# 不知为何，突然想起测度论里的不可测度的维塔利集合

Originally published at 狗和留美者不得入内. You can comment here or there.

$\mu^*(E) = \inf\{\displaystyle\sum_{k=1}^{\infty} I_k : (I_k)_{n \in \mathbb{N}} \text{as open intervals}, \displaystyle\bigcup_{k=1}^{\infty} I_k \supset E\}$

$\displaystyle \mu ^{*}(A)=\mu ^{*}(A\cap E)+\mu ^{*}(A\cap E^{c})$

• 测度平移守恒，那就是 $\mu(S) = \mu(x+S), \forall x \in \mathbb{R}$
• $\{A_k\}_{k \in \mathbb{N}}$ 互相不交则 $\mu(\bigcup_{k=1}^{\infty} A_k) = \sum_{k=1}^{\infty} \mu(A_k)$
• $S \subset T$$\mu(S) \leq \mu(T)$

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