| import OAI.Analysis.FreeEntropy.GaussianDifferential |
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| namespace OAI |
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| universe u1 u2 |
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| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped Topology |
| namespace FiniteEntropySeparation |
| variable {E : Type u1} {F : Type u2} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
| omit [NormedSpace ℝ E] [BorelSpace E] [NormedSpace ℝ F] in |
| theorem integrable_of_compact_ae (μ : Measure E) [IsFiniteMeasure μ] {K : Set E} (hK : IsCompact K) |
| (hμ : ∀ᵐ x ∂μ,x∈K) {f : E → F} (hm : AEStronglyMeasurable f μ) (hf : ContinuousOn f K) : Integrable f μ := by |
| obtain ⟨C,hC⟩ := hK.exists_bound_of_continuousOn hf |
| apply (integrable_const C).mono' hm |
| filter_upwards [hμ] with x hx |
| exact hC x hx |
| theorem hasDerivAt_integral_compact {E : Type u1} {F : Type u2} |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] [SecondCountableTopology E] (μ : Measure E) [IsFiniteMeasure μ] {K : Set E} (hK : IsCompact K) |
| (hμ : ∀ᵐ x ∂μ,x∈K) (f f' : ℝ → E → F) {a b t : ℝ} (ha : a<t) (hb : t<b) |
| (hf : ∀s∈Icc a b,Continuous (f s)) (hf' : ∀s∈Icc a b,Continuous (f' s)) |
| (hc : ContinuousOn (fun z : ℝ × E => f' z.1 z.2) (Icc a b ×ˢ K)) |
| (hd : ∀s∈Icc a b,∀x∈K,HasDerivAt (fun u => f u x) (f' s x) s) : |
| Integrable (f' t) μ ∧ HasDerivAt (fun u => ∫ x,f u x ∂μ) (∫ x,f' t x ∂μ) t := by |
| obtain ⟨C,hC⟩ := (isCompact_Icc.prod hK).exists_bound_of_continuousOn hc |
| apply hasDerivAt_integral_of_dominated_loc_of_deriv_le (s := Icc a b) (bound := fun _ => C) (Icc_mem_nhds ha hb) |
| · filter_upwards [Icc_mem_nhds ha hb] with s hs |
| exact (hf s hs).aestronglyMeasurable |
| · exact integrable_of_compact_ae μ hK hμ (hf t ⟨ha.le,hb.le⟩).aestronglyMeasurable (hf t ⟨ha.le,hb.le⟩).continuousOn |
| · exact (hf' t ⟨ha.le,hb.le⟩).aestronglyMeasurable |
| · filter_upwards [hμ] with x hx s hs |
| exact hC (s,x) ⟨hs,hx⟩ |
| · exact integrable_const _ |
| · filter_upwards [hμ] with x hx s hs |
| exact hd s hs x hx |
| end FiniteEntropySeparation |
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| end |
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| end OAI |
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