| import Mathlib |
| import OAI.Geometry.IntegralFillings.Currents.Summation |
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| namespace OAI |
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| section |
| open Set MeasureTheory Measure Filter Module |
| open Set Filter MeasureTheory Measure ContinuousLinearMap |
| open scoped Topology Convolution NNReal |
| open Set Filter MeasureTheory Measure Metric |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open Set MeasureTheory Filter |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
| open Filter Set |
| open scoped Topology NNReal |
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
| open Matrix |
| open scoped RealInnerProductSpace MatrixOrder |
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|
| namespace SharpIntegralFillings |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
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| variable {X : Type*} [MeasurableSpace X] {μ : Measure X} |
| lemma intCast_tsum_exists (a : ℕ → ℤ) : |
| ∃ z : ℤ, (z : ℝ) = ∑' i, (a i : ℝ) := by |
| by_cases h : Summable (fun i => (a i : ℝ)) |
| · have hm : (∑' i, (a i : ℝ)) ∈ Set.range ((↑) : ℤ → ℝ) := by |
| apply Real.isClosed_range_intCast.mem_of_tendsto h.hasSum |
| exact Filter.Eventually.of_forall fun s => ⟨∑ i ∈ s, a i,by simp⟩ |
| exact hm |
| · exact ⟨0,by simpa only [Int.cast_zero] using (tsum_eq_zero_of_not_summable h).symm⟩ |
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| noncomputable def summedMultiplicity (θ : ℕ → X → ℤ) (x : X) : ℤ := |
| (intCast_tsum_exists (fun i => θ i x)).choose |
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| omit [MeasurableSpace X] in |
| lemma summedMultiplicity_cast (θ : ℕ → X → ℤ) (x : X) : |
| (summedMultiplicity θ x : ℝ) = ∑' i, (θ i x : ℝ) := |
| (intCast_tsum_exists (fun i => θ i x)).choose_spec |
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| lemma summedMultiplicity_integrable {θ : ℕ → X → ℤ} |
| (hθ : ∀ i, Integrable (fun x => (θ i x : ℝ)) μ) |
| (hS : Summable (fun i => ∫ x, |(θ i x : ℝ)| ∂μ)) : |
| Integrable (fun x => (summedMultiplicity θ x : ℝ)) μ := by |
| simp_rw [summedMultiplicity_cast] |
| exact integrable_series hθ (by simpa only [Real.norm_eq_abs] using hS) |
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| lemma summedMultiplicity_integral_weight {θ : ℕ → X → ℤ} |
| (hθ : ∀ i, Integrable (fun x => (θ i x : ℝ)) μ) |
| (hS : Summable (fun i => ∫ x, |(θ i x : ℝ)| ∂μ)) |
| {g : X → ℝ} (hg : AEStronglyMeasurable g μ) |
| {M : ℝ} (hM : ∀ᵐ x ∂μ, ‖g x‖ ≤ M) : |
| (∫ x, (summedMultiplicity θ x : ℝ) * g x ∂μ) = |
| ∑' i, ∫ x, (θ i x : ℝ) * g x ∂μ := by |
| have hI i := (hθ i).mul_bdd hg hM |
| have hsum : Summable (fun i => ∫ x, ‖(θ i x : ℝ) * g x‖ ∂μ) := by |
| apply (hS.mul_left M).of_nonneg_of_le |
| · intro i |
| exact integral_nonneg (fun _ => norm_nonneg _) |
| · intro i |
| rw [←integral_const_mul] |
| apply integral_mono_ae (hI i).norm ((hθ i).abs.const_mul M) |
| filter_upwards [hM] with x hx |
| simpa only [norm_mul,Real.norm_eq_abs,mul_comm] using |
| mul_le_mul_of_nonneg_left hx (norm_nonneg (θ i x : ℝ)) |
| rw [integral_tsum_of_summable_integral_norm hI hsum] |
| apply integral_congr_ae |
| filter_upwards with x |
| rw [summedMultiplicity_cast,tsum_mul_right] |
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| end SharpIntegralFillings |
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| end |
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| end OAI |
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