Theorems · Theorem · measure theory

TopologicalGroup.IsSES.inducedMeasure_lt_of_injOn

∀ {A : Type u_1} {B : Type u_2} {C : Type u_3} [inst : Group A] [inst_1 : Group B] [inst_2 : Group C]
  [inst_3 : TopologicalSpace A] [inst_4 : TopologicalSpace B] [inst_5 : TopologicalSpace C] {φ : A →* B} {ψ : B →* C}
  (H : TopologicalGroup.IsSES φ ψ) [inst_6 : IsTopologicalGroup A] [inst_7 : IsTopologicalGroup B]
  [inst_8 : MeasurableSpace A] [inst_9 : BorelSpace A] (μA : MeasureTheory.Measure A) [hμA : μA.IsHaarMeasure]
  [inst_10 : IsTopologicalGroup C] [inst_11 : LocallyCompactSpace B] [inst_12 : MeasurableSpace C]
  [inst_13 : BorelSpace C] (μC : MeasureTheory.Measure C) [hμC : μC.IsHaarMeasure] [inst_14 : T2Space B]
  [inst_15 : MeasurableSpace B] [inst_16 : BorelSpace B] {U : Set B},
  IsOpen U → ∀ [DiscreteTopology A], Set.InjOn (⇑ψ) U → (H.inducedMeasure μA μC) U ≤ μC Set.univ * μA {1}

If φ : A →* B and ψ : B →* C define a short exact sequence of topological groups, and if ψ is injective on an open set U, then the induced measure on U is bounded above by μC Set.univ * μA {1} (possibly infinite).

Defined in

Mathlib.MeasureTheory.Measure.Haar.Extension

Cited by

0 results in Mathlib

Foundations

Depth 266 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

GroupGroupGroupTopologicalSpaceTopologicalSpaceTopologicalSpaceIsTopologicalGroupIsTopologicalGroupMeasurableSpaceBorelSpaceMeasureTheory.Measure.IsHaarMeasureIsTopologicalGroupLocallyCompactSpaceMeasurableSpaceBorelSpaceMeasureTheory.Measure.IsHaarMeasureT2SpaceMeasurableSpaceBorelSpaceDiscreteTopology