Theorems · Theorem · measure theory
MeasureTheory.Measure.AbsolutelyContinuous.smul
∀ {α : Type u_1} {R : Type u_5} {mα : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [inst : SMul R ENNReal]
[inst_1 : IsScalarTower R ENNReal ENNReal], μ.AbsolutelyContinuous ν → ∀ (c : R), (c • μ).AbsolutelyContinuous (c • ν)If μ ≪ ν, then c • μ ≪ c • ν.
Earlier, this name was used for what's now called AbsolutelyContinuous.smul_left.
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 175 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SMulIsScalarTower
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.
- DFunLike.coeproof · cited by 62,936
- Setproof · 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
- IsScalarTowerstatement and proof · cited by 3,896
- smul_eq_mulproof · cited by 357
- MeasureTheory.Measure.AbsolutelyContinuousstatement and proof · cited by 325
- mul_eq_zeroproof · cited by 94
- smul_one_smulproof · cited by 43
- MeasureTheory.Measure.smul_applyproof · cited by 28
Cited by4
Results whose statement or proof uses this declaration.
- MeasureTheory.Measure.AbsolutelyContinuous.smul_rightproof · cited by 2
- MeasureTheory.mul_le_integral_rnDeriv_of_acproof · cited by 1
- MeasureTheory.Measure.QuasiMeasurePreserving.smul_measureproof · cited by 1
- InformationTheory.klDiv_smul_sameproof · cited by 0