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

χ_(E)f->0 in measure 可否稍為說明一下

｜∫_E_n f |>=ε的

### 1 個解答

• 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