| import OAI.Analysis.FiniteFactor.OrbitSampling.DiagonalContribution |
| import Mathlib.MeasureTheory.Integral.Lebesgue.Markov |
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|
| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped ENNReal Topology |
| noncomputable section |
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| / |
| 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 |
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| / |
| 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⟩) |
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| / |
| 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)) |
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| end |
| end FiniteFactor |
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| end OAI |
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