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c4cdbbd | 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 | import OAI.Probability.InvariantIsing.Core.FiniteSpinEntropy
namespace OAI
/-! The finite probability form of Cauchy--Schwarz, including zero atoms. -/
noncomputable section
open MeasureTheory
open scoped BigOperators
namespace InvariantIsing
lemma finite_weighted_sum_sq_le {X : Type*} [Fintype X]
(p f : X → ℝ) (hp : ∀ x, 0 ≤ p x) (hpSum : ∑ x, p x = 1) :
(∑ x, p x * f x) ^ 2 ≤ ∑ x, p x * f x ^ 2 := by
have hc := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ
(fun x => Real.sqrt (p x)) (fun x => Real.sqrt (p x) * f x)
have hm (x : X) : Real.sqrt (p x) * (Real.sqrt (p x) * f x) = p x * f x := by
rw [← mul_assoc, ← pow_two, Real.sq_sqrt (hp x)]
simp_rw [hm, mul_pow, Real.sq_sqrt (hp _)] at hc
simpa only [hpSum, one_mul] using hc
lemma finite_weighted_sum_abs_le_sqrt {X : Type*} [Fintype X]
(p f : X → ℝ) (hp : ∀ x, 0 ≤ p x) (hpSum : ∑ x, p x = 1) :
|∑ x, p x * f x| ≤ Real.sqrt (∑ x, p x * f x ^ 2) := by
have hn : 0 ≤ ∑ x, p x * f x ^ 2 :=
Finset.sum_nonneg (fun x _ => mul_nonneg (hp x) (sq_nonneg _))
apply (sq_le_sq₀ (abs_nonneg _) (Real.sqrt_nonneg _)).mp
rw [sq_abs, Real.sq_sqrt hn]
exact finite_weighted_sum_sq_le p f hp hpSum
lemma finite_integral_abs_le_sqrt_sq {X : Type*} [Fintype X] [MeasurableSpace X]
[MeasurableSingletonClass X] (μ : Measure X) [IsProbabilityMeasure μ] (F : X → ℝ) :
(∫ x, |F x| ∂μ) ≤ Real.sqrt (∫ x, F x ^ 2 ∂μ) := by
let p := fun x : X => μ.real {x}
have hp : ∀ x, 0 ≤ p x := fun x => measureReal_nonneg
have hsum : ∑ x, p x = 1 := by
have he := integral_fintype (μ := μ) (f := fun _ : X => (1 : ℝ)) Integrable.of_finite
simpa [p, measureReal_def] using he.symm
have he := finite_weighted_sum_abs_le_sqrt p (fun x => |F x|) hp hsum
have hn : 0 ≤ ∑ x, p x * |F x| :=
Finset.sum_nonneg (fun x _ => mul_nonneg (hp x) (abs_nonneg _))
simpa only [integral_fintype Integrable.of_finite, smul_eq_mul, p,
abs_of_nonneg hn, sq_abs] using he
end InvariantIsing
end
end OAI
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