English
A continuous surjective monoid homomorphism between locally compact groups with compact codomain preserves Haar measure when total masses match.
Русский
Непрерывеный сюръективный моноид-гомоморфизм между локально компактными группами с компактной кодом сохраняет меру Хаара при совпадении суммарной массы.
LaTeX
$$$\\forall G,H \\, [Group G] \\, [Group H] \\, [TopologicalSpace H] \\, [IsTopologicalGroup H] \\, [CompactSpace H] \\, [MeasurableSpace H] \\, [BorelSpace H] \\, (μ : \\mathrm{Measure}(G)) [IsHaarMeasure μ] \\, (ν : \\mathrm{Measure}(H)) [IsHaarMeasure ν] \\, (f : G \\to^* H) \\, (hcont : \\text{Continuous } f) \\, (hsurj : \\text{Surjective } f) \\, (huniv : μ(\\mathrm{univ}) = ν(\\mathrm{univ})) \\\\Rightarrow \\text{MeasurePreserving } f \\, μ \\, ν$$
Lean4
@[to_additive]
theorem zpow [CompactSpace G] [RootableBy G ℤ] {n : ℤ} (hn : n ≠ 0) {X : Type*} [MeasurableSpace X] {μ' : Measure X}
{f : X → G} (hf : MeasurePreserving f μ' μ) : MeasurePreserving (fun x => f x ^ n) μ' μ :=
(measurePreserving_zpow μ hn).comp hf
