Theorems · Theorem · measure theory

MeasureTheory.measure_isOpen_pos_of_smulInvariant_of_compact_ne_zero

∀ (G : Type u) {α : Type w} {m : MeasurableSpace α} [inst : Group G] [inst_1 : MulAction G α]
  {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] [inst_3 : TopologicalSpace α]
  [ContinuousConstSMul G α] [MulAction.IsMinimal G α] {K U : Set α},
  IsCompact K → μ K ≠ 0 → IsOpen U → U.Nonempty → 0 < μ U

If measure μ is invariant under a group action and is nonzero on a compact set K, then it is positive on any nonempty open set. In case of a regular measure, one can assume μ ≠ 0 instead of μ K ≠ 0, see MeasureTheory.measure_isOpen_pos_of_smulInvariant_of_ne_zero.

Defined in

Mathlib.MeasureTheory.Group.Action

Cited by

1 results in Mathlib

Foundations

Depth 174 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

GroupMulActionMeasureTheory.SMulInvariantMeasureTopologicalSpaceContinuousConstSMulMulAction.IsMinimal