Theorems · Theorem · measure theory

MeasureTheory.Measure.hausdorffMeasure_zero_or_top

∀ {X : Type u_2} [inst : EMetricSpace X] [inst_1 : MeasurableSpace X] [inst_2 : BorelSpace X] {d₁ d₂ : ℝ},
  d₁ < d₂ →
    ∀ (s : Set X), (MeasureTheory.Measure.hausdorffMeasure d₂) s = 0 ∨ (MeasureTheory.Measure.hausdorffMeasure d₁) s = ⊤

If d₁ < d₂, then for any set s we have either μH[d₂] s = 0, or μH[d₁] s = ∞.

Defined in: Mathlib.MeasureTheory.Measure.Hausdorff
Cited by: 3 results in Mathlib
Foundations: Depth 204 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: EMetricSpace MeasurableSpace BorelSpace

Mathlib docs Source Loogle LeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites47

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
Setstatement and proof · cited by 53,352
Realstatement and proof · cited by 25,697
MeasurableSpacestatement and proof · cited by 13,106
MeasureTheory.Measurestatement · cited by 10,939
ENNRealstatement and proof · cited by 9,879
Top.topstatement and proof · cited by 9,680
NNRealproof · cited by 4,310
LE.le.transproof · cited by 3,151
LT.lt.leproof · cited by 2,189
le_reflproof · cited by 2,061
Nat.cast_zeroproof · cited by 1,870

Cited by3

Results whose statement or proof uses this declaration.

hausdorffMeasure_of_dimH_ltproof · cited by 5
MeasureTheory.Measure.hausdorffMeasure_monoproof · cited by 2
MeasureTheory.Measure.euclideanHausdorffMeasure_zero_or_topproof · cited by 0