Scharze space 發問時間: 科學數學 · 1 0 年前

實分析10(integration)

Let f be an integrable function on the measure space (X,B,μ)

Show that given ε>0,there is a δ>0,such that for each measurable set E with μE<δ

we have |∫(E)f|<ε

已更新項目:

不好意思:

χ_(E)f->0 in measure 可否稍為說明一下

另外,最後的矛盾是否是:

|∫_E_n f |>=ε的

1 個解答

評分
  • Eric
    Lv 6
    1 0 年前
    最佳解答

    Proof. Suppose otherwise; then there exists ε > 0 such that for all n, we can find En in B with μ(En) < 1/n and |∫En f| ≥ ε. Then 1Enf→0 in measure, so by taking an a.e. convergent subsequence 1Enk and applying the dominated convergence theorem, we obtain ∫Enk f = ∫ 1Enkf → 0, which is absurd. :

    2007-06-23 23:56:35 補充:

    (1)

    1Enf→0 in measure: for all δ > 0,

    μ({x:|1En f| ≥ δ}) ≤ μ(En) → 0

    (2)

    是的

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