Theorems · Theorem · measure theory
MeasureTheory.integral_eq_zero_iff_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 → (∫ (x : α), f x ∂μ = 0 ↔ f =ᵐ[μ] 0)- Cited by
- 7 results in Mathlib
- Foundations
- Depth 254 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
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
- Filter.Eventuallyproof · cited by 3,134
- MeasureTheory.aestatement and proof · cited by 2,352
- Filter.EventuallyEqstatement and proof · cited by 1,912
- MeasureTheory.integralstatement · cited by 1,779
- MeasureTheory.Integrablestatement and proof · cited by 1,367
- MeasureTheory.lintegralproof · cited by 1,152
- ENNReal.ofRealproof · cited by 863
- Filter.EventuallyLEstatement and proof · cited by 383
- Measurable.comp_aemeasurableproof · cited by 99
Cited by7
Results whose statement or proof uses this declaration.
- MeasureTheory.integral_pos_iff_support_of_nonneg_aeproof · cited by 3
- intervalIntegral.integral_eq_zero_iff_of_le_of_nonneg_aeproof · cited by 1
- MeasureTheory.setIntegral_eq_zero_iff_of_nonneg_aeproof · cited by 1
- MeasureTheory.integral_eq_zero_iff_of_nonnegproof · cited by 1
- MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurableproof · cited by 1
- MeasureTheory.eLpNorm_one_le_of_leproof · cited by 1
- MeasureTheory.integral_eq_iff_of_ae_leproof · cited by 0