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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 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | import OAI.Analysis.FiniteFactor.OrbitSampling.DiagonalContribution
import Mathlib.MeasureTheory.Integral.Lebesgue.Markov
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
open scoped ENNReal Topology
noncomputable section
/-- Tonelli converts summable expectations of nonnegative functions into
pointwise summability on a conull event. -/
theorem sampling_ae_summable_of_integrals {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) (f : ℕ → Ω → ℝ) (hm : ∀ n, Measurable (f n))
(hi : ∀ n, Integrable (f n) μ) (hn : ∀ n x, 0 ≤ f n x)
(hs : Summable (fun n => ∫ x, f n x ∂μ)) :
∀ᵐ x ∂μ, Summable (fun n => f n x) := by
have he (n : ℕ) : (∫⁻ x, ENNReal.ofReal (f n x) ∂μ) = ENNReal.ofReal (∫ x, f n x ∂μ) :=
(ofReal_integral_eq_lintegral_ofReal (hi n) (Eventually.of_forall (hn n))).symm
have hfin : (∫⁻ x, ∑' n, ENNReal.ofReal (f n x) ∂μ) ≠ ∞ := by
rw [lintegral_tsum (fun n => (hm n).ennreal_ofReal.aemeasurable)]
simp_rw [he]
exact hs.tsum_ofReal_ne_top
have hm' : Measurable (fun x => ∑' n, ENNReal.ofReal (f n x)) :=
Measurable.tsum (fun n => (hm n).ennreal_ofReal)
filter_upwards [ae_lt_top hm' hfin] with x hx
exact (ENNReal.summable_toReal hx.ne).congr (fun n => ENNReal.toReal_ofReal (hn n x))
theorem sourceSamplingDiagonalContribution_measurable (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool)
(E G : SourceCoordinate → Set ℝ) (hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) :
Measurable (sourceSamplingDiagonalContribution stage r b E G) := by
apply Measurable.div_const
apply Finset.measurable_sum
intro n _
exact ((sourceSamplingProduct_measurable stage r b E G hE hG n).pow_const 2).stronglyMeasurable.integral_prod_right.measurable
/-- A genuine conull conclusion for the actual diagonal contribution.
This does not assert convergence of the full empirical sampling error. -/
theorem sourceSamplingDiagonalContribution_ae (r : Bool) (b : SourceCoordinate → Bool)
(E G : SourceCoordinate → Set ℝ) (hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) :
∀ᵐ θ ∂sourceAngleLaw,
Summable (fun stage => sourceSamplingDiagonalContribution stage r b E G θ) ∧
Tendsto (fun stage => sourceSamplingDiagonalContribution stage r b E G θ) atTop (𝓝 0) := by
have ha := sampling_ae_summable_of_integrals sourceAngleLaw
(fun stage θ => sourceSamplingDiagonalContribution stage r b E G θ)
(fun stage => sourceSamplingDiagonalContribution_measurable stage r b E G hE hG)
(fun stage => sourceSamplingDiagonalContribution_integrable stage r b E G hE hG)
(fun stage θ => sourceSamplingDiagonalContribution_nonneg stage r b E G θ)
(sourceSamplingDiagonalContribution_summable_integral r b E G hE hG)
exact ha.mono (fun θ hθ => ⟨hθ,hθ.tendsto_atTop_zero⟩)
/-- One conull event handles every member of any countable family of source
tests, orientations and triangular-coordinate choices for the diagonal term. -/
theorem sourceSamplingDiagonalContribution_countable_event {ι : Type*} [Countable ι]
(r : ι → Bool) (b : ι → SourceCoordinate → Bool) (E G : ι → SourceCoordinate → Set ℝ)
(hE : ∀ j i, MeasurableSet (E j i)) (hG : ∀ j i, MeasurableSet (G j i)) :
∀ᵐ θ ∂sourceAngleLaw, ∀ j,
Summable (fun stage => sourceSamplingDiagonalContribution stage (r j) (b j) (E j) (G j) θ) ∧
Tendsto (fun stage => sourceSamplingDiagonalContribution stage (r j) (b j) (E j) (G j) θ) atTop (𝓝 0) :=
(ae_all_iff).mpr (fun j => sourceSamplingDiagonalContribution_ae (r j) (b j) (E j) (G j) (hE j) (hG j))
end
end FiniteFactor
end OAI
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