| import OAI.Probability.PosteriorReplicas.Model |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace |
| noncomputable section |
| universe u |
| namespace PosteriorRegression |
| open scoped Topology |
| open InnerProductSpace |
| section LimitSet |
| variable {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] |
| [TopologicalSpace Y] [BorelSpace Y] |
| (ρ : Measure X) (τ : Measure Y) |
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| lemma map_apply_open_le_of_tendsto (F : X → Y) (Fₖ : ℕ → X → Y) |
| (hF : Measurable F) (hFₖ : ∀ k, Measurable (Fₖ k)) |
| (ht : ∀ x, Tendsto (fun k => Fₖ k x) atTop (𝓝 (F x))) |
| (C : ℝ≥0∞) (α : ℝ) |
| (hbound : ∀ k U, IsOpen U → (ρ.map (Fₖ k)) U ≤ C * (τ U)^α) |
| (U : Set Y) (hU : IsOpen U) : |
| (ρ.map F) U ≤ C * (τ U)^α := by |
| let f : ℕ → X → ℝ≥0∞ := fun k => (Fₖ k ⁻¹' U).indicator (fun _ => 1) |
| have hf : ∀ k, Measurable (f k) := fun k => |
| measurable_const.indicator (hU.measurableSet.preimage (hFₖ k)) |
| have hind : ∀ x, (F ⁻¹' U).indicator (fun _ => (1 : ℝ≥0∞)) x ≤ |
| atTop.liminf (fun k => f k x) := by |
| intro x |
| by_cases hx : F x ∈ U |
| · rw [Set.indicator_of_mem (show x ∈ F ⁻¹' U from hx)] |
| apply le_liminf_of_le (by isBoundedDefault) |
| filter_upwards [(ht x).eventually (hU.mem_nhds hx)] with k hk |
| change 1 ≤ (Fₖ k ⁻¹' U).indicator (fun _ => (1 : ℝ≥0∞)) x |
| rw [Set.indicator_of_mem (show x ∈ Fₖ k ⁻¹' U from hk)] |
| · rw [Set.indicator_of_notMem (show x ∉ F ⁻¹' U from hx)] |
| exact zero_le |
| calc |
| (ρ.map F) U = ∫⁻ x, (F ⁻¹' U).indicator (fun _ => (1 : ℝ≥0∞)) x ∂ρ := by |
| rw [Measure.map_apply hF hU.measurableSet, |
| lintegral_indicator (hU.measurableSet.preimage hF)] |
| simp |
| _ ≤ ∫⁻ x, atTop.liminf (fun k => f k x) ∂ρ := lintegral_mono hind |
| _ ≤ atTop.liminf (fun k => ∫⁻ x, f k x ∂ρ) := lintegral_liminf_le hf |
| _ ≤ _ := by |
| apply liminf_le_of_frequently_le (hu_le := by isBoundedDefault) |
| apply Filter.Eventually.frequently |
| filter_upwards [] with k |
| have hk := hbound k U hU |
| simpa [f, lintegral_indicator (hU.measurableSet.preimage (hFₖ k)), |
| Measure.map_apply (hFₖ k) hU.measurableSet] using hk |
| end LimitSet |
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| end PosteriorRegression |
| end |
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| end OAI |
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