| import OAI.Analysis.FiniteFactor.Sampling.WeightTransport |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal |
| noncomputable section |
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| / |
| theorem samplingIdealSecond_lintegral (h s : ℝ) (hh : 0 < h) (hh1 : h < 1) : |
| ENNReal.ofReal (samplingIdealSecond h s) = |
| ∫⁻ v, ENNReal.ofReal (samplingTransportDensity v s) ∂samplingVelocityMeasure h := by |
| have := samplingVelocity_probability h hh |
| have hv : ∀ᵐ v ∂samplingVelocityMeasure h, v ∈ Set.Ioc (0 : ℝ) h := ae_cond_mem measurableSet_Ioc |
| have hm : Measurable (fun v => samplingTransportDensity v s) := |
| samplingTransportDensity_measurable.comp (measurable_id.prodMk measurable_const) |
| have hqi : Integrable (fun v => samplingTransportDensity v s) (samplingVelocityMeasure h) := |
| (integrable_const (1/(1-h))).mono' hm.aestronglyMeasurable (by |
| filter_upwards [hv] with v hv |
| rw [Real.norm_eq_abs,abs_of_nonneg (samplingTransportDensity_bounds h v s hh.le hh1 hv.1.le hv.2).1] |
| exact (samplingTransportDensity_bounds h v s hh.le hh1 hv.1.le hv.2).2.1) |
| exact ofReal_integral_eq_lintegral_ofReal hqi (hv.mono (fun v hv => |
| (samplingTransportDensity_bounds h v s hh.le hh1 hv.1.le hv.2).1)) |
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| / |
| theorem samplingTransport_average (h : ℝ) (hh : 0 < h) (hh1 : h < 1) |
| (F : ℝ → ℝ≥0∞) (hF : Measurable F) : |
| (∫⁻ z, ∫⁻ v, F (samplingTransport v z) ∂samplingVelocityMeasure h ∂unitIntervalMeasure) = |
| ∫⁻ s, ENNReal.ofReal (samplingIdealSecond h s)*F s ∂unitIntervalMeasure := by |
| have := samplingVelocity_probability h hh |
| have hv : ∀ᵐ v ∂samplingVelocityMeasure h, v ∈ Set.Ioc (0 : ℝ) h := ae_cond_mem measurableSet_Ioc |
| have htm : Measurable (fun p : ℝ × ℝ => samplingTransport p.2 p.1) := by |
| unfold samplingTransport samplingDistance |
| fun_prop |
| have hswap := lintegral_lintegral_swap (μ := unitIntervalMeasure) (ν := samplingVelocityMeasure h) (f := fun z v => F (samplingTransport v z)) (hF.comp htm).aemeasurable |
| rw [hswap] |
| calc |
| _ = ∫⁻ v, (∫⁻ s, ENNReal.ofReal (samplingTransportDensity v s)*F s ∂unitIntervalMeasure) |
| ∂samplingVelocityMeasure h := by |
| apply lintegral_congr_ae |
| filter_upwards [hv] with v hv |
| have hm : Measurable (samplingTransport v) := by unfold samplingTransport samplingDistance; fun_prop |
| rw [← lintegral_map hF hm,samplingTransport_density v hv.1.le (hv.2.trans_lt hh1)] |
| have hmd : Measurable (fun s => ENNReal.ofReal (samplingTransportDensity v s)) := |
| (samplingTransportDensity_measurable.comp (measurable_const.prodMk measurable_id)).ennreal_ofReal |
| exact lintegral_withDensity_eq_lintegral_mul unitIntervalMeasure hmd hF |
| _ = ∫⁻ s, (∫⁻ v, ENNReal.ofReal (samplingTransportDensity v s)*F s ∂samplingVelocityMeasure h) |
| ∂unitIntervalMeasure := lintegral_lintegral_swap |
| (samplingTransportDensity_measurable.ennreal_ofReal.mul (hF.comp measurable_snd)).aemeasurable |
| _ = _ := by |
| apply lintegral_congr |
| intro s |
| have hmd : Measurable (fun v => ENNReal.ofReal (samplingTransportDensity v s)) := |
| (samplingTransportDensity_measurable.comp (measurable_id.prodMk measurable_const)).ennreal_ofReal |
| rw [lintegral_mul_const _ hmd,← samplingIdealSecond_lintegral h s hh hh1] |
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| / |
| identification. It holds against every measurable nonnegative test. -/ |
| theorem samplingWeight_endpoint_domination (h α : ℝ) |
| (hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) |
| (F : ℝ → ℝ≥0∞) (hF : Measurable F) : |
| (∫⁻ z, ∫⁻ t, ENNReal.ofReal (samplingWeight false h α z t)*F (z+t) |
| ∂unitIntervalMeasure ∂unitIntervalMeasure) ≤ |
| ∫⁻ s, ENNReal.ofReal (samplingIdealSecond h s)*F s ∂unitIntervalMeasure := by |
| rw [← samplingTransport_average h hh hh1 F hF] |
| apply lintegral_mono_ae |
| filter_upwards [show ∀ᵐ z ∂unitIntervalMeasure, z ∈ Set.Ico (0 : ℝ) 1 from |
| ae_restrict_mem measurableSet_Ico] with z hz |
| exact samplingWeight_transport_le h α z hh hh1 hα hz F hF |
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| end |
| end FiniteFactor |
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| end OAI |
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