File size: 3,316 Bytes
f61e547 | 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 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | import OAI.Probability.FreeIsing.Energy.DiscrepancyRowEnergy
import OAI.Probability.FreeIsing.Estimates.ProgressivePrimitiveFubini
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.DiscrepancyEnergy
variable {Ω ι : Type*} [mΩ : MeasurableSpace Ω] [Countable ι]
theorem density_congr_ae {μ : Measure Ω}
{c c' b b' : ι → Ω → ℝ} {X Y m m' k k' : Ω → ℝ}
(hc : ∀ i, c i =ᵐ[μ] b i) (hc' : ∀ i, c' i =ᵐ[μ] b' i)
(hX : X =ᵐ[μ] Y) (hm : m =ᵐ[μ] k) (hm' : m' =ᵐ[μ] k') :
density c c' X m m' =ᵐ[μ] density b b' Y k k' := by
filter_upwards [ae_all_iff.mpr hc, ae_all_iff.mpr hc', hX, hm, hm']
with ω hω hω' hXω hmω hmω'
simp only [density, hω, hω', hXω, hmω, hmω']
theorem isStronglyProgressive_density
{F : Filtration ℝ≥0 mΩ}
{c c' : ι → ℝ≥0 → Ω → ℝ} {X m m' : ℝ≥0 → Ω → ℝ}
(hc : ∀ i, IsStronglyProgressive F (c i))
(hc' : ∀ i, IsStronglyProgressive F (c' i))
(hX : IsStronglyProgressive F X)
(hm : IsStronglyProgressive F m) (hm' : IsStronglyProgressive F m') :
IsStronglyProgressive F (fun t =>
density (fun i => c i t) (fun i => c' i t) (X t) (m t) (m' t)) := by
intro T
let : MeasurableSpace Ω := F T
have hnoise : StronglyMeasurable (fun p : Set.Iic T × Ω =>
∑' i : ι, (c i p.1 p.2 - c' i p.1 p.2) ^ 2) := StronglyMeasurable.tsum
(fun i => ((hc i T).sub (hc' i T)).pow 2)
have hleft : StronglyMeasurable (fun p : Set.Iic T × Ω =>
∑' i : ι, c i p.1 p.2 ^ 2) := StronglyMeasurable.tsum (fun i => (hc i T).pow 2)
have hright : StronglyMeasurable (fun p : Set.Iic T × Ω =>
∑' i : ι, c' i p.1 p.2 ^ 2) := StronglyMeasurable.tsum (fun i => (hc' i T).pow 2)
exact hnoise.add (((stronglyMeasurable_const.mul (hX T)).mul
(((hm T).mul hleft).sub ((hm' T).mul hright))))
theorem ae_all_primitive_density_eq_of_modification {μ : Measure Ω} [SFinite μ]
{F G : Filtration ℝ≥0 mΩ}
{c c' b b' : ι → ℝ≥0 → Ω → ℝ}
{X Y m m' k k' : ℝ≥0 → Ω → ℝ}
(hc : ∀ i, IsStronglyProgressive F (c i))
(hc' : ∀ i, IsStronglyProgressive F (c' i))
(hX : IsStronglyProgressive F X)
(hm : IsStronglyProgressive F m) (hm' : IsStronglyProgressive F m')
(hb : ∀ i, IsStronglyProgressive G (b i))
(hb' : ∀ i, IsStronglyProgressive G (b' i))
(hY : IsStronglyProgressive G Y)
(hk : IsStronglyProgressive G k) (hk' : IsStronglyProgressive G k')
(hcb : ∀ i t, c i t =ᵐ[μ] b i t)
(hcb' : ∀ i t, c' i t =ᵐ[μ] b' i t)
(hXY : ∀ t, X t =ᵐ[μ] Y t)
(hmk : ∀ t, m t =ᵐ[μ] k t) (hmk' : ∀ t, m' t =ᵐ[μ] k' t) :
∀ᵐ ω ∂μ, ∀ t,
nnrealTimePrimitive (fun s => density (fun i => c i s)
(fun i => c' i s) (X s) (m s) (m' s)) t ω =
nnrealTimePrimitive (fun s => density (fun i => b i s)
(fun i => b' i s) (Y s) (k s) (k' s)) t ω := by
apply ae_all_nnrealTimePrimitive_eq_of_modification
(isStronglyProgressive_density hc hc' hX hm hm')
(isStronglyProgressive_density hb hb' hY hk hk')
intro t
exact density_congr_ae (fun i => hcb i t) (fun i => hcb' i t)
(hXY t) (hmk t) (hmk' t)
end Problem367.DiscrepancyEnergy
end
end OAI
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