Theorems · Theorem · measure theory
MeasureTheory.withDensity_apply
∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ENNReal) {s : Set α},
MeasurableSet s → (μ.withDensity f) s = ∫⁻ (a : α) in s, f a ∂μ- Cited by
- 73 results in Mathlib
- Foundations
- Depth 203 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Setstatement and proof · cited by 53,352
- MeasurableSpacestatement and proof · cited by 13,106
- MeasureTheory.Measurestatement and proof · cited by 10,939
- ENNRealstatement and proof · cited by 9,879
- MeasurableSetstatement and proof · cited by 3,075
- MeasureTheory.Measure.restrictstatement · cited by 1,646
- MeasureTheory.lintegralstatement · cited by 1,152
- MeasureTheory.Measure.withDensitystatement · cited by 265
- MeasureTheory.Measure.ofMeasurable_applyproof · cited by 5
Cited by73
Results whose statement or proof uses this declaration.
- MeasureTheory.withDensity_absolutelyContinuousproof · cited by 40
- MeasureTheory.withDensity_congr_aeproof · cited by 19
- MeasureTheory.restrict_withDensityproof · cited by 15
- MeasureTheory.withDensity_apply'proof · cited by 9
- MeasureTheory.withDensity_eq_iff_of_sigmaFiniteproof · cited by 9
- MeasureTheory.withDensity_oneproof · cited by 8
- MeasureTheory.withDensity_smulproof · cited by 8
- MeasureTheory.setLIntegral_condLExpproof · cited by 8
- MeasureTheory.withDensity_eq_iffproof · cited by 6
- MeasureTheory.withDensity_absolutelyContinuous'proof · cited by 5
- MeasureTheory.withDensity_indicatorproof · cited by 5
- MeasureTheory.withDensity_mul₀proof · cited by 5