| import Mathlib |
| import OAI.Computability.MinUncut.Estimates.Basic |
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|
| namespace OAI |
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| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped Topology NNReal |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Polynomial Filter |
| open scoped BigOperators Topology |
| open MeasureTheory ProbabilityTheory WithLp |
| open scoped BigOperators RealInnerProductSpace |
| namespace MinUncut.GaussianHermite |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators |
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|
| lemma integral_expect {T Ω : Type*} [Fintype T] [MeasurableSpace Ω] {μ : Measure Ω} |
| (f : T → Ω → ℝ) (hf : ∀ t, Integrable (f t) μ) : |
| (∫ x, (𝔼 t, f t x) ∂μ) = 𝔼 t, ∫ x, f t x ∂μ := by |
| simp only [Finset.expect_eq_sum_div_card] |
| rw [integral_div, integral_finsetSum _ (fun t _ => hf t)] |
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| lemma integral_norm_mul_le_sqrt {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} |
| {f g : Ω → ℝ} (hf : MemLp f 2 μ) (hg : MemLp g 2 μ) : |
| (∫ x, |f x| * |g x| ∂μ) ≤ Real.sqrt (∫ x, f x^2 ∂μ) * Real.sqrt (∫ x, g x^2 ∂μ) := by |
| have h := integral_mul_norm_le_Lp_mul_Lq (p := 2) (q := 2) (by rw [Real.holderConjugate_iff]; norm_num : (2:ℝ).HolderConjugate 2) |
| (by simpa using hf) (by simpa using hg) |
| simpa only [Real.norm_eq_abs, Real.rpow_two, sq_abs, ← Real.sqrt_eq_rpow] using h |
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| variable {ι : Type*} [Fintype ι] |
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| lemma psi_l1 (I : ι → ℕ) : (∫ x, |psi I x| ∂γpi ι) ≤ 1 := by |
| have h := integral_norm_mul_le_sqrt (memLp_psi I) |
| (memLp_const (1:ℝ) (p:=2) (μ:=γpi ι)) |
| have hn : (∫ x, psi I x^2 ∂γpi ι) = 1 := by simpa [sq] using psi_product I I |
| simpa [hn] using h |
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| end MinUncut.GaussianHermite |
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| end |
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| end OAI |
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