Theorems · Theorem · measure theory

IsFoelner.amenable

∀ {G : Type u_1} {X : Type u_2} [inst : MeasurableSpace X] {μ : MeasureTheory.Measure X} [inst_1 : Group G]
  [inst_2 : MulAction G X] {ι : Type u_3} {l : Filter ι} {F : ι → Set X} [MeasureTheory.SMulInvariantMeasure G X μ]
  [l.NeBot],
  IsFoelner G μ l F →
    ∃ m,
      m Set.univ = 1 ∧
        (∀ (s t : Set X), MeasurableSet t → Disjoint s t → m (s ∪ t) = m s + m t) ∧
          ∀ (g : G) (s : Set X), m (g • s) = m s

If there exists a non-trivial Følner filter with respect to some group G acting on a measure space X, then there exists a G-invariant finitely additive probability measure on X.

Defined in

Mathlib.MeasureTheory.Group.FoelnerFilter

Cited by

1 results in Mathlib

Foundations

Depth 192 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceGroupMulActionMeasureTheory.SMulInvariantMeasureFilter.NeBot