Theorems · Theorem · measure theory

VitaliFamily.le_mul_withDensity

∀ {α : Type u_1} [inst : PseudoMetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
  (v : VitaliFamily μ) [inst_1 : SecondCountableTopology α] [inst_2 : BorelSpace α]
  [inst_3 : MeasureTheory.IsLocallyFiniteMeasure μ] {ρ : MeasureTheory.Measure α}
  [inst_4 : MeasureTheory.IsLocallyFiniteMeasure ρ] (hρ : ρ.AbsolutelyContinuous μ) {s : Set α},
  MeasurableSet s → ∀ {t : NNReal}, 1 < t → ρ s ≤ ↑t * (μ.withDensity (v.limRatioMeas hρ)) s

As an intermediate step to show that μ.withDensity (v.limRatioMeas hρ) = ρ, we show here that ρ ≤ t μ.withDensity (v.limRatioMeas hρ) for any t > 1.

Defined in

Mathlib.MeasureTheory.Covering.Differentiation

Cited by

1 results in Mathlib

Foundations

Depth 212 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PseudoMetricSpaceSecondCountableTopologyBorelSpaceMeasureTheory.IsLocallyFiniteMeasureMeasureTheory.IsLocallyFiniteMeasure