Theorems · Theorem · measure theory
MeasureTheory.ofReal_integral_eq_lintegral_ofReal
∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ},
MeasureTheory.Integrable f μ → 0 ≤ᵐ[μ] f → ENNReal.ofReal (∫ (x : α), f x ∂μ) = ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ- Cited by
- 22 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.
Cites19
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
- ENNRealstatement · cited by 9,879
- Norm.normproof · cited by 5,413
- MeasureTheory.aestatement and proof · cited by 2,352
- Filter.EventuallyEqproof · cited by 1,912
- MeasureTheory.integralstatement · cited by 1,779
- MeasureTheory.Integrablestatement and proof · cited by 1,367
- MeasureTheory.lintegralstatement and proof · cited by 1,152
- ENNReal.ofRealstatement and proof · cited by 863
- Filter.Eventually.monoproof · cited by 646
Cited by22
Results whose statement or proof uses this declaration.
- MeasureTheory.tendsto_of_integral_tendsto_of_monotoneproof · cited by 3
- ProbabilityTheory.IsCondKernelCDF.setLIntegralproof · cited by 3
- ProbabilityTheory.gaussianReal_apply_eq_integralproof · cited by 2
- MeasureTheory.integral_le_measureproof · cited by 2
- InformationTheory.klDiv_eq_lintegral_klFunproof · cited by 2
- MeasureTheory.integrable_sum_measureproof · cited by 2
- ProbabilityTheory.Kernel.setLIntegral_densityproof · cited by 2
- MeasureTheory.Integrable.measure_le_integralproof · cited by 2
- ProbabilityTheory.setIntegral_stieltjesOfMeasurableRatproof · cited by 2
- NumberField.mixedEmbedding.fundamentalCone.setLIntegral_paramSet_expproof · cited by 1