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Volumes of definable sets in o-minimal expansions and affine GAGA theorems

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.