Theorems · Theorem · measure theory
MeasureTheory.Measure.rnDeriv_withDensity
∀ {α : Type u_1} {m : MeasurableSpace α} (ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite ν] {f : α → ENNReal},
Measurable f → (ν.withDensity f).rnDeriv ν =ᵐ[ν] fThe Radon-Nikodym derivative of f ν with respect to ν is f.
- Cited by
- 13 results in Mathlib
- Foundations
- Depth 214 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- MeasureTheory.SigmaFinite
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.
- MeasurableSpacestatement and proof · cited by 13,106
- MeasureTheory.Measurestatement and proof · cited by 10,939
- ENNRealstatement and proof · cited by 9,879
- MeasureTheory.aestatement · cited by 2,352
- Filter.EventuallyEqstatement · cited by 1,912
- Measurablestatement and proof · cited by 1,499
- MeasureTheory.SigmaFinitestatement and proof · cited by 526
- Measurable.aemeasurableproof · cited by 304
- MeasureTheory.Measure.withDensitystatement · cited by 265
- MeasureTheory.Measure.rnDerivstatement · cited by 234
- MeasureTheory.Measure.rnDeriv_withDensity₀proof · cited by 1
Cited by13
Results whose statement or proof uses this declaration.
- VitaliFamily.ae_tendsto_rnDerivproof · cited by 4
- MeasureTheory.Measure.inv_rnDerivproof · cited by 3
- ProbabilityTheory.rnDeriv_measure_compProd_leftproof · cited by 2
- MeasureTheory.Measure.rnDeriv_withDensity_leftproof · cited by 2
- VitaliFamily.ae_tendsto_lintegral_div'proof · cited by 2
- MeasureTheory.Measure.rnDeriv_withDensity_rightproof · cited by 1
- MeasureTheory.MeasurePreserving.rnDeriv_comp_aeEqproof · cited by 1
- MeasureTheory.Measure.inv_rnDeriv_auxproof · cited by 1
- VitaliFamily.ae_tendsto_rnDeriv_of_absolutelyContinuousproof · cited by 1
- MeasureTheory.Measure.rnDeriv_pos'proof · cited by 1
- MeasureTheory.Measure.rnDeriv_restrict_selfproof · cited by 1
- ProbabilityTheory.rnDeriv_gaussianRealproof · cited by 0