Theorems · Theorem · measure theory

MeasureTheory.IsAddFundamentalDomain.measure_eq_card_smul_of_vadd_ae_eq_self

∀ {G : Type u_1} {α : Type u_3} [inst : AddGroup G] [inst_1 : AddAction G α] [inst_2 : MeasurableSpace α] {s : Set α}
  {μ : MeasureTheory.Measure α} [MeasurableConstVAdd G α] [MeasureTheory.VAddInvariantMeasure G α μ] [Countable G]
  [Finite G],
  MeasureTheory.IsAddFundamentalDomain G s μ → ∀ (t : Set α), (∀ (g : G), g +ᵥ t =ᵐ[μ] t) → μ t = Nat.card G • μ (t ∩ s)

Given a measure space with an action of a finite additive group G, the measure of any G-invariant set is determined by the measure of its intersection with a fundamental domain for the action of G.

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.VAddInvariantMeasureCountableFinite