English
If f is an isometry, the Hausdorff measure is preserved up to the conditions on d (d ≤ 0 or surjective).
Русский
Если f является изометрией, мера Хаусдорфа сохраняется при условии on d (d ≤ 0 или сюръекция).
LaTeX
$$$f\text{ is isometry and }(d\ge 0 \Rightarrow \text{?}) \; μ_H^{d}(f(S)) = μ_H^{d}(S)$$$
Lean4
theorem hausdorffMeasure_image (hf : Isometry f) (hd : 0 ≤ d ∨ Surjective f) (s : Set X) : μH[d] (f '' s) = μH[d] s :=
by
simp only [hausdorffMeasure, ← OuterMeasure.coe_mkMetric, ← OuterMeasure.comap_apply]
rw [OuterMeasure.isometry_comap_mkMetric _ hf (hd.imp_left _)]
exact ENNReal.monotone_rpow_of_nonneg