Theorems · Theorem · measure theory

VitaliFamily.mul_measure_le_of_subset_lt_limRatioMeas

∀ {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) {q : NNReal} {s : Set α},
  s ⊆ {x | ↑q < v.limRatioMeas hρ x} → ↑q * μ s ≤ ρ s

If, for all x in a set s, one has frequently q < ρ a / μ a, then q * μ s ≤ ρ s, as proved in measure_le_of_frequently_le. Since ρ a / μ a tends almost everywhere to v.limRatioMeas hρ x, the same property holds for sets s on which q < v.limRatioMeas hρ.

Defined in

Mathlib.MeasureTheory.Covering.Differentiation

Cited by

2 results in Mathlib

Foundations

Depth 210 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PseudoMetricSpaceSecondCountableTopologyBorelSpaceMeasureTheory.IsLocallyFiniteMeasureMeasureTheory.IsLocallyFiniteMeasure