# NBV 函数的导数与测度的关系　`prop.nbv-derivative`
命题：$\mu_F \perp m \iff F' = 0 \text{a.e.}$；$\mu_F \ll m \iff F(x) = \int_{-\infty}^{x} F'$
layer 20 · 命题 · 有界变差与绝对连续 · 分析学
[在星图里打开](../#n=prop.nbv-derivative)　·　[节点索引](../index.md)

设 $F \
> 陈述续见 [nodes/prop.nbv-derivative.2.md](./prop.nbv-derivative.2.md)


## 为什么成立（入边，证明在 proofs/）
- …另有入边，续页见 [nodes/prop.nbv-derivative.4.md](./prop.nbv-derivative.4.md)

## 它能推出什么 / 谁在用它
- …另有出边，续页见 [nodes/prop.nbv-derivative.6.md](./prop.nbv-derivative.6.md)

## English abstract · **draft** · machine-drafted, pending author review; Chinese is authoritative
> English abstract 续见 [nodes/prop.nbv-derivative.3.md](./prop.nbv-derivative.3.md)

refs: Folland, Real Analysis, Theorem 3.35

外部：Wikipedia https://en.wikipedia.org/wiki/Bounded_variation · nLab https://ncatlab.org/nlab/show/absolutely+continuous+function

> 说明见 `notes/prop.nbv-derivative.md`

引用：`prop.nbv-derivative`（内容 v0.9.0）· 条款见 [LICENSE.txt](../LICENSE.txt)