withDensity_ae_eq — Mathlib

English

Under AEMeasurable density d, two functions f,g are almost-everywhere equal with respect to μ.withDensity d iff they are almost-everywhere equal with respect to μ.

Русский

При плотности, а именно AEMeasurable d: f =ᵃᵉ g с μ.withDensity d эквивалентно f =ᵉ g с μ.

LaTeX

$$$f =ᵐ[μ.withDensity d] g \iff f =ᵐ[μ] g$ under assumptions on d.$$

Lean4

theorem withDensity_ae_eq {β : Type} {f g : α → β} {d : α → ℝ≥0∞} (hd : AEMeasurable d μ)
    (h_ae_nonneg : ∀ᵐ x ∂μ, d x ≠ 0) : f =ᵐ[μ.withDensity d] g ↔ f =ᵐ[μ] g :=
  Iff.intro (fun h ↦ Measure.AbsolutelyContinuous.ae_eq (withDensity_absolutelyContinuous' hd h_ae_nonneg) h)
    (fun h ↦ Measure.AbsolutelyContinuous.ae_eq (withDensity_absolutelyContinuous μ d) h)

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