Theorems · Theorem · measure theory

VitaliFamily.measure_le_mul_of_subset_limRatioMeas_lt

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

If, for all x in a set s, one has frequently ρ a / μ a < p, then ρ s ≤ p * μ 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 v.limRatioMeas hρ < p.

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