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+ |y| n . 5 f n (x, y) = n · ln 1 + | x | + |y| n . 5 f n (x, y) = exp(x + 1 n y 2 ). 5 f n (x, y) = exp(sin n x + cos n y ). %&% f n : A → R " ! _ ε > 0 ! A ε ⊂ A ! & μ (A\A ε ) < ε {f n } " A ε & ! 5 f n (x) = cos n (x), A = [0, 2π], ε = 10 −1 . 5 f n (x) = x n 1 + x n , A = [0, 1], ε = 10 −2 . !5 f n (x) = 2nx 1 + n 2 x 2 , A = [0, 1], ε = 10 −3 . 5 f n (x) = x n − x 2n , A = [0, 1], ε = 10 −4 . 5 f n (x) = n 2 |sin πx| 1 + n 2 |sin πx| , A = [−1, 1], ε = 10 −5 . 5 f n (x) = exp(n(x − 2)), A = [0, 2], ε = 10 −6 . %&& f n : R → R, f n (x) = χ [−n, n] (x) " f (x) ≡ 1 {f n } " f (x) ≡ 1 ! ! ' / 37a8" 5 c 2 78W"7W>" f n : R → R " ! ! ! _ ! & %&' f n (x) = χ[ √ n, √ n +1 ](x). %&. f n (x) = sin n x · χ [2πn, 2πn+π] (x). %'/ f n (x) = ∞ k =n χ [k, k+k −2 ] (x). 7W#"7Wb" / [0, 1] ! ' K − = & %' f (x) = x · χ [0, 1]\Q (x). %' f (x) = D(x) + R(x). %' f (x) = ⎧ ⎨ ⎩ x, x ∈ K 1 + x 2 , x ∈ [0, 1]\K . %' f (x) = ⎧ ⎨ ⎩ sin x, x ∈ [0, 1]\(K ∪ Q) 1 + x 2 , x ∈ K ∪ Q . %'$ f n (x) = χ [n, n+1] (x) " ! x ∈ R lim n →∞ f n (x) = 0 ' " ! ! " c %'% * {f n } ! " f : E → R x ∈ E g : R → R ! {ϕ n = g(f n )} " g (f) %'& * {f n } ! " f : E → R " ! ! g : R → R ! {ϕ n = g(f n )} " g (f) E & ! ! & §. "0 # * " "4 - "4 ' & ! A & ! f μ (A) < +∞ &7, ! f : A → R f \ ! y 1 , y 2 , . . . , y n ! f ! / (; k ∈ {1, 2, . . . , n} ! ! - A k = {x ∈ A : f(x) = y k }. (7.1) & 7, 7 y 1 , y 2 , . . . , y n f : A → R 5 n k =1 y k μ (A k ) f A 1 $ A f (x)dμ = n k =1 y k μ (A k ) . [ y 1 , y 2 , . . . , y n , . . . ! f : A → R f ! ∞ k =1 y k μ (A k ) (7.2) A k 38#5 & 7, ! 38%5 f A 1 38%5 f A 1 $ A f (x)dμ = ∞ n =1 y n μ (A n ) . 2 ( 38%5 * ! " " 3 Z 5 8`" ( y n / y n ! / (; & 7, A = k B k , B i B j = ∅, i = j B k f c k ! k c k μ (B k ) (7.3) f A 1 38`5 f A 1 [ f ! &$7, ! A f {f n } f A 1 lim n →∞ $ A f n (x)dμ = $ A f (x)dμ . (7.4) / ( d " ! A & ! ! f : A → R ' m M + ! x ∈ A m ≤ f(x) ≤ M + [m, M] m = y 1 < y 2 < · · · < y n −1 < y n = M n ' [y k −1 , y k ), k = 1, 2, . . . , n − 1 A k = {x ∈ A : y k −1 ≤ f(x) < y k } & A n = {x ∈ A : y n −1 ≤ f(x) ≤ y n } & ' / " & - s (f) = n k =1 y k −1 μ (A k ), S (f) = n k =1 y k μ (A k ). / s (f) ! " 3 M ·μ(A) 5 S (f) ! " 3 m · μ(A) 5 2 ! + ! - L ∗ (f) = sup s (f), L ∗ (f) = inf S (f). (7.5) 38a5 ! [m, M] ! ! ! L ∗ (f) f A & ! L ∗ (f) f A & ! &%7, ! L ∗ (f) = L ∗ (f) f A 1 L ∗ (f) L ∗ (f) f A 1 $ A f (x)dμ = L ∗ (f) = L ∗ (f). < 0 ≤ S (f) − s (f) ≤ λ n · μ(A), λ n = max 0≤k≤n−1 (y k +1 − y k ) ! ! A & ! " ! f / ( ! " ! ' / + & d ! " ! / [ ! ! A & ! f : A → R / (; \ f A & ; ( ∀x ∈ A ! f (x) ≥ 0 ' f : A → R {f n } " ! - f n (x) = ⎧ ⎨ ⎩ f (x), f (x) ≤ n n, f (x) > n ' f n A ! ! ( " ( f n (x) ≤ f n +1 (x), ∀x ∈ A, ∀n ∈ N. 2 ! 3! ! 5 + lim n →∞ $ A f n (x)dμ. (7.6) &&7, ! 3875 - f A + f A 3875 $ A f (x)dμ = lim n →∞ $ A f n (x)dμ. [ ! f : A → R A ' f A & ! f (x) = f + (x) − f − (x), f + (x) = 1 2 (|f(x)| + f(x)) ≥ 0, f − (x) = 1 2 (|f(x)| − f(x)) ≥ 0 (7.7) &'7, ! A f + f − f A $ A f + (x)dμ − $ A f − (x)dμ $ A f (x)dμ = $ A f + (x)dμ − $ A f − (x)dμ. &"" $1 σ − & 2 A A 1 , A 2 , . . . , A n , . . . Download 1.57 Mb. Do'stlaringiz bilan baham: 
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