Theorems · Theorem · measure theory
MeasureTheory.lintegral_coe_eq_integral
∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → NNReal),
MeasureTheory.Integrable (fun x => ↑(f x)) μ → ∫⁻ (a : α), ↑(f a) ∂μ = ENNReal.ofReal (∫ (a : α), ↑(f a) ∂μ)- Cited by
- 11 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.
Cites20
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement · cited by 25,697
- MeasurableSpacestatement and proof · cited by 13,106
- MeasureTheory.Measurestatement and proof · cited by 10,939
- ENNRealstatement and proof · cited by 9,879
- Top.topproof · cited by 9,680
- NNRealstatement and proof · cited by 4,310
- MeasureTheory.integralstatement · cited by 1,779
- MeasureTheory.Integrablestatement and proof · cited by 1,367
- ENNReal.ofNNRealstatement and proof · cited by 1,279
- NNReal.toRealstatement and proof · cited by 1,260
- MeasureTheory.lintegralstatement and proof · cited by 1,152
- ENNReal.ofRealstatement and proof · cited by 863
Cited by11
Results whose statement or proof uses this declaration.
- integral_withDensity_eq_integral_smulproof · cited by 4
- VitaliFamily.ae_tendsto_average_norm_subproof · cited by 3
- MeasureTheory.hasSum_integral_of_summable_integral_normproof · cited by 2
- MeasureTheory.integrableOn_iUnion_of_summable_integral_normproof · cited by 1
- MeasureTheory.tendsto_integral_thickenedIndicator_of_isClosedproof · cited by 1
- MeasureTheory.lintegral_coe_le_coe_iff_integral_leproof · cited by 1
- volume_regionBetween_eq_integral'proof · cited by 1
- intervalIntegral.integrableOn_deriv_right_of_nonnegproof · cited by 1
- NNRealRMK.lintegral_rieszMeasureproof · cited by 0
- ProbabilityTheory.evariance_def'proof · cited by 0