令 $A \subset \mathbb{R}^*$ ,定義 outer measure $m^*: \mathcal{P}(\mathbb{R}) \to [0,\infty]$ 滿足 \[{m^*}(A): = \mathop {\inf }\limits_{A \subset \bigcup\limits_j^{} {\left[ {{a_j},{b_j}} \right]} } \sum\limits_j^{} {\left( {{b_j} - a_j^{}} \right)} \] 以下我們檢驗幾個性質 Property of Outer Measure on R: 1. $m^*(\emptyset) = 0$ 2. Monotonicity : 若 $A \subset B$ 則 $m^*(A) \subset m^*(B)$ 3. Subadditivity : $m^*(\cup_j^\infty A_j) \leq \sum_j^\infty m^*(A_j)$ Proof: 1. 取 $A:= \emptyset$ 則任意區間必定涵蓋 $\emptyset$,故 $m^*(\emptyset) = 0$。 2. 若 $A \subset B$ 則存在一組區間 $I_{j} := [a_{j}, b_{j}]$, $j=1,2...$ 使得 $A \subset B \subset \cup_{j=1}^\infty [a_{j}, b_{j}]$,故由定義可知 $m^*(A) \subset m^*(B)$。 3. 首先觀察 $m^*(A_j)$ 定義中有 infimum,故給定 $\varepsilon>0$ 可知必定存在一組區間 $I_{j,k} := [a_{j,k}, b_{j,k}]$, $k=1,2...$ 使得 $A_j \subset \cup_{k=1}^\infty [a_{j,k}, b_{j,k}]$ \[ \sum\limits_k^{} {\left( {{b_{j,k}} - a_{j,k}^{}} \right)} < {m^*}(A_j) + \frac{\varepsilon}{10^j} \]由此可知 \[ \sum\limits_{j,k}^{} {
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