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67aee1d | 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 | import OAI.Analysis.FreeEntropy.GaussianDifferential
namespace OAI
universe u1 u2
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
end
end OAI
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