English
A variant of the zero equivalence: from equality of integrals over all measurable subsets on a sigma-finite measure, conclude a.e. equality to zero.
Русский
Вариант нулевой эквивалентности: из равенства интегралов по всем измеримым подмножествам на сигма-конечной мере следует a.e. равенство нулю.
LaTeX
$$$f = 0 \ a.e. \ μ$ when analogous integrals vanish on all measurable sets.$$
Lean4
theorem ae_eq_of_forall_setIntegral_eq_of_sigmaFinite [SigmaFinite μ] {f g : α → E}
(hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
(hg_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn g s μ)
(hfg_eq : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ) : f =ᵐ[μ] g :=
by
rw [← sub_ae_eq_zero]
have hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, (f - g) x ∂μ) = 0 :=
by
intro s hs hμs
rw [integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs), sub_eq_zero.mpr (hfg_eq s hs hμs)]
have hfg_int : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn (f - g) s μ := fun s hs hμs =>
(hf_int_finite s hs hμs).sub (hg_int_finite s hs hμs)
exact ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite hfg_int hfg
