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dd14c26 | 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 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 | import Mathlib
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling.CompactExpectation
open Set MeasureTheory Filter Function
open scoped Topology
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[MeasurableSpace E] [BorelSpace E] {μ : Measure E} [IsFiniteMeasureOnCompacts μ]
lemma hasDerivAt_setIntegral {f : ℝ × E → ℝ} (hf : ContDiff ℝ 1 f)
{K : Set E} (hK : IsCompact K) (t : ℝ) :
HasDerivAt (fun t => ∫y in K,f (t,y) ∂μ)
(∫y in K,fderiv ℝ f (t,y) (1,0) ∂μ) t := by
let D := fun p : ℝ × E => fderiv ℝ f p (1,0)
have hD : Continuous D := (hf.continuous_fderiv (by norm_num)).clm_apply continuous_const
obtain ⟨C,hC⟩ := ((isCompact_closedBall t 1).prod hK).exists_bound_of_continuousOn hD.continuousOn
have hh := hasDerivAt_integral_of_dominated_loc_of_deriv_le (μ:=μ.restrict K)
(F:=fun t y => f (t,y)) (F':=fun t y => D (t,y))
(Metric.ball_mem_nhds t zero_lt_one)
(Eventually.of_forall fun s => (hf.continuous.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable)
((hf.continuous.comp (continuous_const.prodMk continuous_id)).continuousOn.integrableOn_compact hK)
((hD.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable)
(by
filter_upwards [ae_restrict_mem hK.measurableSet] with y hy s hs
exact hC (s,y) ⟨Metric.ball_subset_closedBall hs,hy⟩)
(integrableOn_const (μ:=μ) hK.measure_ne_top)
(by
filter_upwards [] with y s _hs
exact (hf.differentiable (by norm_num) (s,y)).hasFDerivAt.comp_hasDerivAt s
((hasDerivAt_id s).prodMk (hasDerivAt_const s y)))
exact hh.2
theorem compact_test_hasDerivAt {A : ℝ × E → ℝ} (hA : ContDiff ℝ 1 A)
{φ : E → ℝ} (hφ : ContDiff ℝ 1 φ) (hφc : HasCompactSupport φ) (t : ℝ) :
HasDerivAt (fun t => ∫y,A (t,y)*φ y ∂μ)
(∫y,(fderiv ℝ A (t,y) (1,0))*φ y ∂μ) t := by
let B := fun p : ℝ × E => A p*φ p.2
have hB : ContDiff ℝ 1 B := hA.mul (hφ.comp contDiff_snd)
have hD (s : ℝ) (y : E) : fderiv ℝ B (s,y) (1,0)=fderiv ℝ A (s,y) (1,0)*φ y := by
have hd := (hA.differentiable (by norm_num) (s,y)).hasFDerivAt.mul
((hφ.differentiable (by norm_num) y).hasFDerivAt.comp (s,y)
(ContinuousLinearMap.snd ℝ ℝ E).hasFDerivAt)
change HasFDerivAt B _ _ at hd
rw [hd.fderiv]
simp [ContinuousLinearMap.snd,mul_comm]
have he (s : ℝ) : (∫y in tsupport φ,B (s,y) ∂μ)=∫y,A (s,y)*φ y ∂μ := by
apply setIntegral_eq_integral_of_forall_compl_eq_zero
intro y hy
simp only [B,image_eq_zero_of_notMem_tsupport hy,mul_zero]
have hg : (∫y in tsupport φ,fderiv ℝ B (t,y) (1,0) ∂μ)=
∫y,fderiv ℝ A (t,y) (1,0)*φ y ∂μ := by
simp only [hD]
apply setIntegral_eq_integral_of_forall_compl_eq_zero
intro y hy
simp only [image_eq_zero_of_notMem_tsupport hy,mul_zero]
simpa only [he,hg] using hasDerivAt_setIntegral (μ:=μ) hB hφc t
end LogConcaveSampling.CompactExpectation
end
end
end
end OAI
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