Theorems · Theorem · measure theory
MeasureTheory.integral_pos_iff_support_of_nonneg_ae
∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ},
0 ≤ᵐ[μ] f → MeasureTheory.Integrable f μ → (0 < ∫ (x : α), f x ∂μ ↔ 0 < μ (Function.support f))- Cited by
- 3 results in Mathlib
- Foundations
- Depth 255 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
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 · 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 · cited by 9,879
- Set.ofPredproof · cited by 6,101
- MeasureTheory.aestatement and proof · cited by 2,352
- MeasureTheory.integralstatement and proof · cited by 1,779
- MeasureTheory.Integrablestatement and proof · cited by 1,367
- Function.supportstatement and proof · cited by 610
- Filter.EventuallyLEstatement and proof · cited by 383
Cited by3
Results whose statement or proof uses this declaration.
- MeasureTheory.integral_pos_iff_support_of_nonnegproof · cited by 9
- MeasureTheory.setIntegral_pos_iff_support_of_nonneg_aeproof · cited by 6