Theorems · Theorem · measure theory
MeasureTheory.integral_eq_zero_iff_of_nonneg
∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ},
0 ≤ f → MeasureTheory.Integrable f μ → (∫ (x : α), f x ∂μ = 0 ↔ f =ᵐ[μ] 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.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- MeasurableSpacestatement and proof · cited by 13,106
- MeasureTheory.Measurestatement and proof · cited by 10,939
- MeasureTheory.aestatement · cited by 2,352
- Filter.EventuallyEqstatement · cited by 1,912
- MeasureTheory.integralstatement · cited by 1,779
- MeasureTheory.Integrablestatement and proof · cited by 1,367
- Filter.Eventually.of_forallproof · cited by 526
- MeasureTheory.integral_eq_zero_iff_of_nonneg_aeproof · cited by 7
Cited by1
Results whose statement or proof uses this declaration.
- ProbabilityTheory.ae_eq_integral_of_variance_eq_zeroproof · cited by 1