# 可测函数　`def.measurable-function`
可测函数（Measurable Function）
layer 6 · 定义 · 可测函数与收敛 · 分析学
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设 $(X, \mathcal{M})$ 与 $(Y, \mathcal{N})$ 是可测空间。函数 $f : X \to Y$ 是 **$(\mathcal{M}, \mathcal{N})$可测的**，当且仅当每个可测集的原像可测：

$$f^{-1}(E) \in \mathcal{M}\quad  \forall E \in \mathcal{N}$$

## 为什么成立（入边，证明在 proofs/）
- [`def.sigma-algebra`](../nodes/def.sigma-algebra.md) σ-代数：用到了定义 σ-代数　[proofs/def-dep.sigma-algebra-measurable-fn.md](../proofs/def-dep.sigma-algebra-measurable-fn.md)

- …另有入边，续页见 [nodes/def.measurable-function.2.md](./def.measurable-function.2.md)

## 它能推出什么 / 谁在用它
- …另有出边，续页见 [nodes/def.measurable-function.3.md](./def.measurable-function.3.md)

refs: Folland, Real Analysis, §2.1；Halmos, Measure Theory, §18

> 说明见 `notes/def.measurable-function.md`

引用：`def.measurable-function`（内容 v0.9.0）· 条款见 [LICENSE.txt](../LICENSE.txt)