Stability Slice Bound

Supporting estimates for the Navier–Stokes development.

theorem CKN.stability_essSup_lintegral_le_of_ae_tendsto {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure β) (F : ℕ → β → α → ENNReal) (G : β → α → ENNReal) (M : ENNReal) (hF : ∀ (n : ℕ), ∀ᵐ (t : β) ∂ν, AEMeasurable (F n t) μ) (hconv : ∀ᵐ (t : β) ∂ν, ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun (n : ℕ) => F n t x) Filter.atTop (nhds (G t x))) (hbound : ∀ (n : ℕ), ∀ᵐ (t : β) ∂ν, ∫⁻ (x : α), F n t x ∂μ ≤ M) : essSup (fun (t : β) => ∫⁻ (x : α), G t x ∂μ) ν ≤ M

A uniform almost-everywhere bound on slice energies passes to an almost-everywhere pointwise limit by Fatou's lemma (thm:stability).