Re: [分析] 兩題實變

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

※ 引述《k6416337 (とある煞氣の光希)》之銘言：
: 1.Prove that for all ε>0,there exists δ>0 s.t. if E is measurable and |E|<δ,
:   then
:                         ∫f<ε,
:                          E
:   where f is a nonnegative,measurable and integrable function.
: 2.Suppose that Eㄈ|R^n is measurable and |E|<∞.For f,g屬於L(E),define
:                                      |f-g|
:                          d(f,g)=∫-----------=d(g,f).
:                                  E  1+|f-g|
: (a)Show that d(f,g)=0 <=> f=g almost everywhere on E and d(f,g)≦d(f,h)+d(h,g)
:    for f,g,h屬於L(E).That is d is a metric on L(E).
: (b)We say f_k → f in (L(E),d) if f_k,f屬於L(E) and d(f_k,f)→0.
:    Show that f_k → f in (L(E),d) <=> f_k → f in measure.
: 這兩題我怎麼想都沒頭緒，也不太知道該用什麼定理來證
: 請知道的神人能夠指引一下，謝謝

(1) 這應該是很標準的定理 (絕對連續). (略過…)
(2) (a) 略過…
    (b) 你找你要的就好… (我剪貼一下我以前寫的, 懶得改了).

(1) In a finite measure pace X, a seqence {f_n} of a.e. real-valued measurable
    functions is convergent in measure to zero iff

        |f_n|
   ∫ ---------- dμ → 0 as n → ∞.
      1 ＋ |f_n|

                                         _
(2) Denote by Z the space of all classes f of a.e. real-valued measurable
    functions f on a finite measure space (X,Ａ,μ), with
    _    _
    f ＝ g iff f ＝ g a.e. Define, on Z,

      _  _                      |f－g|
   ρ(f, g) ＝ ρ(f,g) ＝ ∫ ----------- dμ.
                             1 ＋ |f－g|

    Prove that Z is a complete metric space with metric ρ.

Proof.

(1) (＝＞) Assume that f_n → 0 in measure, then |f_n| → 0 in measure.
           So, 1＋|f_n| → 1 in measure. Since m(X) ＜∞, we know that

           |f_n|/(1＋|f_n|) → 0 in measure.

           Since |f_n|/(1＋|f_n|)≦ 1 in L(X), then by LDCT, we know that

           ∫ |f_n|/(1＋|f_n|) dμ → 0 as n → ∞.

    (＜＝) Assume that ∫ |f_n|/(1＋|f_n|) dμ → 0 as n → ∞, we will
           show that f_n → 0 in measure as follows. Consider given ε＞0,
           and {x: |f_n| ≧ε}：＝ E_n. We will show that μ(E_n) → 0 as
           n → ∞.

           Note that for x in E_n, we have |f_n|/(1＋|f_n|) ≧ ε/(1＋ε).
           Hence,

          ∫_X |f_n|/(1＋|f_n|) dμ ≧ ∫_E_n |f_n|/(1＋|f_n|) dμ

                                    ≧ ∫_E_n ε/(1＋ε) dμ

                                    ＝ {ε/(1＋ε)} μ(E_n)

          Since ∫ |f_n|/(1＋|f_n|) dμ → 0, we get μ(E_n) → 0. □

NOTE. 上述習題等價於

 f_n → f in measure ＜＝＞ ∫ |f_n－f|/(1＋|f_n－f|) dμ → 0 as n → ∞.

(2) Under the assumption m(X)＜∞, we have the following diagram.

f_n → f in measure    ＜＝＞ ∫ |f_n－f|/(1＋|f_n－f|) dμ→0 as n →∞.
         ︿                                      ︿
         ||                                      ||
         ﹀                                      ﹀
f_n: Cauchy in measure ＜＝＞∫ |f_n－f_m|/(1＋|f_n－f_m|) dμ→0 as n,m →∞.

   So, it is clear that (X,ρ) is complete.


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→ math1209    :?? 只要分母不是 0 就行了...
→ math1209    :關於這一些, 你大可看 Zygmund 習題, 他應該有要求.
→ math1209    :你要的話, 我就把我以前寫的弄一份給你看.
推 k6416337    :習題是寫g_k->g而已，點收斂
→ math1209    :???
→ math1209    :不懂你的意思...
→ k6416337    :我在BS2的P_Apostol看到這篇文，你是版主嗎?
→ math1209    :嗯...
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