| import Mathlib |
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| namespace OAI |
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| /-! Compact Parameter Integral. -/ |
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| noncomputable section |
| open Set Filter Topology MeasureTheory |
| namespace KahlerCalculus |
| variable {E : Type*} [TopologicalSpace E] [T2Space E] |
| [MeasurableSpace E] [BorelSpace E] {μ : Measure E} [IsFiniteMeasureOnCompacts μ] |
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| lemma hasDerivAt_integral_compact {F F' : ℝ → E → ℝ} {K : Set E} |
| (hK : IsCompact K) (hF : Continuous (Function.uncurry F)) |
| (hF' : Continuous (Function.uncurry F')) |
| (hs : ∀ t, Function.support (F t) ⊆ K) |
| (hs' : ∀ t, Function.support (F' t) ⊆ K) |
| (hd : ∀ t z, HasDerivAt (fun s => F s z) (F' t z) t) (t : ℝ) : |
| HasDerivAt (fun s => ∫ z, F s z ∂μ) (∫ z, F' t z ∂μ) t := by |
| have hfc (s : ℝ) : Continuous (F s) := hF.comp (continuous_const.prodMk continuous_id) |
| have hfc' (s : ℝ) : Continuous (F' s) := hF'.comp (continuous_const.prodMk continuous_id) |
| have hfi (s : ℝ) : Integrable (F s) μ := (hfc s).integrable_of_hasCompactSupport |
| (HasCompactSupport.of_support_subset_isCompact hK (hs s)) |
| obtain ⟨C, hC⟩ := (isCompact_Icc (a := t - 1) (b := t + 1)).prod hK |
| |>.exists_bound_of_continuousOn hF'.continuousOn |
| let B : E → ℝ := K.indicator (fun _ => max C 0) |
| have hB : Integrable B μ := by |
| apply (integrable_indicator_iff hK.measurableSet).mpr |
| exact integrableOn_const hK.measure_lt_top.ne |
| refine (hasDerivAt_integral_of_dominated_loc_of_deriv_le |
| (Ioo_mem_nhds (by linarith : t - 1 < t) (by linarith : t < t + 1)) |
| (Filter.Eventually.of_forall (fun s => (hfc s).aestronglyMeasurable)) (hfi t) |
| (hfc' t).aestronglyMeasurable ?_ hB ?_).2 |
| · filter_upwards [] with z s hst |
| by_cases hz : z ∈ K |
| · exact (hC (s, z) ⟨⟨hst.1.le, hst.2.le⟩, hz⟩).trans |
| (by simpa only [B, Set.indicator_of_mem hz] using le_max_left C 0) |
| · have he : F' s z = 0 := Function.notMem_support.mp (fun hn => hz (hs' s hn)) |
| simp only [he, norm_zero, B, Set.indicator_of_notMem hz, le_refl] |
| · exact Filter.Eventually.of_forall (fun z s _ => hd s z) |
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| end KahlerCalculus |
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| end |
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| end OAI |
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