Theorems · Theorem · measure theory

MeasureTheory.IsAddFundamentalDomain.measure_le_of_pairwise_disjoint

∀ {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 →
      Pairwise (Function.onFun (MeasureTheory.AEDisjoint μ) fun g => (g +ᵥ t) ∩ s) → μ t ≤ μ s

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 such that the sets g +ᵥ t ∩ s are pairwise a.e.-disjoint has measure at most μ s.

Defined in

Mathlib.MeasureTheory.Group.FundamentalDomain

Cited by

1 results in Mathlib

Foundations

Depth 212 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddGroupAddActionMeasurableSpaceMeasurableConstVAddMeasureTheory.VAddInvariantMeasureCountable