Re: [分析] 兩題實變

[smartlwj (我要征服實變)]

※ 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.

since f is nonnegative, then for all ε> 0

there is measurable simple function φ_k(x) such that φ_k(x)≦f(x) , x \in E

and limφ_k(x)=f(x).  By MCT => |∫f(x)-φ_k(x)| < ε/2

Assume that |φ_k(x)|≦M (M \in R), taking δ =  ε/(2M)

then for |E| < δ , we have

|∫ f| = |∫f -∫φ_k + ∫φ_k|
  E        E    E        E
      ≦ |∫f -∫φ_k | + |∫φ_k|
           E    E           E
      <  ε/2 + M|E|

      =ε

: 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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