Re: [分析] 博士班入學考題一題(Lebesgue Integral)

[math1209 (.......................)]

※ 引述《k6416337 (とある煞氣の光希)》之銘言：
: http://www.math.ccu.edu.tw/chinese/test-paper/Phd/95_function.pdf

Suppose μ is a positive measure on X, f:X→[0,∞] is measurable,

∫ f ＝ c , where 0＜c＜∞, and α is a constant. Prove that
 X

 lim   ∫  n log [ 1 ＋ (f/n)^α ] dμ ＝ c  if α＝1,
n→∞    X                                0  if 1＜α＜∞,
                                         ∞  if 0＜α＜1.
Proof.

(1) For 1≦α, we notice that n log [ 1 ＋ (f/n)^α ] ≦αf (proved later＊).
    In addition,

            lim  n log [ 1 ＋ (f/n)^α ] ＝ f if α＝1,
           n→∞                         ＝ 0 if 1＜α＜∞.

    So, by LDCT we get

            lim   ∫  n log [ 1 ＋ (f/n)^α ] dμ ＝ c  if α＝1,
           n→∞    X                                0  if 1＜α＜∞.


(2) For 0＜α＜1, we notice that n log [ 1 ＋ (f/n)^α ] →∞ as n→∞.

    Then by Fatou's lemma we get

            lim inf ∫  n log [ 1 ＋ (f/n)^α ] dμ
             n→∞    X

        ≧  ∫   lim inf n log [ 1 ＋ (f/n)^α ] dμ ＝ ∞.
              X   n→∞

    So,  lim   ∫  n log [ 1 ＋ (f/n)^α ] dμ ＝ ∞.
        n→∞    X

NOTE. (1) 出處: Real And Complex Analysis-Water Rudin, exer. 9, p. 32.
      (2) 也因為 Rudin 書上給的 hint. 我才找得到界定函數 αf.
      (3) 至於那個不等式，你就自己證明了…

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