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f0f05de | 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 | import Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut
import Mathlib.Probability.Moments.SubGaussian
namespace OAI
open MeasureTheory ProbabilityTheory Real
open scoped ENNReal NNReal
namespace Balanced
open scoped _root_.Balanced
theorem integral_mul_le_of_condExp_le
{Ω : Type*} {mΩ : MeasurableSpace Ω} {μ : Measure Ω}
[IsProbabilityMeasure μ] {m : MeasurableSpace Ω} (hm : m ≤ mΩ)
{F G : Ω → ℝ} {K : ℝ}
(hFm : StronglyMeasurable[m] F) (hF : Integrable F μ)
(hG : Integrable G μ) (hFG : Integrable (F * G) μ)
(hFnonneg : ∀ᵐ ω ∂μ, 0 ≤ F ω)
(hGcond : ∀ᵐ ω ∂μ, μ[G | m] ω ≤ K) :
(∫ ω, F ω * G ω ∂μ) ≤ K * ∫ ω, F ω ∂μ := by
have hpull := condExp_mul_of_stronglyMeasurable_left hFm hFG hG
have hprod : Integrable (fun ω => F ω * μ[G | m] ω) μ :=
integrable_condExp.congr hpull
calc
(∫ ω, F ω * G ω ∂μ) = ∫ ω, μ[F * G | m] ω ∂μ :=
(integral_condExp hm).symm
_ = ∫ ω, F ω * μ[G | m] ω ∂μ := integral_congr_ae hpull
_ ≤ ∫ ω, F ω * K ∂μ := by
apply integral_mono_ae hprod (hF.mul_const K)
filter_upwards [hFnonneg, hGcond] with ω hFω hGω
exact mul_le_mul_of_nonneg_left hGω hFω
_ = K * ∫ ω, F ω ∂μ := by rw [integral_mul_const]; ring
theorem mgf_add_le_of_condExp_exp_le
{Ω : Type*} {mΩ : MeasurableSpace Ω} {μ : Measure Ω}
[IsProbabilityMeasure μ] {m : MeasurableSpace Ω} (hm : m ≤ mΩ)
{S Z : Ω → ℝ} {θ K : ℝ}
(hSm : StronglyMeasurable[m] S)
(hSexp : Integrable (fun ω => exp (θ * S ω)) μ)
(hZexp : Integrable (fun ω => exp (θ * Z ω)) μ)
(hSumexp : Integrable (fun ω => exp (θ * (S ω + Z ω))) μ)
(hZcond : ∀ᵐ ω ∂μ, μ[fun ω => exp (θ * Z ω) | m] ω ≤ K) :
mgf (S + Z) μ θ ≤ K * mgf S μ θ := by
have hproduct : Integrable ((fun ω => exp (θ * S ω)) *
(fun ω => exp (θ * Z ω))) μ := by
convert hSumexp using 1
ext ω
simp only [Pi.mul_apply, mul_add, exp_add]
simpa only [mgf, Pi.add_apply, mul_add, exp_add] using
(integral_mul_le_of_condExp_le hm
(Real.continuous_exp.comp_stronglyMeasurable (hSm.const_mul θ))
hSexp hZexp hproduct
(Filter.Eventually.of_forall fun ω => (exp_pos (θ * S ω)).le) hZcond)
end Balanced
end OAI
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