| import OAI.Geometry.PinnedDistances.Model |
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| namespace OAI |
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| universe u1 u2 u3 |
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| section |
| noncomputable section |
| open scoped BigOperators Topology ENNReal |
| open Filter MeasureTheory |
| namespace WeakPinned.Bounded |
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| variable {I : Type u1} {X : Type u2} [Fintype I] [MeasurableSpace X] [MeasurableSingletonClass X] |
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| def finiteLaw (w : I → ℝ) (U : I → X) : Measure X := |
| ∑ i,ENNReal.ofReal (w i) • Measure.dirac (U i) |
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| omit [MeasurableSingletonClass X] in |
| lemma finiteLaw_univ (w : I → ℝ) (U : I → X) (hw : ∀ i,0≤ w i) : |
| finiteLaw w U Set.univ=ENNReal.ofReal (∑ i,w i) := by |
| simp [finiteLaw,Measure.finsetSum_apply,ENNReal.ofReal_sum_of_nonneg (fun i _ => hw i)] |
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| omit [MeasurableSingletonClass X] in |
| lemma finiteLaw_probability (w : I → ℝ) (U : I → X) |
| (hw : ∀ i,0≤ w i) (hW : ∑ i,w i=1) : IsProbabilityMeasure (finiteLaw w U) := by |
| constructor |
| rw [finiteLaw_univ w U hw,hW,ENNReal.ofReal_one] |
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| lemma integral_finiteLaw (w : I → ℝ) (U : I → X) (hw : ∀ i,0≤ w i) (f : X → ℝ) : |
| ∫ x,f x ∂ finiteLaw w U=∑ i,w i*f (U i) := by |
| rw [finiteLaw,integral_finsetSum_measure] |
| · simp only [integral_smul_measure,integral_dirac,ENNReal.toReal_ofReal (hw _),smul_eq_mul] |
| · intro i _ |
| exact (integrable_dirac (by simp)).smul_measure ENNReal.ofReal_ne_top |
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| lemma finiteLaw_ae (w : I → ℝ) (U : I → X) {p : X → Prop} |
| (hp : ∀ i,w i≠0 → p (U i)) : ∀ᵐ x ∂ finiteLaw w U,p x := by |
| rw [finiteLaw,ae_finsetSum_measure_iff] |
| intro i _ |
| by_cases h : w i=0 |
| · simp [h] |
| · exact Measure.ae_smul_measure (by simpa only [ae_dirac_eq,Filter.eventually_pure] using hp i h) _ |
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| omit [MeasurableSingletonClass X] in |
| lemma finiteLaw_map {Y : Type u3} [MeasurableSpace Y] (w : I → ℝ) (U : I → X) |
| (f : X → Y) (hf : Measurable f) : (finiteLaw w U).map f=finiteLaw w (f ∘ U) := by |
| simp only [finiteLaw, Measure.map_finset_sum' hf.aemeasurable, |
| Measure.map_smul _ hf.aemeasurable, Measure.map_dirac' hf, Function.comp_def] |
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| def finiteProbability (w : I → ℝ) (U : I → X) (hw : ∀ i,0≤ w i) (hW : ∑ i,w i=1) : |
| ProbabilityMeasure X := ⟨finiteLaw w U,finiteLaw_probability w U hw hW⟩ |
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| end WeakPinned.Bounded |
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| end |
| end |
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| end OAI |
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