Theorems · Theorem · measure theory

MeasureTheory.IsAddFundamentalDomain.exists_ne_zero_vadd_eq

∀ {G : Type u_1} {α : Type u_3} [inst : AddGroup G] [inst_1 : AddAction G α] [inst_2 : MeasurableSpace α] {s t : Set α}
  {μ : MeasureTheory.Measure α} [MeasurableConstVAdd G α] [MeasureTheory.VAddInvariantMeasure G α μ] [Countable G],
  MeasureTheory.IsAddFundamentalDomain G s μ →
    MeasureTheory.NullMeasurableSet t μ → μ s < μ t → ∃ x ∈ t, ∃ y ∈ t, ∃ g, g ≠ 0 ∧ g +ᵥ x = y

If the additive action of a countable group G admits an invariant measure μ with a fundamental domain s, then every null-measurable set t of measure strictly greater than μ s contains two points x y such that g +ᵥ x = y for some g ≠ 0.

Defined in

Mathlib.MeasureTheory.Group.FundamentalDomain

Cited by

0 results in Mathlib

Foundations

Depth 213 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddGroupAddActionMeasurableSpaceMeasurableConstVAddMeasureTheory.VAddInvariantMeasureCountable