<aside> 🎎 Generalized and formalization theory of volumes, and the probability of events.  Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Constantin Carathéodory, and Maurice Fréchet

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measure space 测度空间 $(X,\mathcal A,\mu)$ consists of a measurable space $(X,\mathcal A)$ together with a measure on it.

• An underlying set 一个集合$X$ →样本空间
• $\sigma$-algebra over $X$ 集合$X$的子集构成的集合$\mathcal A$, or $\Sigma$ → 事件
• method for measuring (the measure) 计算测度的方法$\mu(\cdot)$ → 事件的概率$\mu(X)=1$
  • non-negativity: $\mu(e)\ge 0, \forall e\in\Sigma$
  • $\mu(\emptyset) =0$
  • $\sigma$-additivity: $\mu (\bigcup _{k=1}^{\infty }E_k)=\sum {k=1}^{\infty }\mu (E_k)$: For all countable collections$\{E_k\}{k=1}^\infty$of pairwise disjoint sets两两不相交 in $\Sigma$

complex measure

Lebesgue measure

Lebesgue Integral

Measures that take values in Banach spaces.