Theorems · Theorem · measure theory
MeasureTheory.Measure.exists_subset_measure_lt_top
∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [MeasureTheory.SigmaFinite μ]
{r : ENNReal}, MeasurableSet s → r < μ s → ∃ t, MeasurableSet t ∧ t ⊆ s ∧ r < μ t ∧ μ t < ⊤In a σ-finite space, any measurable set of measure > r contains a measurable subset of
finite measure > r.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 202 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- MeasureTheory.SigmaFinite
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites20
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
- MeasurableSpacestatement and proof · cited by 13,106
- MeasureTheory.Measurestatement and proof · cited by 10,939
- ENNRealstatement and proof · cited by 9,879
- Top.topstatement · cited by 9,680
- MeasurableSetstatement and proof · cited by 3,075
- MeasureTheory.Measure.restrictproof · cited by 1,646
- LE.le.trans_ltproof · cited by 795
- MeasureTheory.SigmaFinitestatement and proof · cited by 526
- Set.inter_subset_leftproof · cited by 360
- MeasureTheory.measure_monoproof · cited by 338
Cited by2
Results whose statement or proof uses this declaration.
- MeasureTheory.SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegralproof · cited by 1
- MeasureTheory.Measure.eventually_nonempty_inter_smul_of_density_oneproof · cited by 1