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1 & ! & $ & A = {(x, y) : 0 ≤ x < 7, 0 ≤ y < 7} , A 1 = {(x, y) : 4 ≤ x < 6, 3 ≤ y < 5} . $ ' A = {(x, y) : 0 ≤ x ≤ 7, 0 ≤ y ≤ 7} , A 1 = {(x, y) : 3 ≤ x ≤ 7, 0 ≤ y < 7} . $ . A = {(x, y) : 0 ≤ x ≤ 7, 0 ≤ y ≤ 7} , A 1 = {(x, y) : 0 ≤ x < 7, 0 < y < 6} . aa>"a7b" $$/ A ⊂ R & ! ! ε > 0 G (G ⊃ A) ! & + μ ∗ (G\A) < ε + $$ * A ⊂ R ! & ε > 0 ! G (G ⊂ A) & & + μ ∗ (A\G) < ε " + $$ * A ⊂ B μ ∗ (A) ≤ μ ∗ (B) $$ * A − & μ ∗ (A) = m (A) $$ * ! {A n } & ! A ⊂ n A n μ ∗ (A) ≤ n μ ∗ (A n ) . $$$ * A ⊂ [0, 1] ! & [0, 1]\A ! $$% * A B & ! A ∪B, A∩B, AΔB, A\B & ! $$& . ! & U(R) $$' d ! & ! & $$. * A B ! & μ (A ∪ B) = μ(A) + μ(B) $%/ * A B ! & - 5 μ (A ∪ B) = μ(A) + μ(B) − μ(A ∩ B), 5 μ (AΔB) = μ(A) + μ(B) − 2μ(A ∩ B). $% A B & ! - |μ ∗ (A) − μ ∗ (B)| ≤ μ ∗ (AΔB). $% A ⊂ E = [0, 1] ! & ! μ (E\A) = 1 − μ(A) $% 2 ! & ! & $% . ! & U(R) σ − $%$ . ! σ − & " ! &- 5 A = ∞ n =1 [n, n + 2 −n ) ; 5 A = ∞ n =1 n − 2 3 n , n + 2 3 n ; !5 A = ∞ n =1 2 −n , 2 1−n 4 5 A = ∞ n =1 1 (n + 1)! , 1 n ! $%% . ! + & ! &- 5 A = ∞ n =1 (1 + 1 n ) n , 11 − (1 + 1 n ) n ; 5 B = ∞ n =0 1 − 8 2 n , 4 + 16 2 n ; !5 A = ∞ n =1 1 − (1 + 1 n ) n , 5 + (1 + 1 n ) n ; 5 B = ∞ n =0 1 − n n + 1 , 4 + n n + 1 . $%& d ! & a7W"a8a" aW" + $%' . ! & ! c $%. . ! & ! c $&/ . ! & ! c $& A ⊂ E = [0, 10] ! & B ⊂ E ! & , ! c A ⊂ B $& A ⊂ E ! & B ⊂ E ! & A ∩ B & ! c A ∩ B = ∅ B ⊂ A A ⊂ B $& A ⊂ E ! & B ⊂ E ! & A \B & ! c A ⊂ B B ⊂ A $& A ⊂ E ! & B ⊂ E ! & B \A & ! ! ! A ∩ B = ∅ B ⊂ A A ⊂ B $&$ A ⊂ E ! & B ⊂ E ! & A ΔB & ! c A ∩ B = ∅ B ⊂ A A ⊂ B $&% A 1 ⊃ A 2 ⊃ · · · ⊃ A n ⊃ · · · ! & " ! μ ( ∞ ∩ n =1 A n ) = lim n →∞ μ (A n ) $&& . ! & A 1 ⊂ A 2 ⊂ · · · ⊂ A n ⊂ · · · " ! μ ( ∞ ∪ n =1 A n ) = lim n →∞ μ (A n ) $&' * U(R)− / ( ! & " [a, b] [a, b] ! ! & σ − $&. = & ! $'/ A ⊂ R & ! μ (A) = 0 $' A B ! & A ∪ B A ΔB ! & $' A ⊂ R ! & B ⊂ R & ! μ (A ∩ B ) = 0 $' 2 ! ! & σ δ $' 2 ! ! & && $'$ 2 ! & $'% [0, 1] ! > $ [0, 1] ! ! & $'& [0, 1] ! # [0, 1] ! ! & + c $'' [0, 1] ! [0, 1] ! " & $'. A ⊂ [0, 1] ! ! ! & & A ! & c $./ 2 ! A ⊂ R B ⊂ R & A + B = {c : c = a + b, a ∈ A, b ∈ B} ! $. 2 A ⊂ [0, 1] B ⊂ [0, 1] & + - μ (A) = μ(B) = 0, μ (A + B) = 2. $. F (x) = 2x + 1 μ F − / "2 ! ! $. μ F , F (x) = 2x + 1 ! ! A = (1, 5] & ! & $. μ F , F (x) = [x] / "2 ! ! A = (1, 5] ∪ {7, 8} & ! & < = & −K 3`a" 5 ) 3a8"! 5 = & K R ! - K(x) = 0, x ∈ (−∞, 0] K(x) = 1, x ∈ [1, ∞) K [0, 1]\K ! K 1 = 1 3 , 2 3 & ! 3a8"! 5 K(x) = 1 2 , x ∈ 1 3 , 2 3 . K 2 = K 21 ∪ K 22 = 1 9 , 2 9 7 9 , 8 9 & ! K(x) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 4 , agar x ∈ 1 9 , 2 9 , 3 4 , agar x ∈ 7 9 , 8 9 . [ K 3 = 4 k =1 K 3k = 1 3 3 , 2 3 3 7 3 3 , 8 3 3 19 3 3 , 20 3 3 25 3 3 , 26 3 3 & ! K(x) = 2k − 1 2 3 , x ∈ K 3k , k = 1, 2, 3, 4. ')# K n = 2 n−1 k =1 K nk & k − " ! K(x) = 2k − 1 2 n , x ∈ K nk , k = 1, 2, 3, . . . , 2 n −1 . 2 K n & ! K " ' ∞ ∪ n =1 K n = [0, 1]\K & [0, 1] ! [ x 0 ∈ K K K (x 0 ) = sup K(x) : x < x 0 , x ∈ ∞ ∪ n =1 K n ) $.$ = & K [0, 1] " $.% = & F (x) = K (x) / " 2 ! −μ K 5 μ K ! 5 μ K (K) = 1 ' K − = & !5 A = & ! (K ⊂ A) & μ K (A) = 1 $.& 5 F (x) = x / "2 ! ! c 5 F (x) = 2[x] + 1 / "2 ! ! c !5 2 / "2 ! $.' * σ − ! $.. A ⊂ R & m (A) = n ∈N∩A 1 2 n m & ! A = (−∞, 0) B = [1, 4] & ! & - 0 # - " * < ! + # A ΔB = (A ∪ B)\(A ∩ B), % A ΔB = (A ∪ B)Δ(A ∩ B), ` A Δ B = (A\B) ∪ (B\A). *5 # % '5 % ` d5 # % ` \5 # ` E − & & #5 E \ ∪ α A α = ∩ α (E\A α ), %5 E \ ∩ α A α = ∪ α (E\A α ) `5 A ΔB = (A ∪ B)\(A ∩ B), b5 (E\A)Δ(E\B) = AΔB *5 # % '5 % ` d5 ` b \5 # b |
