File size: 2,997 Bytes
9eeb9c7 | 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 59 60 61 62 63 64 | import OAI.Analysis.IrrationalRotation.Analysis.MaskedCauchySchwarz
import OAI.Analysis.IrrationalRotation.Analysis.MaskedTail
namespace OAI
open MeasureTheory
open scoped ENNReal
namespace InvariantProjection
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω}
theorem lintegral_tsum_masked_mul_le_of_condExp
{ι : Type*} [mι : MeasurableSpace ι] [MeasurableSingletonClass ι] [Countable ι]
[IsFiniteMeasure μ]
{a b : Ω × ι → ℝ} (ha : Measurable a) (hb : Measurable b)
(ha_nonneg : ∀ z, 0 ≤ a z) (hb_nonneg : ∀ z, 0 ≤ b z)
{H : Set (Ω × ι)} (hH : MeasurableSet H)
{m : MeasurableSpace Ω} (hm : m ≤ mΩ) {δ : ℝ} (hδ : 0 ≤ δ)
(ha_endpoint : ∀ i, StronglyMeasurable[m] (fun ω => a (ω, i) ^ 2))
(hcond : ∀ i, μ[({ω | (ω, i) ∈ H}).indicator (fun _ => (1 : ℝ)) | m]
≤ᵐ[μ] fun _ => δ)
(ha_energy : (∫⁻ ω, ∑' i, ENNReal.ofReal (a (ω, i) ^ 2) ∂μ) ≤ 1)
(hb_energy : (∫⁻ ω, ∑' i, ENNReal.ofReal (b (ω, i) ^ 2) ∂μ) ≤ 1) :
(∫⁻ ω, ∑' i, H.indicator
(fun z => ENNReal.ofReal (a z) * ENNReal.ofReal (b z)) (ω, i) ∂μ) ≤
ENNReal.ofReal (Real.sqrt δ) := by
have ha_sq_int : ∀ i, Integrable (fun ω => a (ω, i) ^ 2) μ := by
intro i
refine ⟨((ha.comp (@measurable_prodMk_right Ω ι mΩ mι i)).pow_const 2).aestronglyMeasurable, ?_⟩
apply (hasFiniteIntegral_iff_ofReal (Filter.Eventually.of_forall fun _ => sq_nonneg _)).2
calc
(∫⁻ ω, ENNReal.ofReal (a (ω, i) ^ 2) ∂μ) ≤
∫⁻ ω, ∑' j, ENNReal.ofReal (a (ω, j) ^ 2) ∂μ :=
lintegral_mono fun ω => ENNReal.le_tsum (f := fun j => ENNReal.ofReal (a (ω, j) ^ 2)) i
_ ≤ 1 := ha_energy
_ < ⊤ := by simp
have hs : ∀ i, MeasurableSet[mΩ] {ω | (ω, i) ∈ H} := fun i =>
hH.preimage (@measurable_prodMk_right Ω ι mΩ mι i)
have hsmall := lintegral_tsum_indicator_le_of_condExp_indicator_le_of_energy_le_one
(mΩ := mΩ) hm hs hδ ha_sq_int ha_endpoint
(fun _ => Filter.Eventually.of_forall fun _ => sq_nonneg _) hcond ha_energy
have hmasked : (∫⁻ ω, ∑' i, H.indicator
(fun z => ENNReal.ofReal (a z) ^ (2 : ℕ)) (ω, i) ∂μ) ≤ ENNReal.ofReal δ := by
convert hsmall using 1
apply lintegral_congr
intro ω
apply tsum_congr
intro i
by_cases hi : (ω, i) ∈ H
· simp [hi, ENNReal.ofReal_pow (ha_nonneg (ω, i))]
· simp [hi]
have hb2 : (∫⁻ ω, ∑' i, ENNReal.ofReal (b (ω, i)) ^ (2 : ℕ) ∂μ) ≤ 1 := by
simpa only [← ENNReal.ofReal_pow (hb_nonneg _)] using hb_energy
let : MeasurableSpace Ω := mΩ
have hcs := Invariant233.MaskedTail.lintegral_tsum_indicator_mul_le_sqrt
(μ := μ) hH ha.ennreal_ofReal.aemeasurable hb.ennreal_ofReal.aemeasurable hmasked hb2
simpa only [ENNReal.ofReal_rpow_of_nonneg hδ (by norm_num : (0 : ℝ) ≤ 1 / 2),
← Real.sqrt_eq_rpow] using hcs
end InvariantProjection
end OAI
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