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2fca478 | 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 | import OAI.Geometry.PinnedDistances.Model
namespace OAI
universe u1 u2 u3
section
noncomputable section
open scoped BigOperators Topology ENNReal
open Filter MeasureTheory
namespace WeakPinned.Bounded
variable {I : Type u1} {X : Type u2} [Fintype I] [MeasurableSpace X] [MeasurableSingletonClass X]
def finiteLaw (w : I → ℝ) (U : I → X) : Measure X :=
∑ i,ENNReal.ofReal (w i) • Measure.dirac (U i)
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)]
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]
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
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) _
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]
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⟩
end WeakPinned.Bounded
end
end
end OAI
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