Documentation

PFR.Mathlib.Probability.Independence.Basic

theorem ProbabilityTheory.iIndepFun.inv {Ω : Type u_10} {ι : Type u_11} [MeasurableSpace Ω] {α : ιType u_13} [n : (i : ι) → MeasurableSpace (α i)] {f : (i : ι) → Ωα i} {μ : MeasureTheory.Measure Ω} (i : ι) [Inv (α i)] [MeasurableInv (α i)] [DecidableEq ι] (h : iIndepFun f μ) :
theorem ProbabilityTheory.iIndepFun.neg {Ω : Type u_10} {ι : Type u_11} [MeasurableSpace Ω] {α : ιType u_13} [n : (i : ι) → MeasurableSpace (α i)] {f : (i : ι) → Ωα i} {μ : MeasureTheory.Measure Ω} (i : ι) [Neg (α i)] [MeasurableNeg (α i)] [DecidableEq ι] (h : iIndepFun f μ) :
theorem ProbabilityTheory.iIndepFun.finsets {Ω : Type u_10} {ι : Type u_11} [MeasurableSpace Ω] {β : ιType u_14} [m : (i : ι) → MeasurableSpace (β i)] {μ : MeasureTheory.Measure Ω} {f : (i : ι) → Ωβ i} {J : Type u_15} [Fintype J] (S : JFinset ι) (h_disjoint : Set.univ.PairwiseDisjoint S) (hf_Indep : iIndepFun f μ) (hf_meas : ∀ (i : ι), Measurable (f i)) :
iIndepFun (fun (j : J) (a : Ω) (i : { x : ι // x S j }) => f (↑i) a) μ

If f is a family of mutually independent random variables, (S j)ⱼ are pairwise disjoint finite index sets, then the tuples formed by f i for i ∈ S j are mutually independent, when seen as a family indexed by J.

theorem ProbabilityTheory.iIndepFun.finsets_comp {Ω : Type u_10} {ι : Type u_11} [MeasurableSpace Ω] {β : ιType u_14} [m : (i : ι) → MeasurableSpace (β i)] {μ : MeasureTheory.Measure Ω} {f : (i : ι) → Ωβ i} {J : Type u_15} [Fintype J] (S : JFinset ι) (h_disjoint : Set.univ.PairwiseDisjoint S) (hf_Indep : iIndepFun f μ) (hf_meas : ∀ (i : ι), Measurable (f i)) {γ : JType u_16} { : (j : J) → MeasurableSpace (γ j)} (φ : (j : J) → ((i : { x : ι // x S j }) → β i)γ j) ( : ∀ (j : J), Measurable (φ j)) :
iIndepFun (fun (j : J) (a : Ω) => φ j fun (i : { x : ι // x S j }) => f (↑i) a) μ

If f is a family of mutually independent random variables, (S j)ⱼ are pairwise disjoint finite index sets, and φ j is a function that maps the tuple formed by f i for i ∈ S j to a measurable space γ j, then the family of random variables formed by φ j (f i)_{i ∈ S j} and indexed by J is iIndep.

@[simp]
theorem ProbabilityTheory.indepFun_zero_measure {Ω : Type u_1} {β : Type u_6} { : MeasurableSpace Ω} {α : Type u_11} [MeasurableSpace α] [MeasurableSpace β] {f : Ωα} {g : Ωβ} :
IndepFun f g 0
theorem ProbabilityTheory.indepFun_const {Ω : Type u_1} {β : Type u_6} { : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ωβ} {α : Type u_11} [MeasurableSpace α] [MeasurableSpace β] [MeasureTheory.IsZeroOrProbabilityMeasure μ] (c : α) :
IndepFun f (fun (x : Ω) => c) μ

Random variables are always independent of constants.

theorem ProbabilityTheory.IndepFun.comp_right {Ω : Type u_1} {β : Type u_6} { : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Ω' : Type u_10} {α : Type u_11} [MeasurableSpace Ω'] [MeasurableSpace α] [MeasurableSpace β] {f : Ωα} {g : Ωβ} {i : Ω'Ω} (hi : MeasurableEmbedding i) (hi' : ∀ᵐ (a : Ω) μ, a Set.range i) (hf : Measurable f) (hg : Measurable g) (hfg : IndepFun f g μ) :

Composing independent functions with a measurable embedding of conull range gives independent functions.

theorem ProbabilityTheory.iIndepFun_iff' {Ω : Type u_1} {ι : Type u_2} [MeasurableSpace Ω] {β : ιType u_12} (m : (i : ι) → MeasurableSpace (β i)) (f : (i : ι) → Ωβ i) (μ : MeasureTheory.Measure Ω) :
iIndepFun f μ ∀ (s : Finset ι) ⦃f' : ιSet Ω⦄, (∀ (i : ι), MeasurableSet (f' i))μ (⋂ is, f' i) = is, μ (f' i)
theorem ProbabilityTheory.indepFun_iff_map_prod_eq_prod_map_map' {Ω : Type u_1} {β' : Type u_7} { : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {β : Type u_12} { : MeasurableSpace β} {mβ' : MeasurableSpace β'} {f : Ωβ} {g : Ωβ'} [MeasureTheory.IsFiniteMeasure μ] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
theorem ProbabilityTheory.iIndepFun_iff_pi_map_eq_map {Ω : Type u_1} { : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ι : Type u_12} {β : ιType u_13} [Fintype ι] (f : (x : ι) → Ωβ x) [m : (x : ι) → MeasurableSpace (β x)] [MeasureTheory.IsProbabilityMeasure μ] (hf : ∀ (x : ι), Measurable (f x)) :
iIndepFun f μ (MeasureTheory.Measure.pi fun (i : ι) => MeasureTheory.Measure.map (f i) μ) = MeasureTheory.Measure.map (fun (ω : Ω) (i : ι) => f i ω) μ
theorem ProbabilityTheory.measurable_sigmaCurry {ι : Type u_1} {κ : ιType u_3} {α : (i : ι) → κ iType u_4} [m : (i : ι) → (j : κ i) → MeasurableSpace (α i j)] :
theorem Finset.prod_univ_prod {ι : Type u_1} {κ : ιType u_3} [Fintype ι] [(i : ι) → Fintype (κ i)] {β : Type u_5} [CommMonoid β] (f : (i : ι) → κ iβ) :
ij : (i : ι) × κ i, f ij.fst ij.snd = i : ι, j : κ i, f i j
theorem Finset.sum_univ_sum {ι : Type u_1} {κ : ιType u_3} [Fintype ι] [(i : ι) → Fintype (κ i)] {β : Type u_5} [AddCommMonoid β] (f : (i : ι) → κ iβ) :
ij : (i : ι) × κ i, f ij.fst ij.snd = i : ι, j : κ i, f i j
theorem Finset.prod_univ_prod' {ι : Type u_1} {κ : ιType u_3} [Fintype ι] [(i : ι) → Fintype (κ i)] {β : Type u_5} [CommMonoid β] (f : (i : ι) × κ iβ) :
ij : (i : ι) × κ i, f ij = i : ι, j : κ i, f i, j
theorem Finset.sum_univ_sum' {ι : Type u_1} {κ : ιType u_3} [Fintype ι] [(i : ι) → Fintype (κ i)] {β : Type u_5} [AddCommMonoid β] (f : (i : ι) × κ iβ) :
ij : (i : ι) × κ i, f ij = i : ι, j : κ i, f i, j
theorem ProbabilityTheory.iIndepFun.pi {Ω : Type u_2} [MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_5} {κ : ιType u_6} [(i : ι) → Fintype (κ i)] {α : (i : ι) → κ iType u_7} {f : (i : ι) → (j : κ i) → Ωα i j} [m : (i : ι) → (j : κ i) → MeasurableSpace (α i j)] (f_meas : ∀ (i : ι) (j : κ i), Measurable (f i j)) (hf : iIndepFun (fun (ij : (i : ι) × κ i) => f ij.fst ij.snd) μ) :
iIndepFun (fun (i : ι) (ω : Ω) (j : κ i) => f i j ω) μ

If a family of functions (i, j) ↦ f i j is independent, then the family of function tuples i ↦ (f i j)ⱼ is independent.

theorem ProbabilityTheory.iIndepFun.pi' {Ω : Type u_2} [MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_5} {κ : ιType u_6} [(i : ι) → Fintype (κ i)] {α : (i : ι) → κ iType u_7} [m : (i : ι) → (j : κ i) → MeasurableSpace (α i j)] {f : (ij : (i : ι) × κ i) → Ωα ij.fst ij.snd} (f_meas : ∀ (i : (i : ι) × κ i), Measurable (f i)) (hf : iIndepFun f μ) :
iIndepFun (fun (i : ι) (ω : Ω) (j : κ i) => f i, j ω) μ

If a family of functions (i, j) ↦ f i j is independent, then the family of function tuples i ↦ (f i j)ⱼ is independent.

theorem ProbabilityTheory.iIndepFun.prod {Ω : Type u_2} [MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {ι : Type u_8} {ι' : Type u_9} {α : ιType u_10} {n : (i : ι) → MeasurableSpace (α i)} {f : (i : ι) → Ωα i} {hf : ∀ (i : ι), Measurable (f i)} {ST : ι'Finset ι} (hS : Pairwise (Function.onFun Disjoint ST)) (h : iIndepFun f μ) :
let β := fun (k : ι') => (i : { x : ι // x ST k }) → α i; iIndepFun (fun (k : ι') (x : Ω) (i : { x : ι // x ST k }) => f (↑i) x) μ
theorem ProbabilityTheory.EventuallyEq.finite_iInter {ι : Type u_14} {α : Type u_2} {l : Filter α} (s : Finset ι) {E F : ιSet α} (h : is, E i =ᶠ[l] F i) :
is, E i =ᶠ[l] is, F i

The new Mathlib tool Finset.eventuallyEq_iInter will supersede this result.

theorem ProbabilityTheory.iIndepFun.ae_eq {Ω : Type u_13} { : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ι : Type u_14} {β : ιType u_15} {m : (i : ι) → MeasurableSpace (β i)} {f g : (i : ι) → Ωβ i} (hf_Indep : iIndepFun f μ) (hfg : ∀ (i : ι), f i =ᵐ[μ] g i) :

TODO: a kernel version of this theorem