  In this mostly expository note, I give a very quick proof of the definable
Chow theorem of Peterzil and Starchenko using the Bishop-Stoll theorem and a
volume estimate for definable sets due to Nguyen and Valette. The volume
estimate says that any $d$-dimensional definable subset of
$S\subseteq\mathbb{R}^n$ in an o-minimal expansion of the ordered field of real
numbers satisfies the inequality $\mathcal{H}^d(\{x\in S:\lVert
x\rVert<r\})\leq Cr^d$, where $\mathcal{H}^d$ denotes the $d$-dimensional
Hausdorff measure on $\mathbb{R}^n$ and $C$ is a constant depending on $S$. A
closely related volume estimate for subanalytic sets goes back to Kurdyka and
Raby. Since this note is intended to be helpful to algebraic geometers not
versed in o-minimal structures and definable sets, I review these notions and
also prove the main volume estimate from scratch.

Algebraic Geometry;Logic