Theorems · Theorem · measure theory
IsOpen.eq_empty_of_measure_zero
∀ {X : Type u_1} [inst : TopologicalSpace X] {m : MeasurableSpace X} {μ : MeasureTheory.Measure X} [μ.IsOpenPosMeasure]
{U : Set X}, IsOpen U → μ U = 0 → U = ∅An open null set w.r.t. an IsOpenPosMeasure is empty.
- Defined in
- Mathlib.MeasureTheory.Measure.OpenPos
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 175 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- IsOpenstatement and proof · cited by 2,400
- MeasureTheory.Measure.IsOpenPosMeasurestatement and proof · cited by 80
- IsOpen.measure_eq_zero_iffproof · cited by 6
Cited by2
Results whose statement or proof uses this declaration.
- MeasureTheory.Measure.eqOn_open_of_ae_eqproof · cited by 2
- MeasureTheory.Measure.interior_eq_empty_of_nullproof · cited by 2