Theorems · Theorem · measure theory
MeasureTheory.integral_pos_iff_support_of_nonneg
∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ},
0 ≤ f → MeasureTheory.Integrable f μ → (0 < ∫ (x : α), f x ∂μ ↔ 0 < μ (Function.support f))- Cited by
- 9 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.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · 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
- MeasureTheory.integralstatement · cited by 1,779
- MeasureTheory.Integrablestatement and proof · cited by 1,367
- Function.supportstatement · cited by 610
- Filter.Eventually.of_forallproof · cited by 526
- MeasureTheory.integral_pos_iff_support_of_nonneg_aeproof · cited by 3
Cited by9
Results whose statement or proof uses this declaration.
- MeasureTheory.integral_exp_posproof · cited by 8
- MeasureTheory.Measure.addHaarScalarFactor_eq_mulproof · cited by 6
- ExistsContDiffBumpBase.u_int_posproof · cited by 4
- ContDiffBump.integral_posproof · cited by 3
- MeasureTheory.Measure.haarScalarFactor_eq_mulproof · cited by 3
- ExistsContDiffBumpBase.y_pos_of_mem_ballproof · cited by 1
- MeasureTheory.integral_pos_of_integrable_nonneg_nonzeroproof · cited by 1