# shrinking　`def.shrinks-nicely`
English entry · root `../`
[in graph](../#n=def.shrinks-nicely)　·　[Chinese page (authoritative)](./def.shrinks-nicely.md)

> English: **draft** · machine-drafted, pending author review; the Chinese page is authoritative.

layer 15 · definition · cluster 微分定理 (`analysis.differentiation`) · Analysis

## Abstract
A family $\{E_r\}_{r > 0}$ of Borel subsets of $\mathbb{R}^n$ shrinks nicely to $x$ when $E_r \subseteq B(r,x)$ for every $r$ and there is $\alpha > 0$ with $m(E_r) > \alpha \cdot m(B(r,x))$ for all $r$. The two conditions say "fits inside the ball" and "is not too thin". Balls themselves qualify, as does any fixed shape rescaled inside them; a thin sliver does not. The notion exists so that differentiation theorems do not depend on the shape of the sets being shrunk.

> 续见 nodes/def.shrinks-nicely.en.2.md