We introduce the Hausdorff measure for definable sets in an o-minimal structure, and prove the Cauchy—Crofton and co-area formulae for the o-minimal Hausdorff measure. We also prove that every definable set can be partitioned into “basic rectifiable sets”, and that the Whitney arc property holds for basic rectifiable sets.
"Hausdorff measure on o-minimal structures." J. Symbolic Logic 77 (2) 631 - 648, June 2012. https://doi.org/10.2178/jsl/1333566642