| import Mathlib | |
| namespace OAI | |
| open MeasureTheory Set Filter | |
| open scoped Topology | |
| namespace CompactControls | |
| variable {Z U : Type*} [TopologicalSpace Z] [MeasurableSpace Z] [BorelSpace Z] | |
| [SecondCountableTopology Z] | |
| [TopologicalSpace U] [MeasurableSpace U] [BorelSpace U] | |
| [SecondCountableTopology U] | |
| lemma ci {X : Type*} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X] | |
| [CompactSpace X] (μ : Measure X) [IsFiniteMeasure μ] {f : X → ℝ} (hf : Continuous f) : | |
| Integrable f μ := hf.integrable_of_hasCompactSupport (isCompact_univ.of_isClosed_subset isClosed_closure (subset_univ _)) | |
| omit [BorelSpace Z] [SecondCountableTopology Z] [BorelSpace U] | |
| [SecondCountableTopology U] in | |
| lemma support_prod (μ : Measure Z) (ν : Measure U) [SFinite μ] [SFinite ν] | |
| {z : Z} {u : U} (hz : z∈μ.support) (hu : u∈ν.support) : (z,u)∈(μ.prod ν).support := by | |
| rw [Measure.mem_support_iff_forall] | |
| intro S hS | |
| obtain ⟨A,hA,B,hB,hAB⟩ := mem_nhds_prod_iff.mp hS | |
| have ha := (Measure.mem_support_iff_forall z).mp hz A hA | |
| have hb := (Measure.mem_support_iff_forall u).mp hu B hB | |
| apply lt_of_lt_of_le _ (measure_mono hAB) | |
| rw [Measure.prod_prod] | |
| exact ENNReal.mul_pos ha.ne' hb.ne' | |
| lemma support_map {X Y : Type*} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X] | |
| [TopologicalSpace Y] [MeasurableSpace Y] [BorelSpace Y] | |
| (μ : Measure X) (T : X → Y) (hT : Continuous T) {x : X} (hx : x∈μ.support) : | |
| T x∈(Measure.map T μ).support := by | |
| rw [Measure.support_eq_forall_isOpen] | |
| intro O hxO hO | |
| rw [Measure.map_apply hT.measurable hO.measurableSet] | |
| exact (Measure.mem_support_iff_forall x).mp hx _ (hT.continuousAt.preimage_mem_nhds (hO.mem_nhds hxO)) | |
| noncomputable def step (ν : Measure U) [IsProbabilityMeasure ν] | |
| (T : Z×U → Z) (hT : Continuous T) (μ : ProbabilityMeasure Z) : ProbabilityMeasure Z := | |
| ⟨Measure.map T ((μ:Measure Z).prod ν), | |
| (Measure.isProbabilityMeasure_map_iff hT.measurable.aemeasurable).mpr inferInstance⟩ | |
| omit [SecondCountableTopology Z] in | |
| lemma step_integral (ν : Measure U) [IsProbabilityMeasure ν] (T : Z×U → Z) (hT : Continuous T) | |
| (μ : ProbabilityMeasure Z) (f : Z → ℝ) (hf : Continuous f) : | |
| (∫ z,f z ∂(step ν T hT μ))=∫ z,f (T z) ∂((μ:Measure Z).prod ν) := by | |
| exact integral_map hT.measurable.aemeasurable hf.aestronglyMeasurable | |
| omit [BorelSpace Z] [BorelSpace U] in | |
| lemma integral_fst (ν : Measure U) [IsProbabilityMeasure ν] (μ : ProbabilityMeasure Z) | |
| (f : Z → ℝ) (_hf : Continuous f) : | |
| (∫ z : Z×U,f z.1 ∂((μ:Measure Z).prod ν))=∫ z,f z ∂μ := by | |
| simpa using integral_fun_fst (μ := (μ:Measure Z)) (ν := ν) f | |
| variable [CompactSpace Z] [CompactSpace U] | |
| theorem exists_zero {I : Type*} (ν : Measure U) [IsProbabilityMeasure ν] | |
| (hν : ∀ u : U,u∈ν.support) (u₀ : U) | |
| (T : I → Z×U → Z) (hT : ∀ i,Continuous (T i)) | |
| (K : Set (ProbabilityMeasure Z)) (hK : IsCompact K) (hne : K.Nonempty) | |
| (hpres : ∀ μ∈K,∀ i,step ν (T i) (hT i) μ∈K) | |
| (d H : Z → ℝ) (hd : Continuous d) (hH : Continuous H) | |
| (hdec : ∀ i z u,d (T i (z,u))≤d z) | |
| (hdown : ∀ i z,H (T i (z,u₀))≤H z) | |
| (hfixed : ∀ z,(∀ i,H (T i (z,u₀))=H z) → d z=0) : | |
| ∃ μ∈K,∀ᵐ z∂(μ:Measure Z),d z=0 := by | |
| classical | |
| let cost : ProbabilityMeasure Z → ℝ := fun μ => ∫ z,d z ∂μ | |
| have hcost : Continuous cost := ProbabilityMeasure.continuous_integral_continuousMap (⟨d,hd⟩ : C(Z,ℝ)) | |
| obtain ⟨μ₀,hμ₀,hmin⟩ := hK.exists_isMinOn hne hcost.continuousOn | |
| let A : Set Z := {z | ∃ μ : ProbabilityMeasure Z,μ∈K ∧ cost μ=cost μ₀ ∧ z∈(μ:Measure Z).support} | |
| let S : Set Z := closure A | |
| have hopt (μ : ProbabilityMeasure Z) (hμ : μ∈K) (he : cost μ=cost μ₀) (i : I) : | |
| cost (step ν (T i) (hT i) μ)=cost μ₀ ∧ | |
| ∀ z∈(μ:Measure Z).support,∀ u,d (T i (z,u))=d z := by | |
| have hf := ci ((μ:Measure Z).prod ν) (hd.comp (hT i)) | |
| have hg := ci ((μ:Measure Z).prod ν) (hd.comp continuous_fst) | |
| have hl : cost (step ν (T i) (hT i) μ)≤cost μ := by | |
| rw [show cost (step ν (T i) (hT i) μ)=∫ z,d (T i z) ∂((μ:Measure Z).prod ν) from step_integral ν _ _ μ d hd, | |
| show cost μ=∫ z,d z ∂μ from rfl, ← integral_fst ν μ d hd] | |
| exact integral_mono hf hg (fun z => hdec i z.1 z.2) | |
| have he' : cost (step ν (T i) (hT i) μ)=cost μ₀ := le_antisymm (hl.trans_eq he) (hmin (hpres μ hμ i)) | |
| refine ⟨he',?_⟩ | |
| have heq : (fun z : Z×U => d (T i z))=ᵐ[(μ:Measure Z).prod ν] (fun z => d z.1) := by | |
| apply (integral_eq_iff_of_ae_le hf hg (Eventually.of_forall (fun z => hdec i z.1 z.2))).mp | |
| change (∫ z,d (T i z) ∂((μ:Measure Z).prod ν))=∫ z,d z.1 ∂((μ:Measure Z).prod ν) | |
| rw [← step_integral ν (T i) (hT i) μ d hd,integral_fst ν μ d hd] | |
| exact he'.trans he.symm | |
| have hsub := Measure.support_subset_of_isClosed (μ := (μ:Measure Z).prod ν) | |
| (isClosed_eq (hd.comp (hT i)) (hd.comp continuous_fst)) heq | |
| intro z hz u | |
| exact hsub (support_prod _ _ hz (hν u)) | |
| have hA (i : I) (u : U) (z : Z) (hz : z∈A) : T i (z,u)∈A ∧ d (T i (z,u))=d z := by | |
| obtain ⟨μ,hμ,he,hz⟩ := hz | |
| obtain ⟨he',hd'⟩ := hopt μ hμ he i | |
| exact ⟨⟨step ν (T i) (hT i) μ,hpres μ hμ i,he',support_map _ _ (hT i) (support_prod _ _ hz (hν u))⟩,hd' z hz u⟩ | |
| have hS (i : I) (u : U) (z : Z) (hz : z∈S) : T i (z,u)∈S ∧ d (T i (z,u))=d z := by | |
| have ht : Continuous (fun z => T i (z,u)) := (hT i).comp (continuous_id.prodMk continuous_const) | |
| constructor | |
| · exact closure_minimal (s := A) (t := (fun z => T i (z,u)) ⁻¹' S) | |
| (fun z hz => subset_closure (hA i u z hz).1) (isClosed_closure.preimage ht) hz | |
| · exact closure_minimal (fun z hz => (hA i u z hz).2) (isClosed_eq (hd.comp ht) hd) hz | |
| have hz0 (z : Z) (hz : z∈S) : d z=0 := by | |
| let L : Set Z := S∩{w | d w=d z} | |
| have hlc : IsCompact L := (isClosed_closure.inter (isClosed_eq hd continuous_const)).isCompact | |
| obtain ⟨w,hw,hwm⟩ := hlc.exists_isMinOn ⟨z,hz,rfl⟩ hH.continuousOn | |
| have hw0 : d w=0 := hfixed w (fun i => le_antisymm (hdown i w) | |
| (hwm ⟨(hS i u₀ w hw.1).1,(hS i u₀ w hw.1).2.trans hw.2⟩)) | |
| exact hw.2.symm.trans hw0 | |
| refine ⟨μ₀,hμ₀,?_⟩ | |
| filter_upwards [Measure.support_mem_ae (μ := (μ₀:Measure Z))] with z hz | |
| exact hz0 z (subset_closure ⟨μ₀,hμ₀,rfl,hz⟩) | |
| end CompactControls | |
| end OAI | |