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2fca478 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 | import Mathlib
namespace OAI
/-! Compact Parameter Integral. -/
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]
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)
end KahlerCalculus
end
end OAI
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