Theorems · Theorem · measure theory
MeasureTheory.setIntegral_pos_iff_support_of_nonneg_ae
∀ {X : Type u_1} {mX : MeasurableSpace X} {s : Set X} {μ : MeasureTheory.Measure X} {f : X → ℝ},
0 ≤ᵐ[μ.restrict s] f →
MeasureTheory.IntegrableOn f s μ → (0 < ∫ (x : X) in s, f x ∂μ ↔ 0 < μ (Function.support f ∩ s))- Cited by
- 6 results in Mathlib
- Foundations
- Depth 256 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
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 and proof · cited by 10,939
- ENNRealstatement and proof · cited by 9,879
- Compl.complproof · cited by 2,925
- MeasureTheory.aestatement and proof · cited by 2,352
- MeasureTheory.integralstatement · cited by 1,779
- MeasureTheory.Measure.restrictstatement and proof · cited by 1,646
- Function.supportstatement and proof · cited by 610
- MeasureTheory.IntegrableOnstatement and proof · cited by 548
Cited by6
Results whose statement or proof uses this declaration.
- Real.Gamma_pos_of_posproof · cited by 17
- ProbabilityTheory.mgf_pos'proof · cited by 9
- MeasureTheory.measure_le_setAverage_posproof · cited by 5
- intervalIntegral.integral_pos_iff_support_of_nonneg_ae'proof · cited by 2
- MeasureTheory.setIntegral_gt_gtproof · cited by 1