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&$ [−1, 1] Z ( ! " Z ( ! 8$" ! f &$ 38#%5 f ϕ (x) ≡ 1 [−1, 1] , [−1, 1] " / Z ( ! &$$ * f A & ! f A & ! A ! &$ * + A | f(x) | dμ = 0 ! x ∈ A ! f (x) = 0 &$$ 1 * f A (μ(A) < ∞) & ! n ∈ N ! m ∈ N + μ (D) < m −1 ! ! D ⊂ A & ! ## ## $ D f (x) dμ ## ## < 1 n + &$% [0, 1] ! / ( ! &$& f (x) = [x 2 ] A = [0, 2] & ! / &$' [a, b] c &$. \ Z c , [0, 4] & ! / 87>"87a" f : A → R ( X Z / 8b" &%/ f (x) = 2x + 1, x ∈ [0, 3]. &% f (x) = 6x − 3, x ∈ [−1, 2]. &% f (x) = 3x 2 − 2x, x ∈ [−1, 1]. &% f (x) = 6x 2 + 4x − 5, x ∈ [−1, 1]. &% f (x) = 2 x + 3, x ∈ [0, 2]. &%$ f (x) = e x + 3x, x ∈ [0, 1]. &%% < 5 + [−3, 3] sign(cos πx)dμ; 5 + (0, 1] sign(sin π x )dμ; !5 ++ [0, 2]×[0, 2] [x + y]dμ; 5 ++ x ≤y≤4 " [y − x]dμ. 878" / (; ' D− \ R− Z K− = &%& 5 + [0, 1] x · χ [0, 1]\Q (x)dμ; 5 + [0, 2] (1 + 2x) dμ; !5 + [0, 2] 3x 2 + 1 dμ ; 5 + [0, 1] (2 x + 2) dμ; 5 + [0, 1] (ln 3 + e x ) dμ; 5 + [0, 1] K(x) dμ; 5 + [0, 1] x (1 − D(x))dμ; 5 + [0, 1] x (1 − R(x))dμ; 5 + [0, 1] (x + K(x)) dμ; +5 + [0, 1] x · K(x) dμ; 5 + [0, 1] x 2 · R(x) dμ. &%' n ∈ N ! f n : [0, 1] → R ! f n x ∈ [0, 1] x ! n − f 2 (0, 10010 . . .) = 0, f 3 (0, 10110 . . .) = 1 ' " ! - $ [0,1] f n (x)f m (x)dμ = 1 4 , n = m, $ [0,1] (f n (x)) 2 dμ = 1 2 . &%. n ∈ N ! g n : [0, 1] → R ! * x ∈ [0, 1] ! n − # g n (x) = 1 n − > g n (x) = −1 ' " ! - $ [0,1] g n (x)g m (x)dμ = 0, n = m, $ [0,1] (g n (x)) 2 dμ = 1. ' § " "4 - "4 "4 4 " & ! ! ' ! lim n →∞ $ A f n (x)dμ = $ A lim n →∞ f n (x)dμ := $ A f (x)dμ (8.1) ( c " ( 2 ! ! 3W#5 + * {f n } ! " " A & f " 3W#5 , ( '" ! ' [0, π] " f n (x) = ⎧ ⎨ ⎩ n sin nx, x ∈ , 0, π n 0, x ∈ , π n , π - . (8.2) ' " ' " ! 3W#5 c ! "# x ∈ [0, π] ! lim n →∞ f n (x) = 0 " [ f n [0, π] ! - $ π 0 f n (x)dμ = n $ π n 0 sin nxdμ = 2. ! lim n →∞ $ π 0 f n (x)dμ = 2 = $ π 0 θ (x)dμ = 0. \ " ! < '"" $1 & ! {f n } A f n ∈ N |f n (x) | ≤ ϕ (x) ϕ A f A lim n →∞ $ A f n (x)dμ = $ A f (x)dμ. '-5 ! | f n (x) | ≤ M = const x ∈ A lim n →∞ f n (x) = f (x) lim n →∞ $ A f n (x)dμ = $ A f (x)dμ. 0 ! & " ( ! W#" {f n } " f | f(x) | ≤ ϕ(x) ! x ! + ' "" $1 & A f 1 (x) ≤ f 2 (x) ≤ · · · ≤ f n (x) ≤ · · · , {f n } n ∈ N $ A f n (x)dμ ≤ K 5 A lim n →∞ f n (x) = f(x) f A lim n →∞ $ A f n (x) dμ = $ A f (x) dμ. ' -5 ! ψ n (x) ≥ 0 ∞ n =1 $ A ψ n (x)dμ < +∞ A ∞ n =1 ψ n (x) $ A ∞ n =1 ψ n (x) dμ = ∞ n =1 $ A ψ n (x) dμ . ' "" $ & ! - {f n } A f $ A f n (x) dμ ≤ K f A $ A f (x) dμ ≤ K 2 & ! ! ! (μ(A) < ∞) & / / & ! ! ! & R = (−∞, ∞) / ' X & ! ! X n & ! '7, ! X μ X X μ σ − σ − ! ! / ! ' 7, ! {X n } (X n ⊂ X n +1 ) #5 ∞ n =1 X n = X, %5 n ∈ N μ (X n ) < ∞, {X n } X ' 7, X σ − μ X - f ! f A ⊂ X {X n } lim n →∞ $ X n f (x)dμ f X $ X f (x)dμ = lim n →∞ $ X n f (x)dμ f X 1 [ f , ; " ( f (x) = f + (x) − f − (x) f + f − 3885 ' 7, ! 3885 f + f − - X f X $ X f (x)dμ = $ X f + (x)dμ − $ X f − (x)dμ. / Z * [a, b] f Z ( ! f [a, b] / ( ! 38b" 5 * f [0, 1] ( Z ! " ! / ( ! $ 1 0 sin 1 t dt t (8.3) Z ( + 3 ! 5 ! $ 1 0 sin 1 Download 1.57 Mb. Do'stlaringiz bilan baham: 
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