ae_eq_restrict_iff_indicator_ae_eq

English

If 𝒜 is measure-dense and c ∈ E, then constant-indicatorLp functions with sets from 𝒜 are dense in the corresponding Lp space.

Русский

Если 𝒜 плотна по мере и константный индикаторLp-выражения образуют множество, они плотны в пространстве Lp.

LaTeX

$$$ {indicatorConstLp p hs hμs c | (s) ∈ 𝒜} \\subseteq \\overline{ {indicatorConstLp p (h𝒜.measurable s hs) hμs c | (s) ∈ 𝒜} } $$$

Lean4

theorem ae_eq_restrict_iff_indicator_ae_eq {g : α → β} (hs : MeasurableSet s) :
    f =ᵐ[μ.restrict s] g ↔ s.indicator f =ᵐ[μ] s.indicator g :=
  by
  rw [Filter.EventuallyEq, ae_restrict_iff' hs]
  refine ⟨fun h => ?_, fun h => ?_⟩ <;> filter_upwards [h] with x hx
  · by_cases hxs : x ∈ s
    · simp [hxs, hx hxs]
    · simp [hxs]
  · intro hxs
    simpa [hxs] using hx

Похожие утверждения