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dd14c26 | 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 | import Mathlib
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling.RMSIntegral
open MeasureTheory
variable {Ω E : Type*} [MeasurableSpace Ω] {ν : Measure Ω}
[NormedAddCommGroup E]
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‖)]
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
end
end
end
end OAI
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