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a2ee19a | 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 | import OAI.Probability.PosteriorReplicas.Model
namespace OAI
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)
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
end PosteriorRegression
end
end OAI
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