Theorems · Theorem · measure theory
MeasureTheory.integral_nonneg
∀ {α : Type u_1} {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {m : MeasurableSpace α}
{μ : MeasureTheory.Measure α} [inst_2 : PartialOrder E] [IsOrderedAddMonoid E] [IsOrderedModule ℝ E]
[ClosedIciTopology E] {f : α → E}, 0 ≤ f → 0 ≤ ∫ (x : α), f x ∂μ- Cited by
- 52 results in Mathlib
- Foundations
- Depth 253 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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
- NormedAddCommGroupstatement and proof · cited by 15,752
- MeasurableSpacestatement and proof · cited by 13,106
- NormedSpacestatement and proof · cited by 12,499
- MeasureTheory.Measurestatement and proof · cited by 10,939
- PartialOrderstatement and proof · cited by 6,410
- MeasureTheory.integralstatement · cited by 1,779
- IsOrderedAddMonoidstatement and proof · cited by 1,659
- IsOrderedModulestatement and proof · cited by 156
- ClosedIciTopologystatement and proof · cited by 156
- MeasureTheory.ae_of_allproof · cited by 137
- MeasureTheory.integral_nonneg_of_aeproof · cited by 17
Cited by52
Results whose statement or proof uses this declaration.
- MeasureTheory.Integrable.convolution_integrandproof · cited by 6
- ProbabilityTheory.mgf_nonnegproof · cited by 5
- integral_withDensity_eq_integral_smulproof · cited by 4
- InformationTheory.toReal_klDiv_eq_integral_klFunproof · cited by 3
- VitaliFamily.ae_tendsto_average_norm_subproof · cited by 3
- InformationTheory.integral_llr_add_sub_measure_univ_nonnegproof · cited by 3
- ContDiffBump.nonneg_normedproof · cited by 3
- MeasureTheory.tilted_tiltedproof · cited by 2
- integral_gaussianproof · cited by 2
- MeasureTheory.hasSum_integral_of_summable_integral_normproof · cited by 2