eventually_filterAt_integrableOn — Mathlib

English

For p < q there exist measurable sets a,b such that the limRatio sublevel {x: limRatio x < p} is contained in a, and the limRatio superlevel {x: q < limRatio x} is contained in b, with μ(a ∩ b) = 0.

Русский

Пусть p < q; существуют измеримые множества a,b такие, что подмерки limRatio x < p содержатся в a, limRatio x > q содержатся в b, и μ(a ∩ b) = 0.

LaTeX

$$$\exists a,b\ MeasurableSet a, MeasurableSet b \,∧\ {x\mid v.limRatio ρ x < p} ⊆ a ∧ {x\mid q < v.limRatio ρ x} ⊆ b ∧ μ(a ∩ b) = 0.$$$

Lean4

theorem eventually_filterAt_integrableOn (x : α) {f : α → E} (hf : LocallyIntegrable f μ) :
    ∀ᶠ a in v.filterAt x, IntegrableOn f a μ :=
  by
  rcases hf x with ⟨w, w_nhds, hw⟩
  filter_upwards [v.eventually_filterAt_subset_of_nhds w_nhds] with a ha
  exact hw.mono_set ha

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