ae_nonneg_restrict_of_forall_setIntegral_nonneg_inter

English

Restricting to a measurable set t with finite measure preserves the ae-nonnegativity of f provided the integrals over intersactions with t are nonnegative.

Русский

Ограничение функции на измеримое множество t сохраняет ae-негативность, если интегралы по пересечениям с t неотрицательны.

LaTeX

$$$0 \leq_{a.e.} f \text{ on } t\text{ provided } f|_{t}$ integrals satisfy nonnegativity on intersections.$$

Lean4

theorem ae_nonneg_restrict_of_forall_setIntegral_nonneg_inter {f : α → ℝ} {t : Set α} (hf : IntegrableOn f t μ)
    (hf_zero : ∀ s, MeasurableSet s → μ (s ∩ t) < ∞ → 0 ≤ ∫ x in s ∩ t, f x ∂μ) : 0 ≤ᵐ[μ.restrict t] f :=
  by
  refine ae_nonneg_of_forall_setIntegral_nonneg hf fun s hs h's => ?_
  simp_rw [Measure.restrict_restrict hs]
  apply hf_zero s hs
  rwa [Measure.restrict_apply hs] at h's

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