Re: [分析] 兩題實分析

[yutzu903 ()]

※ 引述《wyob (Go Dolphins)》之銘言：
:                                                 ∞
:  (2)f is integrable over [1,∞) if and only if Σ  An converges absolutely?
:                                                 n=1
: 2.Let f be a nonnegative measurable function on R.Show that
:       n
:  lim ∫ f=∫f
: n→∞ -n   R


   Let fn=f1_[-n,n].

   Since f is nonnegative, fn >=0 and is increaing to f  a.s.

   by monotone conv. thereom
          n
    lim  ∫ f = lim  ∫ fn = ∫ lim   fn   = ∫ f
   n→∞  -n    n→∞           n→∞         R

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