Theorems · Theorem · measure theory

MeasureTheory.IsFundamentalDomain.quotientMeasureEqMeasurePreimage_quotientMeasure

∀ {G : Type u_1} {α : Type u_3} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : MeasurableSpace α]
  {ν : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α ν] [Countable G] [MeasurableConstSMul G α]
  {s : Set α},
  MeasureTheory.IsFundamentalDomain G s ν →
    MeasureTheory.QuotientMeasureEqMeasurePreimage ν
      (MeasureTheory.Measure.map (Quotient.mk (MulAction.orbitRel G α)) (ν.restrict s))

Given a measure upstairs (i.e., on α), and a choice s of fundamental domain, there's always an artificial way to generate a measure downstairs such that the pair satisfies the QuotientMeasureEqMeasurePreimage typeclass.

Defined in

Mathlib.MeasureTheory.Group.FundamentalDomain

Cited by

2 results in Mathlib

Foundations

Depth 214 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

GroupMulActionMeasurableSpaceMeasureTheory.SMulInvariantMeasureCountableMeasurableConstSMul