[分析] 請教幾題

[kemowu (小展)]

                                               r
1. If (X,M,μ) is a measure space and if f in L (X,M,μ) for some 0＜r＜∞.

                    p
Show that lim ∫ |f| dμ = μ{x in X | f(x)≠0}.
         p→0  X
                                       n
2. Let f in C[0,1] and let f_n(x) = f(x ), n = 1,2,.... Assume {f_n} is

equicontinuous on [0,1], show that f ≡ constant.

            1                          ∞
3. If f in L (-∞,∞), show that lim   ∫f(x)cos(nx)dx = 0.
                                n→∞ -∞

4.Suppose f, f_n are of bounded variationon [a,b], n = 1,2,..., and

V(f_n-f;a,b)→0, where V(f;a,b) is the total variation of f on [a,b]. Prove

that there is a subsequence n_k→∞ such that f'  → f' a.e..
                                               n_k

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◆ From: 140.115.221.107
→ VFresh      :第四題用推文的...
→ VFresh      :V(fn-f) 這個東西...可以表示成微分積分的型式
→ VFresh      :沒給絕對連續...所以是一個不等號
→ VFresh      :考慮 ∫|f'-fn'| -> 0
→ VFresh      :因此這邊可以考慮 convergent in measure
→ VFresh      :在 finite measure set 上面 有子序列收斂
推 ppia        :＠＠ V(f_n-f_m;a,b)≦V(f_n-f;a,b)+V(f-f_m;a,b)
→ ppia        :故存在{f_n}子序列{h_n}使得 V(h_n-f;a,b)<1/2^n
→ ppia        :上行打錯
→ ppia        :故有{f_n}子序列{h_n}使得 V(h_{n+1}-h_n;a,b)<1/2^n
→ ppia        :令g_n = h_{n+1}-h_n, 由Fubini逐項微分定理知:
→ ppia        :Σ(g_n')=(Σg_n)' a.e..