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Theorems · Theorem · measure theory

MeasureTheory.Measure.setIntegral_toReal_rnDeriv

∀ {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ]
  [MeasureTheory.SigmaFinite ν],
  μ.AbsolutelyContinuous ν → ∀ (s : Set α), ∫ (x : α) in s, (μ.rnDeriv ν x).toReal ∂ν = μ.real s
Defined in
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym
Cited by
3 results in Mathlib
Foundations
Depth 256 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
MeasureTheory.SigmaFiniteMeasureTheory.SigmaFinite
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

MeasureTheory.toReal_rnDeriv_map · cited by 5MeasureTheory.toReal_rnDe…MeasureTheory.SignedMeasure.withDensityᵥ_rnDeriv_eq · cited by 2SignedMeasure.withDensity…MeasureTheory.Measure.integral_toReal_rnDeriv · cited by 2Measure.integral_toReal_r…Set · cited by 53352SetReal · cited by 25697RealMeasurableSpace · cited by 13106MeasurableSpaceMeasureTheory.Measure · cited by 10939MeasureTheory.MeasureMeasureTheory.integral · cited by 1779MeasureTheory.integralMeasureTheory.Measure.restrict · cited by 1646Measure.restrictENNReal.toReal · cited by 859ENNReal.toRealMeasureTheory.Measure.real · cited by 530Measure.realMeasureTheory.SigmaFinite · cited by 526MeasureTheory.SigmaFiniteMeasureTheory.Measure.AbsolutelyContinuous · cited by 325Measure.AbsolutelyContinu…MeasureTheory.Measure.rnDeriv · cited by 234Measure.rnDerivMeasureTheory.Measure.withDensity_rnDeriv_eq · cited by 31Measure.withDensity_rnDer…MeasureTheory.Measure.setIntegral_toReal_rnDeriv_eq_withDensity · cited by 2Measure.setIntegral_toRea…Measure.setIntegral_toReal_rn…CITED BYCITES

Cites13

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Cited by3

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