l=1 β li Gl (x) F i, G l F G l F i = ( i, l) {x f(x) g(x)} = ( i j (E i F j )) ( i,l (E i G l )) l, i, j m(g l ) = 0 m(e i F i ) = 0. m(e i ) = m(f i ) (1 i n). fdm = gdm. 22

5.24 (1) f, g f = g a.e.x fdm = gdm. (2) f L 1 f = g a.e. x g L 1 gdm = fdm. (3) f f = 0 a.e. x fdm = 0 (1) N = {x f(x) g(x)} m(n) = 0 f n f MCT f n dm = fdm. lim f n (x) = f n (x)i N c(x) f n f(x)i N c(x) lim f n dm = fi N cdm. 5.23 f ndm = f n dm. fdm = fi N cdm. gdm = gi N cdm. x f(x)i N c(x) = g(x)i N c(x) fdm = gdm. (2) f = g a.e. x f + = g + a.e. x f = g a.e. x. f ±dm = g ±dm. (3) f = 0 a.e. x = fdm = 0 (1) A = {x f(x) > 0} m(a) = 0 A n = {x f(x) > 1/n} A n A n+1 (n = 1, 2,...) A = n=1 A n. m(a) = lim m(a n ). m(a) > 0 n m(a n ) > 0. fdm m(a) = 0 fi An dm 1 n I An dm = m(a n) n 5.25 (1) f(x) [a, b] b a f(x)dx = 0 f(x) = 0 x [a, b] (3) (2) N F N c ( ) N c f(x)dm(x) f(x)dm(x) > 0. (Fubini 8.2 (3) (ii) ) 6 R R- f(x)dx, A L- f(x)dx A 23

6.1 f(x) I = [a, b] L- f(x)dx = R- f(x)dx. I I f N (x) := F N,k = 2 N 1 k=0 inf{f(x) [ k 2 N, k + 1 ) 2 N k 2 N x k + 1 2 N }I F N,k (x) ( k = 2 N 1 ) f N (x) f N (x) f(x) L- I f(x)dx = lim N L- I f N (x)dx. L- I f N (x)dx 2N Darboux R- I f(x)dx 6.2 f(x) I f(x) f N (x) inf sup f N (x) (1) Darboux (2) ( 10.3) (3) 7 7.1 7.1 (Fatou ) f n (x) lim inf f n(x)dm(x) lim inf f n (x)dm(x). (7.1) g n (x) = inf k n f k (x) x lim inf f n (x) = lim g n (x) g 1 (x) g 2 (x) g n (x). lim inf f n(x)dm(x) = lim g n(x)dm(x) = lim g n (x)dm(x). g n(x)dm(x) f n(x)dm(x) 7.2 f n (x) x f(x) sup n f n L 1 < f L 1 f L 1 lim inf f n L 1. 24

7.3 ( (Lebesgue s dominated convergence theorem)) {f n (x)} (1), (2) (1) lim f n(x) = f(x) (x ) (2) g L 1 (, m) f n (x) g(x) (x ). lim f n (x)dm(x) = f(x)dx. (2) f n L 1 (, m) (1),(2) f(x) g(x) ( x ) f L 1 (, m) h n (x) = g(x)+f n (x) h n (x) 0 ( x ) lim h n (x) = g(x) + f(x). Fatou (g(x) + f(x)) dm(x) lim inf h n (x)dm(x) = g(x)dm(x) + lim inf f(x)dm(x) lim inf f n, f f n, f f(x)dm(x) lim sup (7.3), (7.4) f n (x)dm(x). (7.2) f n (x)dm(x). (7.3) f n (x)dm(x). f(x)dm(x) lim sup f n (x)dm(x). (7.4) 7.4 3.9 (3) 3.9 (5) 7.5 {x n n = 1, 2,...} t [0, 1] lim e 1tx n = 1 lim x n = 0. 7.6 f(x) [a, b] F (x) = [a,x] f(t)dm L(t) F (x) x 7.3 7.7 {f n (x)} g(x) n=1 f n(x) g(x) (x ) n=1 f n(x) f n (x)dm(x) = f n (x)dm(x). n=1 n=1 25

( ) 7.8 7.7 n=1 a n < +, a n 0 ( n) f(t, n) (t 0, n N) (1), (2) (1) lim f(t, n) = α n. (2) t, n f(t, n) a n. n=1 f(t, n), n=1 α n lim t n=1 7.9 n=1 a 1 n <, a n 0 ( n) lim t t f(t, n) = α n. n=1 log(1 + a n t) = 0 7.10 () f(t, x) (a t b, x ) (1) f(t, x) t C 1 (2) t f(t, ) L 1 (, m). (3) g L 1 (, m) (t, x) f(t, x) t g(x). F (t) = f(t, x)dm(x) t C1 F (t) = f(t, x)dm(x). t lim h 0 n=1 F (t + h) F (t) f(t + h, x) f(t, x) = dm(x) (7.5) h h f(t + h) f(t) = f(t, x) (7.6) h t = f (t + θh, x) t g(x). (7.7) f(t + h, x) f(t, x) h F (t) = f t (t, x)dm(x). F (t) 7.2 0 x lim f n (x) = f(x) ( x ) x lim f n (x) = f(x) x lim f n (x) = f(x) f n (x) f(x) 7.11 (1) (, F, m) f n (x) f(x) m ({ x }) lim f n(x) f(x) = 0 N = {x lim f n (x) f(x)} (2) m ({x f(x) > g(x)}) = 0 f(x) g(x) a.e. x 26

7.3 7.12 ( ( )) {f n (x)} (1), (2) (1) lim f n(x) = f(x) a.e. x. (2) g L 1 (, m) f n (x) g(x) a.e. x. lim f n (x)dm(x) = f(x)dx. { } N = x lim f n(x) f(x) N k = {x f k (x) > g(x)} (k = 1, 2,...) m(n) = m(n k ) = 0 (k = 1, 2,...). Ñ = N ( k=1 N k) m(ñ) = 0(!). fn (x) = f n (x)iñ c(x), f(x) = f(x)iñ c(x) lim fn (x) = f(x) ( x ) f n (x) g(x) ( x ). 7.3 lim f n (x)dm(x) = f(x)dm(x). (7.8) f n (x) = f n (x) a.e. x, f(x) = f(x) a.e. x 5.24 (2) f n(x)dm(x) = f n (x)dm(x), f(x)dm(x) = x f(x)dm(x). (7.8) a.e. 7.13 (1) f(x) = g(x) a.e. x g(x) = h(x) a.e. x f(x) = h(x) a.e. x. (2) f 1 (x) = g 1 (x) a.e. x f 2 (x) = g 2 (x) a.e. x f 1 (x) + f 2 (x) = g 1 (x) + g 2 (x) a.e. x. 8 R n+m z z = (x, y) x R n, y R m n x m y f(z) = f(x, y) R n+m ( ) ( ) f(z)dz = f(x, y)dy dx = f(x, y)dx dy R n+m R n R m R m R n Fubini( ) (1) Fubini (2) Fubini 27

8.1 A R n+m A x = {y R m (x, y) A} (x R n ) A y = {x R n (x, y) A} (y R m ) R n, R m, R n+m m L 8.1 Fubini Fubini 8.2 f(z) (z = (x, y) R n+m, x R n, y R m ) (1) y R m x( R n ) f(x, y) R n x R n R m y f(x, y) (2) () z = (x, y) f(x, y) 0 x, y F (x) = f(x, y)dm L (y), R m G(y) = f(x, y)dm L (x) (8.1) R n (+ ) f(z)dm L (z) = R n+m F (x)dm L (x) = R n G(y)dm L (y) R m (8.2) (3) ( ) f L 1 (R n+m, m L ) (i) m L (N 1 ) = 0 (N 1 B(R n )) x / N 1 f(x, ) L 1 (R m, m L ). m L (N 2 ) = 0 (N 2 B(R m )) y / N 2 f(, y) L 1 (R n, m L ). (ii) F (x) = R f(x, y)dm m L (y) (x N1 c), G(y) = R f(x, y)dm m L (x) (y N2 c) R n, R m f(z)dm L (z) = F (x)dm L (x) R n+m R n = G(y)dm L (y). (8.3) R m 8.3 (ii) statement 5.25 (2) F (x), G(y) 0 N c 1 F (x)dm L (x) R n F (x)dm L (x) 28

Fubini 1. f(x, y) ( 8.4) 2. f(x, y) ( 8.8) 3. f(x, y) ( 8.2) 1 8.4 ( Fubini ) A B(R n+m ) (1) A B(R n+m ) x, y A x B(R m ), A y B(R n ). x( R n ) m L (A x ), y( R m ) m L (A y ) (2) m L (A) = m L (A x )dm L (x) = m L (A y )dm L (y). (8.4) R n R m 8.5 f(x, y) = I A (x, y) (1) f(x, y) = I Ax (y) = I A y(x), (2) R m f(x, y)dm L (y) = m L (A x ), R n f(x, y)dm L (x) = m L (A y ), R n+m f(z)dm L (z) = m L (A) f(x, y) = I A (x, y) Theorem 8.2 (1), (2) (Monotone class theorem) Monotone class theorem 8.6 C (1) A i C, A i A i+1 (i = 1, 2,...) i=1 A i C. (2) A i C, A i+1 A i (i = 1, 2,...) i=1 A i C. 8.7 (Monotone class theorem) A C A C σ(a) C. 8.4 A n+m i=1 (a i, b i ] ( a i b i + ) b i = + (a i, + ] (a i, + ) A E N = ( N, N] n { C = A B(R n+m ) N A E N } (1), (2). (i) C 29

(ii) A C B(R n+m ) = σ(a) C N (A E N ) x, (A E N ) y x( R n ) m L ((A E N ) x ), y( R m ) m L ((A E N ) y ) N (1) m L (A E N ) = m L ((A E N ) x )dm L (x) = R n m L ((A E N ) y )dm L (y). R m (8.5) N (2) (i), (ii) (i) 3.9 (3), (5) A n+m i=1 (a i, b i ] (ii) A = n+m i=1 (a i, b i ] x = (x 1,..., x n ) n+m j=n+1 A x = (a j, b j ] (a i < x i b i 1 n n ) (8.6) ( ) y = (y 1,..., y m ) n A y i=1 = i, b i ] ( ) (a n+j < y j b n+j 1 j m ) (8.7) a i < x i b i (1 n n) m L (A x ) = m j=1 (b n+j a n+j ), a n+j < y j b n+j (1 j m) m L (A y ) = n i=1 (b i a i ). (1),(2) 8.2 8.4 8.8 ( Fubini ) R n+m f(x, y) = k i=1 α ii Ai (x, y) A i B(R n+m ) (1) f(x, y) = k α i I (Ai ) x (y) = i=1 k i=1 α i I (Ai ) y(x) (8.8) x, y y f(x, y), x f(x, y) (2) α i 0 (1 i k) F (x) = R n f(x, y)dm L (y), G(y) = R m f(x, y)dm L (x) 30

F (x) = G(y) = F (x)dm L (x) = R n G(y)dm L (y) = R m k α i m L ((A i ) x ) (8.9) i=1 k α i m L ((A i ) y ) (8.10) i=1 k α i m(a i ) = f(z)dm L (z) (8.11) R n+m k α i m(a i ) = f(z)dm L (z) (8.12) R n+m i=1 i=1 8.2 (1) f + N (x, y) f(x) f + (x, y) f (x, y) f N (x, y) f N(x, y) = f + N (x, y) f N (x, y) f N(x, y) lim f N(x, y) = f(x, y) (x, y) R n+m. N x f n (x, y) B(R n )- x f(x, y) B(R n )- y f(x, y) B(R m )- (2) (1) f N (x, y) f N (x, y) 0 F N (x) = f N (x, y)dm L (y) R m G N (y) = f N (x, y)dm L (x) R n x, y lim N F N (x) = F (x), lim N G N (y) = G(y) f(z)dm L (z) = F (x)dm L (x) (8.13) R n+m R n f(z)dm L (z) = lim f N (z)dm L (z) () R n+m N R n+m = lim F N (x)dm L (x) ( Fubini ) N R N = F (x)dm L (x) (). (8.14) R N G(y) (3) (1) f ± N (x, y), f N(x, y) G(y) statement F (x) F (x) F N,1 (x) = f + N (x, y)dm L(y) (8.15) R m F N,2 (x) = f N (x, y)dm L(y) (8.16) R m 31

F N,1, F N,2 N F 1 (x) = lim N F N,1 (x), F 2 (x) = lim N F N,2 (x) (+!) F 1 (x) = f + (x, y)dm L (y) (8.17) R m F 2 (x) = f (x, y)dm L (y) (8.18) R m x Fubini (2) F 1 (x)dm L (x) = f + (z)dm L (z) < (8.19) R n R n+m F 2 (x)dm L (x) = f (z)dm L (z) <. (8.20) R n R n+m K 1 = {x F 1 (x) = + }, K 2 = {x F 2 (x) = + } m L (K 1 ) = m L (K 2 ) = 0. N 1 = K 1 K 2 m L (N 1 ) = 0 R n+m f(z)dm L (z) = = f + (z)dm L (z) f (z)dm L (z) R n+m R n+m R n (F 1 (x)i K c(x) F 2 (x)i K c(x)) dm L (x) (8.21) x N1 c f(x, y) y F 1 (x) F 2 (x) = f(x, y)dm L (y) = F (x) ( x K c ) R m (8.21) R n F (x)dm L (x) 8.2 Fubini 10.1 8.9 A B L (R n+m ) (1) N 1 R n, N 2 R m x N c 1, y N c 2 A x, A y x( N c 1 ) m L(A x ), y( N c 2 ) m L(A y ) (2) m L (A) = m L (A x )dm L (x) = R n m L (A y )dm L (y). R m (8.22) 10.2 B, C C A B, m L (B \ C) = 0 D = B \ C 8.4 (2) m L (D x )dm L (x) = m L (D y )dm L (y) = 0. R n R m 32

N 1 R n, N 2 R m x N c 1, y N c 2 m L (D x ) = 0, m L (D y ) = 0. x, y m L ((B \ A) x ) = m L ((B \ A) y ) = 0. B x, B y x N c 1, y N c 2 A x, A y x N c 1, y N c 2 m L(A x ) = m L (B x ), m L (A y ) = m L (B y ) x( R n ) m L (B x ), y( R m ) m L (B y ) x( N c 1 ) m L(A x ), y( N c 2 ) m L(A y ) (2) x N c 1, y N c 2 m L(A x ) = m L (B x ), m L (A y ) = m L (B y ) B 8.4 8.10 f(z) (z = (x, y) R n+m, x R n, y R m ) (1) y R m N 1 R n, N 2 R m x N1 c y( Rm ) f(x, y), y N2 c x( R n ) f(x, y) (2) () z = (x, y) f(x, y) 0 x, y f(x, y)dm L (y), f(x, y)dm L (x) (+ ) R m R n ( ) ( ) f(z)dm L (z) = f(x, y)dm L (y) dm L (x) = f(x, y)dm L (x) dm L (y) R n+m R n R m R m R n (8.23) (3) ( ) f L 1 (R n+m, m L ) (i) (ii) m L (N 1 ) = 0 (N 1 B(R n )) x / N 1 f(x, ) L 1 (R m, m L ). m L (N 2 ) = 0 (N 2 B(R m )) y / N 2 f(, y) L 1 (R n, m L ). x(/ N 1 ) f(x, y)dm L (y), y(/ N 2 ) R m f(x, y)dm L (x) R m R n, R m ( ) f(z)dm L (z) = f(x, y)dm L (y) dm L (x) R n+m R n R ( m ) = R m f(x, y)dm L (x) R n dm L (y). (8.24) (1) f + (z), f (z), f(z) 0 f(z) f n (z) A I A x R n, y R m {y R m I A (x, y) = 1} = A x {x R n I A (x, y) = 1} = A y. 8.9 (1) N 1, N 2 (2) f(x, y) f(x, y) f(x, y) = f(x, y) m L a.e.(x, y). N = {(x, y) f(x, y) f(x, y)} m L (N) = 0. N 8.9 (1) 33

0 N 1 R n, N 2 R m (2) N m 1 c L (N x )dm L (x) = N m 2 c L (N y )dm L (y) = 0. N i N i N i m L (N x ) = 0 (x N 1 c), m L(N y ) = 0 (y N 2 c ). x N c 1 f(x, y) = f(x, y) (m L a.e. y) y N c 2 f(x, y) = f(x, y) (m L a.e. x) (8.25) x N c 1 R m f(x, y)dm L (y) = y N c 2 R n f(x, y)dm L (x) = R m R n f(x, y)dml (y) f(x, y)dml (y) R m f(x, y)dm L (y), R n f(x, y)dm L (y) 8.2 (2) (3) (2) 8.2 (3) (8.26) 8.11 (1) f(x, y) R n (R m f(x, y) dm L (y) ) dm L (x) f L 1 (R n+m, m L ) (2) f(x, y) A B L (R n+m ) welldefined ( ) f(z)dm L (z) = f(x, y)dm L (y) dm L (x) A A y A ( x ) = f(x, y)dm L (x) dm L (y). (8.27) A y A x 8.12 I R n J R m f(x, y) I J (1) m L -a.e. x I y f(x, y), m L -a.e. y J x f(x, y) (2) J F (y) R- I F (y)dy, R- I F (y)dy R- J f(x, y)dy, R- J f(x, y)dy I ( ) ( ) R- f(x, y)dy dx = R- f(x, y)dy dx = f(x, y)dxdy I J I J I J 8.13 (2) (1) (1) x R- J f(x, y)dy = R- J f(x, y)dy 34

9 10 10.1 (1) (2) L p - (3) (4) (1) 10.1 ε A G F F A G m L (G \ A) ε, m L (A \ F ) ε (1) A m L (A) < {I i } i=1 A i=1 I i, i=1 m L(I i ) m L (A) + ε 2 I i = n l=1 [a(i) l, b (i) l ] 0 {ε l } J i = n l=1 (a(i) l ε l, b (i) l + ε l ) A i=1 J i m L (J i ) m L (A) + ε. (10.1) i=1 G = j=1 J i G m L (G \ A) = m L (G) m L (A) m L (J i ) m L (A) ε. A E E B = E \ A B G B G m L (G \ B) ε F = E G c F F A m L (A \ F ) = m L (A) m L (F ) = m L (E) m L (B) (m L (E) m L (G E)) i=1 = m L (G E) m L (B) m L (G) m L (B) = m L (G \ B) ε. (10.2) (2) A n B n (B 0 = ) A n = A (B n ) B c n 1 A n A = n=1 A n. A n (1) () 35

10.2 A B, C C A B m L (B\A) = m L (A \ C) = 0 10.3 (1) f(x) R n g(x) f(x) = g(x) m L a.e. x. (10.3) (2) (1) f(x) f N (x) = E N,k = 2 N N k=0 k 2 N I E N,k (x) { x R n E N,2 N N = f 1 ((N, + )) k 2 N < f(x) k + 1 2 N } (0 k 2 N N 1) lim N f N (x) = f(x) (x R n ). E N,k Ẽ N,k ẼN,k E N,k m L (E N,k \ ẼN,k) = 0 f N (x) = 2 N N k k=0 2 N IẼN,k f N (x) f N (x) = f N (x) m L a.e. x. N=1 { f N (x) f N (x)} f N (x) = f N (x) lim N fn (x) = f(x) f(x) lim sup N fn (x) lim sup N fn (x) < + f(x) = 0 lim sup N fn (x) = + (10.4) Borel f(x) = f(x) m L a.e. x (2) f(x) g(x) m L ({f(x) g(x)}) = 0 N = {x R n f(x) g(x)} a R {x R n f(x) > a} = ({f(x) > a} N c ) ({f(x) > a} N) = ({g(x) > a} \ ({g(x) > a} N)) ({f(x) > a} N). (10.5) (10.5) 3 f(x) (2) L p C 0 (R n ) = {f : R n R f(x) R n {x R n f(x) 0} }. (10.6) 10.4 f L p (R n, m L ) (p 1) ε > 0 f ε C 0 (R n ) f f ε L p ε. 36

(3) 10.5 A R n R n v O(n) T A + v := {x + v x A}, T A := {T x x A} m L (A) = m L (A + v), m L (T A) = m L (A). (4) 10.6 (, F, m) A F m(a) = 0 A F 0 10.7 10.8 (1) m L (A) = 0 (2) A m L (A) = 0. (2) (1) (1) (2) B R n m L (A B) + m L (A c B) m L (B) m L (A B) m L (A) = 0 10.2 Carathéodory 10.3 Fubini 37