[分析] 兩題實變

[k6416337 (とある煞氣の光希)]

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

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律:知道嗎?聽說我們的歌被海外的電視台所錄用耶!看來我們離武道館不遠了
唯:真的嗎?那真的是太好了，我一直夢想能在武道館彈著吉太，好高興
釉:小唯能高興真的是太好了，呵呵~
澪:拜託!那個明明是盜用不是錄用，你們怎麼還這麼高興?
律、唯、釉:啊?什麼?                                         輕音部
澪:絕望啦!我對盜用錄用分不清楚的輕音部社員們絕望啦!           邁向武道館之路

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◆ From: 140.113.178.13
推 Dexterity   :第一題起頭重點應該是在f is integrable function
→ Dexterity   :所以∫f < ∞
→ Dexterity   :=> f < ∞ a.e on E
→ Dexterity   :再將E分成{f<∞}和{f=∞}討論就好了
推 Dexterity   :第二題的(a) 有個定理是∫f=0 <=> f=0 a.e on E
→ Dexterity   :套用一下就好了 至於d is a metric 那高微應該有做過
推 Dexterity   :(b)感覺應該可以從定義去處理
→ Dexterity   :(=>)的方向 會用到(a)的結果