Theorems · Theorem · measure theory
IsOpen.measure_pos_iff
∀ {X : Type u_1} [inst : TopologicalSpace X] {m : MeasurableSpace X} (μ : MeasureTheory.Measure X) [μ.IsOpenPosMeasure]
{U : Set X}, IsOpen U → (0 < μ U ↔ U.Nonempty)- Defined in
- Mathlib.MeasureTheory.Measure.OpenPos
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
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
- 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 · cited by 2,627
- IsOpenstatement and proof · cited by 2,400
- LT.lt.ne'proof · cited by 1,417
- MeasureTheory.measure_emptyproof · cited by 169
- Set.nonempty_iff_ne_emptyproof · cited by 96
- MeasureTheory.Measure.IsOpenPosMeasurestatement and proof · cited by 80
Cited by2
Results whose statement or proof uses this declaration.
- IsOpen.measure_eq_zero_iffproof · cited by 6
- MeasureTheory.Measure.measure_Ioo_posproof · cited by 1