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m $"" ! S m m M(S m ) $ S m & m $ 7, ! S m m A 1 , A 2 , . . . , A n , . . . ∈ S m ∞ ∪ k =1 A k ∈ S m m ∞ k =1 A k = ∞ k =1 m (A k ) m σ − 8 # * * a b 2 a ≤ x ≤ b, a ≤ x < b, a < x ≤ b, a < x < b & ' & " ! + 3 a > b 5 ( ∅ & S ! $ 2 ! − S S [a, b], [a, b), (a, b], (a, b) " S a, b " 3 & ! ! 5 P = P ab S [a, a) = ∅ & S & " & ( P ab P cd & P ab ∩ P cd = P nm , n = max {a, c}, m = min {b, d} 3a#" ! 5 '"# * P ab ∈ S, P cd ∈ S P cd ⊂ P ab P ab \P cd = P ac ∪ P db 3a%"! 5 \ S '&# S a, b 3 & ! ! 5 P = P ab ! m (P ) = b−a P & m (P ) = 0 , m : S → R & a#" ( ( m : S → R ! ' ! - P = n ∪ k =1 P k , P i ∩ P k = ∅, i = k, P, P k ∈ S m (P ) = n k =1 m (P k ) M(S) S b#" M(S) A " P k ∈ S ( A = n k =1 P k M(S) [ M(S) & ( & ! ! A = n k =1 P k ∈ M(S) & m (A) = n k =1 m (P k ) (5.1) ! m : M(S) → R m (A) A & " "4 # * \ E = [0, 1] ! &" ! $ 7, # A ⊂ E μ ∗ (A) = inf A ⊂∪ k P k k m (P k ) (5.2) A 3a%5 ! A & & ! ! ! ! 2 ( A ⊂ E & ! + d ; m (P k ) ; ! ! < ! & ! [ A ⊂ E & (; $$7, ! ε > 0 B ∈ M(S) μ ∗ (AΔB) < ε A 1 * A / ( ! & ! ! ! & U(E) " μ ∗ & 1 μ 2 ! & U(E) / ! μ \ A ∈ U(E) ! μ (A) = μ ∗ (A). ' E = [0, 1] ! & ' ! ( R E n = [n, n + 1), n ∈ Z ( R = n ∈Z E n . $%7, ! n ∈ Z A n = A ∩ E n A ! A μ (A) = n ∈Z μ (A n ), (5.3) A 1 * 3a`5 ! A * A 2 ! μ ! ! $ A = 8 n =1 1 n − 1 8 , 1 n + 1 8 & , P 1 , P 2 , . . . , P n " A = n ∪ k =1 P k A &" ! & ! "# P n = 1 n − 1 8 , 1 n + 1 8 , n = 1, 2, . . . , 8 P n P 1 = 7 8 , 9 8 , P 2 = 3 8 , 5 8 , P 3 = 5 24 , 11 24 , . . . , P 8 = 0, 1 4 . a`"! ( P 1 ∩P 2 = ∅, P k ∩P k +1 = ∅ , k = 2, 3, . . . , 8 2 ! A = P 1 ∪ Q 1 , Q 1 = 8 n =2 P n = 0, 5 8 , P 1 , Q 1 ∈ S \ A & ' " - μ (A) = μ(P 1 ) + μ(Q 1 ) = 2 8 + 5 8 = 7 8 . '$# 6""- # * [ & / ! * a, b, c d 1 \ a ≤ x ≤ b, a ≤ x < b, a < x ≤ b, a < x < b c ≤ y ≤ d, c ≤ y < d, c < y ≤ d, c < y < d + & " ' & ! + 3 " a > b c > d 5 S 2 ! ! 2 A ⊂ E 2 = [0, 1] × [0, 1] & 1 (: $&7, # A ⊂ E 2 μ ∗ (A) = inf A ⊂∪ k P k k m (P k ) (5.4) A ' ! A & & ! ! " ! ! ! 1 ! ! −S 2 " 3a%b"a%a 5 S 2 a, b, c, d 3 ! & ! 5 P = P abcd ! ! m (P ) = (b − a)(d − c) P & m (P ) = 0 ' " ! m : S 2 → R & ! 3a#" ( 5 M(S 2 ) S 2 M(S 2 ) $'7, ! ε > 0 B ∈ M(S 2 ) μ ∗ (AΔB) < ε A 1 ! & U(E 2 ) μ ∗ & " 1 μ 2 ! & U(E 2 ) / ! μ \ " A ∈ U(E 2 ) ! μ (A) = μ ∗ (A). ( R 2 E mn = {(x, y) : m < x ≤ m + 1, n < y ≤ n + 1} , n, m ∈ Z - R 2 = m, n ∈Z E mn . $.7, A ⊂ R 2 ! m, n A mn = A ∩ E mn A ! A μ (A) = m, n ∈Z μ (A mn ), (5.5) A 1 * 3aa5 ! A ⊂ R 2 & * A & $ " " $2 σ − & ! {A n } − μ ∞ n =1 A n = ∞ n =1 μ (A n ) $ "" $2 & ! A 1 ⊃ A 2 ⊃ · · · ⊃ A n ⊃ · · · A = ∞ ∩ n =1 A n μ (A) = lim n →∞ μ (A n ) $-5 ! A 1 ⊂ A 2 ⊂ · · · ⊂ A n ⊂ · · · A = ∞ ∪ n =1 A n μ (A) = lim n →∞ μ (A n ) * 3a%5 ! A ⊂ R & & ! " ! B & ! A & X ( j ∗ (A) ( j ∗ (A) = inf B ⊃A m (B), B ∈ M(S). , j ∗ (A) = sup B ⊂A m (B), B ∈ M(S), A & X ( $/7, ! j ∗ (A) = j ∗ (A) A 3 2 ( + Download 1.57 Mb. Do'stlaringiz bilan baham: 
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