Theorems · Theorem · measure theory
IsOpen.measure_ne_zero
∀ {X : Type u_1} [inst : TopologicalSpace X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure]
{U : Set X}, IsOpen U → U.Nonempty → μ U ≠ 0- Defined in
- Mathlib.MeasureTheory.Measure.OpenPos
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 171 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Setstatement and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- MeasurableSpacestatement and proof · cited by 13,106
- MeasureTheory.Measurestatement and proof · cited by 10,939
- ENNRealstatement · cited by 9,879
- Set.Nonemptystatement and proof · cited by 2,627
- IsOpenstatement and proof · cited by 2,400
- MeasureTheory.Measure.IsOpenPosMeasurestatement and proof · cited by 80
- MeasureTheory.Measure.IsOpenPosMeasure.open_posproof · cited by 6
Cited by6
Results whose statement or proof uses this declaration.
- IsOpen.measure_posproof · cited by 18
- Continuous.isOpenPosMeasure_mapproof · cited by 4
- MeasureTheory.Measure.isOpenPosMeasure_smulproof · cited by 2
- MeasureTheory.Measure.AbsolutelyContinuous.isOpenPosMeasureproof · cited by 1
- DenseRange.zpow_of_ergodic_mul_leftproof · cited by 1
- DenseRange.zsmul_of_ergodic_add_leftproof · cited by 1