∀ {G : Type u_1} {H : Type u_2} {α : Type u_3} {β : Type u_4} [inst : Group G] [inst_1 : Group H]
  [inst_2 : MulAction G α] [inst_3 : MeasurableSpace α] [inst_4 : MulAction H β] [inst_5 : MeasurableSpace β]
  {s : Set α} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β},
  MeasureTheory.IsFundamentalDomain G s μ →
    ∀ {f : β → α},
      MeasureTheory.Measure.QuasiMeasurePreserving f ν μ →
        ∀ {e : G → H},
          Function.Bijective e →
            (∀ (g : G), Function.Semiconj f (fun x => e g • x) fun x => g • x) →
              MeasureTheory.IsFundamentalDomain H (f ⁻¹' s) ν