| import OAI.Probability.FreeIsing.Brownian.MeanSquareDescent |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
|
|
| namespace Problem367.PredictableL2 |
|
|
| theorem hasSum_toLp_comp_measurePreserving |
| {Ω Ω' ι : Type*} [MeasurableSpace Ω] [MeasurableSpace Ω'] |
| {P : Measure Ω} {Q : Measure Ω'} {φ : Ω' → Ω} |
| (hφ : MeasurePreserving φ Q P) |
| (f : ι → Ω → ℝ) (hf : ∀ i, MemLp (f i) 2 P) |
| {Y : Ω → ℝ} (hY : MemLp Y 2 P) |
| (hsum : HasSum (fun i => (hf i).toLp (f i)) (hY.toLp Y)) : |
| HasSum (fun i => ((hf i).comp_measurePreserving hφ).toLp (f i ∘ φ)) |
| ((hY.comp_measurePreserving hφ).toLp (Y ∘ φ)) := by |
| apply (hasSum_toLp_iff_meanSquare (fun i => f i ∘ φ) |
| (fun i => (hf i).comp_measurePreserving hφ) |
| (hY.comp_measurePreserving hφ)).mpr |
| exact (BrownianGrid.meanSquare_limit_comp_iff hφ |
| (fun s : Finset ι => (memLp_finsetSum s (fun i _ => hf i)).aestronglyMeasurable) |
| hY.aestronglyMeasurable).mpr |
| (tendsto_integral_sq_error_of_hasSum f hf hY hsum) |
|
|
| end Problem367.PredictableL2 |
|
|
| end |
|
|
| end OAI |
|
|
