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dt t = $ ∞ 1 sin x x dx \ ! 3 f (x) = sin x x ! 5 f (t) = sin 1 t · 1 t (0, 1) / ( ! / ( ! , $ 1 0 ## ##sin 1 t ## ## dt t + ' ! $ 1 0 ## ##sin 1 t ## ## dt t = $ ∞ 1 |sin x| x dx ≥ $ ∞ 1 sin 2 x x dx = $ ∞ 1 1 − cos 2x 2x dx = = $ ∞ 1 dx 2x − $ ∞ 1 cos 2x 2x dx . $ ∞ 1 cos 2x x dx ! '! ! \ $ 1 0 ## ##sin 1 t ## ## dt t ! 2 ( ; Z ( ! / ( ! ' "" ! A 7 y 1 , y 2 , . . . , y n f : A → R A A k = {x ∈ A : f(x) = y k }, k = 1, 2, . . . , n 9 B ⊂ A χ B (x) μ (B) < ∞ '$7, ! A f : A → R y 1 , y 2 , . . . , y n , . . . ! y k A k = {x ∈ A : f(x) = y k } ∞ n =1 y n μ (A n ) f : A → R A 1 ' < R / ( ! c 5 f (x) = ∞ n =1 (−1) n n χ [n, n+1] (x); 5 f (x) = ∞ n =1 sin n n χ [n, n+1] (x); !5 f (x) = ∞ n =1 (−1) n n 2 χ [n 2 , (n+1) 2 ) (x); 5 f (x) = ∞ n =1 cos n · χ[ √ n, √ n +1 )(x); 5 f (x) = ∞ n =1 n 2 (n + 1)! χ [n, n+1) (x); 5 f (x) = ∞ n =1 n 2 2 n χ [n, n+1) (x). ! "# 5 ! ' " A 0 = (−∞, 1) & f (x) = 0 = y 0 f (x) = n 2 2 n , x ∈ A n = [n, n + 1) Wa" ( f : R → R ! ! ∞ n =1 y n μ (A n ) = ∞ n =1 n 2 2 n · 1 ! \ 3 q = 0, 5 5 ! ! \ f R ! ' A = ∞ n =1 [n, n + n −α ) & f (x) = χ A (x) & α R ! c ! "# Wb" f (x) = χ A (x) ! ! A & ! ! A & ! ! σ − μ (A) = ∞ n =1 1 n α ( & α # ! ! \ ! α ∈ (1, ∞) A & χ A R ! ' < " ! 5 lim n →∞ + [0, 1] exp −n x 2 dμ ; 5 lim n →∞ + [0, 1] exp − x 2 n dμ ; !5 lim n →∞ + R sin n x 1 + x 2 dμ ; 5 lim n →∞ + [0, π] exp(− cos n x ) dμ; 5 lim n →∞ + [0, ∞] n exp − x n − 1 dμ 1 + x 4 . ! "# 5 ! / ' f n (x) = exp(−nx 2 ) " [0, 1] ! 3 5 θ (x) ≡ 0 ! ϕ : [0, 1] → R ϕ (x) ≡ 1 , ! x ∈ [0, 1] n ∈ N ! |f n (x)| ≤ ϕ (x) + / + 1 lim n →∞ $ [0, 1] exp(−nx 2 ) dμ = $ [0, 1] θ (x) dμ = 0. '$ f (x) = 1 1 + [x] 2 , x ∈ A = [0, ∞) A ! c ! "# 2 (; f : A → R " A n = [n, n + 1), n = 0, 1, . . . & y n = 1 1 + n 2 ∞ n =0 1 1 + n 2 · 1 ! \ Wa" ( f (x) = 1 1 + [x] 2 A = [0, ∞) ! () * +, * *+- + # - '% < α > 0 & R ! c 5 f (x) = ∞ n =1 (−1) n n α χ [n, n+1] (x); 5 f (x) = ∞ n =1 sin n n α χ [n, n+1) (x); !5 f (x) = ∞ n =1 (−1) n n α χ [n 2 , (n+1) 2 ) (x). '& α f n (x) = nx α nx 2 + 1 , x ∈ [0, 1], " / " c '' < {g n } " / c g n (x) = nx 3 2 nx 2 + 1 , x ∈ [0, 1]. '. + lim n →∞ $ A f n (x)dμ = $ A f (x)dμ c . '/ 5 lim n →∞ + R 2 exp − x 2 + y 2 cos 1 n x · y dx dy ; 5 lim n →∞ + R n (1 + x 4 ) · sin | x | n dμ. ' * ; ( Z ( " ! / ( ! ' 3W`5 ! ' f (x) = 1 1 + [x 2 ] , x ∈ A = [0, ∞) A ! c ' f 1 (x) = x 1 x 2 1 + x 2 2 , f 2 (x) = x 2 x 2 1 + x 2 2 , f + 1 (x) = | x 1 | x 2 1 + x 2 2 , f + 2 (x) = | x 2 | x 2 1 + x 2 2 U 0 (1) = {(x 1 , x 2 ) ∈ R 2 : x 2 1 + x 2 2 ≤ 1} & " ! '$ f i (x) = sin x i 2 − cos x 1 − cos x 2 , i = 1, 2 U 0 (1) = {(x 1 , x 2 ) ∈ R 2 : x 2 1 + x 2 2 ≤ 1} & ! '% f (x) = 1 x 2 1 + x 2 2 + x 2 3 f i (x) = |x i | x 2 1 + x 2 2 + x 2 3 , i = 1, 2, 3 B 0 (1) = {(x 1 , x 2 , x 3 ) ∈ R 3 : x 2 1 + x 2 2 + x 2 3 ≤ 1} & " ! , B 0 (1) & ! '& f (x) = 1 3 − cos x 1 − cos x 2 − cos x 3 B 0 (1) & " ! . § -- * +4 # "4-4- ,-0) $ #>"&: / ' f & * f X ⊂ R ! & " ! $ A f (x) dμ (9.1) ! ! A ⊂ X & ! + f & ' 1 X ' A & X 3$#5 ! A = [a, b] ! & ! $ [a,x] f (t)dμ (9.2) ' ( : ' ! " : -- ,-0) \ ! ! " (: .7, [a, b] f x 1 , x 2 x 1 < x 2 f (x 1 ) ≤ f(x 2 ) (f(x 1 ) ≥ f(x 2 )) f [a, b] $ & . 7, [a, b] f x 1 , x 2 x 1 < x 2 f (x 1 ) < f(x 2 ) (f(x 1 ) > f(x 2 )) (9.3) f [a, b] $ & , ! $# $%" (: '( ! + ! ! ! ! " ! ( / 3$%5 " * Download 1.57 Mb. Do'stlaringiz bilan baham: 
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