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X ( ! & / ( ! & ! $ / ( ! X ( ! & ! "# E = [0, 1] A [0, 1] ! " & A & E ! ! A ! & E 2 ! j ∗ (A) = 1 A & ! B & ! & 2 " ! j ∗ (A) = 0 ' j ∗ (A) = j ∗ (A). \ A & X ( ! ( A & " ! {x 1 , x 2 , . . . , x n , . . . } " ! 2 A ⊂ ∞ ∪ k =1 P k , P k = {x : x k ≤ x ≤ x k } = [x k , x k ]. ! k ∈ N ! m (P k ) = 0. ' μ ∗ (A) = 0 ! 2 ( ! & ! & ' ! & B = ∅ - μ ∗ (AΔB) = μ ∗ (AΔ∅) = μ ∗ (A) = 0 < ε. \ A / ( ! & 2 A / (" ! X ( ! & $ A ⊂ [0, 1] − & ! " a μ (A) & ! "# A & ! ! & [0, 1]\A & ! a [0, 1] ! [0, 5; 0, 6) A 1 " A 1 ! " ! a < $ ! [0, 15; 0, 16) ∪ [0, 25; 0, 26) ∪ · · · ∪ [0, 95; 0, 96) = A 2 A 2 & ! ! a + ! A 1 , A 2 , . . . , A n , . . . & + "+ [0, 1]\A . ! σ − μ ([0, 1]\A) = ∞ n =1 μ (A n ) μ (A 1 ) = 10 −1 , μ (A 2 ) = 9 · 10 −2 , μ (A n ) = 9 n −1 · 10 −n , . . . \ μ ([0, 1]\A) = ∞ n =1 μ (A n ) = ∞ n =1 9 n −1 10 n = 0, 1 1 − 0, 9 = 1. 2 μ (A) + μ([0, 1]\A) = 1 μ (A) = 0 $$ 2 {A n } )' & & ) " + - μ (A n ) = +∞, A n ⊃ A n +1 , n ≥ 1, μ ∞ n =1 A n = 0. ! "# )' & & ) A n = [n, ∞) , n ∈ N 0 + n ∈ N ! μ (A n ) = +∞ A n ⊃ A n +1 " + = ∞ n =1 A n = ∅ ! μ ∞ n =1 A n = 0 < A 1 ⊂ A ! A A 1 & A \A 1 & P 1 , P 2 , . . . , P n ! A \A 1 = n k =1 P k A \A 1 & ! & $% A = {(x, y) : 0 ≤ x ≤ 7, 0 ≤ y ≤ 7} , A 1 = {(x, y) : 4 ≤ x < 7, 3 ≤ y ≤ 7} . ! "# 1 A A 1 & ! ab" ! A \A 1 = P 1 ∪ P 2 ∪ P 3 '%# ' P 1 = [0, 4)×[0, 7), P 2 = [4, 7)×[0, 3], P 3 = {7}×[0, 7]. ' ! . ! μ (A\A 1 ) = μ(P 1 ) + μ(P 2 ) + μ(P 3 ) = 28 + 9 + 0 = 37. $& . ! σ − A = ∞ n =1 1 (n + 1)! , 1 n ! & ! & ! "# A n 1 (n + 1)! , 1 n ! & A n & + "+ ! ! σ − μ (A) = ∞ n =1 μ (A n ) A n & ! μ (A n ) = 1 n ! − 1 (n − 1)! " μ (A) = ∞ n =1 μ (A n ) = 1 1! − 1 2! + 1 2! − 1 3! + · · · + 1 n ! − 1 (n − 1)! + · · · = 1 ) ' " F ' F - m ((a, b )) = F (b − 0) − F (a), m ([a, b]) = F (b) − F (a − 0), m ((a, b]) = F (b) − F (a), m ([a, b)) = F (b − 0) − F (a − 0). Z m & ; _ ! μ F , " μ F ! ! & ; F * R " F ! ! & & " μ F ! ! & $7, + F μ F 1 ( ' / ! μ / "2 ! μ F $ 7, ! μ (A) = 0 μ F (A) = 0 μ F $ 7, ! μ F F μ F $ 7, ! μ F 1 A μ F (R\A) = 0 μ F # *--4 " "4 )# * ' " ! / ! * S m m ! σ − m S m M(S m ) ( ' / ! " ' S m σ − m ! E & S m E ! & & A(E) ! ! μ ∗ & " $$7, # A ⊂ E μ ∗ (A) = inf n m (B n ) A + A {B n } , B n ∈ S m $ "" ! A A 1 , A 2 , . . . , A n , . . . A ⊂ ∞ ∪ n =1 A n μ ∗ (A) ≤ ∞ n =1 μ ∗ (A n ). $%7, ! A ⊂ E ε > 0 B ∈ M(S m ) μ ∗ (AΔB) < ε A $1 & ! & U(E) μ ∗ & " 1 μ ; Z S m M(S m ) & ! ' A ∈ S m B ∈ M(S m ) μ (A) = m(A), μ (B) = m (B). * A ! & μ ∗ (AΔB) < ε ! B ∈ M(S m ) & A ΔB = (E\A)Δ(E\B) A ! & E \A ! ! $$"" 2 U(E) $ " S m " E ! & U(E) ! ! ! & U(E) $%"" 2 U(E) μ $&"" 2 U(E) μ σ − $'"" 1 U(E) E σ − $&7, 2 U(E) U(E) μ ∗ μ m μ = L(m) 1 $."" # m 1 U(E) δ− ( A 1 , A 2 , . . . , A n , . . . A = ∞ ∪ n =1 A n μ n ∪ k =1 A k n / ! $'7, ! μ (A) = 0 A ⊂ A A μ * μ ! A & ! μ (A ) = 0 ! / Download 1.57 Mb. Do'stlaringiz bilan baham: 
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