Theorems · Theorem · measure theory

VitaliFamily.null_of_frequently_le_of_frequently_ge

∀ {α : Type u_1} [inst : PseudoMetricSpace α] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
  (v : VitaliFamily μ) [SecondCountableTopology α] [BorelSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ]
  {ρ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure ρ],
  ρ.AbsolutelyContinuous μ →
    ∀ {c d : NNReal},
      c < d →
        ∀ (s : Set α),
          (∀ x ∈ s, ∃ᶠ (a : Set α) in v.filterAt x, ρ a ≤ ↑c * μ a) →
            (∀ x ∈ s, ∃ᶠ (a : Set α) in v.filterAt x, ↑d * μ a ≤ ρ a) → μ s = 0

A set of points s satisfying both ρ a ≤ c * μ a and ρ a ≥ d * μ a at arbitrarily small sets in a Vitali family has measure 0 if c < d. Indeed, the first inequality should imply that ρ s ≤ c * μ s, and the second one that ρ s ≥ d * μ s, a contradiction if 0 < μ s.

Defined in

Mathlib.MeasureTheory.Covering.Differentiation

Cited by

1 results in Mathlib

Foundations

Depth 204 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PseudoMetricSpaceSecondCountableTopologyBorelSpaceMeasureTheory.IsLocallyFiniteMeasureMeasureTheory.IsLocallyFiniteMeasure