| import Mathlib |
| import OAI.Probability.LogConcave.Model |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
|
|
| section LowerProof |
| open Matrix Topology TopologicalSpace ProbabilityTheory Classical WithLp |
| open scoped Matrix.Norms.Elementwise |
| open WithLp |
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal NNReal |
| open Matrix |
| open Polynomial |
| open scoped BigOperators |
|
|
| namespace LogConcaveSampling |
|
|
| theorem TVAtMost.map {E F : Type*} [MeasurableSpace E] [MeasurableSpace F] |
| {μ ν : Measure E} {ε : ℝ} (h : TVAtMost μ ν ε) |
| {f : E → F} (hf : Measurable f) : TVAtMost (μ.map f) (ν.map f) ε := by |
| intro s hs |
| simpa only [Measure.real, Measure.map_apply hf hs] using h (f ⁻¹' s) (hs.preimage hf) |
|
|
| end LogConcaveSampling |
|
|
| namespace LogConcaveSampling |
| lemma prob_prod_event_lower {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] |
| (μ : Measure X) (ν : Measure Y) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| {s : Set (X × Y)} (hs : MeasurableSet s) {a : ℝ} |
| (h : ∀ y, a ≤ μ.real ((fun x => (x,y)) ⁻¹' s)) : a ≤ (μ.prod ν).real s := by |
| have h' : ENNReal.ofReal a ≤ (μ.prod ν) s := by |
| rw [Measure.prod_apply_symm hs] |
| calc |
| _ = ∫⁻ _ : Y, ENNReal.ofReal a ∂ν := by simp |
| _ ≤ _ := lintegral_mono (fun y => |
| (ENNReal.ofReal_le_iff_le_toReal (measure_ne_top μ _)).mpr (h y)) |
| exact (ENNReal.ofReal_le_iff_le_toReal (measure_ne_top (μ.prod ν) s)).mp h' |
| end LogConcaveSampling |
|
|
| end LowerProof |
| end |
| end |
| end |
|
|
| end OAI |
|
|
