Theorems · Theorem · measure theory
MeasureTheory.setIntegral_eq_zero_iff_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 → (∫ (x : X) in s, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict s] 0)- Cited by
- 1 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.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- 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
- MeasureTheory.aestatement and proof · cited by 2,352
- Filter.EventuallyEqstatement · cited by 1,912
- MeasureTheory.integralstatement · cited by 1,779
- MeasureTheory.Measure.restrictstatement and proof · cited by 1,646
- MeasureTheory.IntegrableOnstatement and proof · cited by 548
- Filter.EventuallyLEstatement and proof · cited by 383
- MeasureTheory.integral_eq_zero_iff_of_nonneg_aeproof · cited by 7
Cited by1
Results whose statement or proof uses this declaration.
- exists_eq_const_mul_setIntegral_of_ae_nonnegproof · cited by 2