Theorems · Theorem · measure theory
MeasureTheory.integral_toReal
∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal},
AEMeasurable f μ → (∀ᵐ (x : α) ∂μ, f x < ⊤) → ∫ (a : α), (f a).toReal ∂μ = (∫⁻ (a : α), f a ∂μ).toReal- Cited by
- 15 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
- ENNRealstatement and proof · cited by 9,879
- Top.topstatement and proof · cited by 9,680
- Filter.Eventuallystatement and proof · cited by 3,134
- MeasureTheory.aestatement and proof · cited by 2,352
- MeasureTheory.integralstatement · cited by 1,779
- MeasureTheory.lintegralstatement and proof · cited by 1,152
- ENNReal.toRealstatement and proof · cited by 859
- AEMeasurablestatement and proof · cited by 840
- Filter.Eventually.of_forallproof · cited by 526
Cited by15
Results whose statement or proof uses this declaration.
- ProbabilityTheory.variance_eq_integralproof · cited by 10
- MeasureTheory.integral_prodproof · cited by 9
- ProbabilityTheory.integral_compProdproof · cited by 4
- MeasureTheory.lpNorm_eq_integral_norm_rpow_toRealproof · cited by 3
- MeasureTheory.toReal_laverageproof · cited by 3
- MeasureTheory.Measure.setIntegral_toReal_rnDeriv_eq_withDensity'proof · cited by 2
- ProbabilityTheory.condDistrib_ae_eq_condExpproof · cited by 2
- MeasureTheory.Integrable.summable_integralproof · cited by 2
- MeasureTheory.Measure.setIntegral_toReal_rnDeriv_eq_withDensityproof · cited by 2
- ProbabilityTheory.setIntegral_preCDF_fstproof · cited by 2
- MeasureTheory.toReal_condLExpproof · cited by 2
- ProbabilityTheory.Kernel.integral_compproof · cited by 1