Theorems · Theorem · measure theory

MeasureTheory.IsFundamentalDomain.essSup_measure_restrict

∀ {G : Type u_1} {α : Type u_3} [inst : Group G] [inst_1 : MulAction G α] [inst_2 : MeasurableSpace α] {s : Set α}
  {μ : MeasureTheory.Measure α} [MeasurableConstSMul G α] [MeasureTheory.SMulInvariantMeasure G α μ] [Countable G],
  MeasureTheory.IsFundamentalDomain G s μ →
    ∀ {f : α → ENNReal}, (∀ (γ : G) (x : α), f (γ • x) = f x) → essSup f (μ.restrict s) = essSup f μ

If f is invariant under the action of a countable group G, and μ is a G-invariant measure with a fundamental domain s, then the essSup of f restricted to s is the same as that of f on all of its domain.

Defined in

Mathlib.MeasureTheory.Group.FundamentalDomain

Cited by

1 results in Mathlib

Foundations

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

Assumes

GroupMulActionMeasurableSpaceMeasurableConstSMulMeasureTheory.SMulInvariantMeasureCountable