| import Mathlib |
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| namespace OAI |
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| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling.RMSIntegral |
| open MeasureTheory |
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| variable {Ω E : Type*} [MeasurableSpace Ω] {ν : Measure Ω} |
| [NormedAddCommGroup E] |
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| lemma norm_sub_sq_le_two (x y : E) : ‖x-y‖^2≤2*‖x‖^2+2*‖y‖^2 := by |
| have h := pow_le_pow_left₀ (norm_nonneg (x-y)) (norm_sub_le x y) 2 |
| nlinarith [sq_nonneg (‖x‖-‖y‖)] |
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| lemma sub_sq_bound {f g : Ω → E} (hf : AEStronglyMeasurable f ν) |
| (hg : AEStronglyMeasurable g ν) |
| (hfi : Integrable (fun z => ‖f z‖^2) ν) (hgi : Integrable (fun z => ‖g z‖^2) ν) |
| {A B : ℝ} (hA : (∫z,‖f z‖^2 ∂ν)≤A) (hB : (∫z,‖g z‖^2 ∂ν)≤B) : |
| Integrable (fun z => ‖f z-g z‖^2) ν ∧ |
| (∫z,‖f z-g z‖^2 ∂ν)≤2*A+2*B := by |
| have hi : Integrable (fun z => 2*‖f z‖^2+2*‖g z‖^2) ν := |
| (hfi.const_mul 2).add (hgi.const_mul 2) |
| have hm : AEStronglyMeasurable (fun z => ‖f z-g z‖^2) ν := (hf.sub hg).norm.pow 2 |
| have hs : Integrable (fun z => ‖f z-g z‖^2) ν := |
| hi.mono' hm (Filter.Eventually.of_forall (fun z => by |
| simpa only [Real.norm_eq_abs,abs_sq] using norm_sub_sq_le_two (f z) (g z))) |
| refine ⟨hs,(integral_mono hs hi (fun z => norm_sub_sq_le_two (f z) (g z))).trans ?_⟩ |
| rw [integral_add (hfi.const_mul 2) (hgi.const_mul 2),integral_const_mul,integral_const_mul] |
| linarith |
| end LogConcaveSampling.RMSIntegral |
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| end |
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| end |
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| end |
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| end OAI |
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