mathlib documentation

probability.independence

Independence of sets of sets and measure spaces (σ-algebras) #

Main statements #

Implementation notes #

We provide one main definition of independence:

Additionally, we provide four corresponding statements for two measurable space structures (resp. sets of sets, sets, functions) instead of a family. These properties are denoted by the same names as for a family, but without a capital letter, for example indep_fun is the version of Indep_fun for two functions.

The definition of independence for Indep_sets uses finite sets (finset). An alternative and equivalent way of defining independence would have been to use countable sets. TODO: prove that equivalence.

Most of the definitions and lemma in this file list all variables instead of using the variables keyword at the beginning of a section, for example lemma indep.symm {Ω} {m₁ m₂ : measurable_space Ω} [measurable_space Ω] {μ : measure Ω} ... . This is intentional, to be able to control the order of the measurable_space variables. Indeed when defining μ in the example above, the measurable space used is the last one defined, here [measurable_space Ω], and not m₁ or m₂.

References #

def probability_theory.Indep_sets {Ω : Type u_1} {ι : Type u_2} [measurable_space Ω] (π : ι → set (set Ω)) (μ : measure_theory.measure Ω . "volume_tac") :
Prop

A family of sets of sets π : ι → set (set Ω) is independent with respect to a measure μ if for any finite set of indices s = {i_1, ..., i_n}, for any sets f i_1 ∈ π i_1, ..., f i_n ∈ π i_n, then μ (⋂ i in s, f i) = ∏ i in s, μ (f i). It will be used for families of pi_systems.

Equations
def probability_theory.indep_sets {Ω : Type u_1} [measurable_space Ω] (s1 s2 : set (set Ω)) (μ : measure_theory.measure Ω . "volume_tac") :
Prop

Two sets of sets s₁, s₂ are independent with respect to a measure μ if for any sets t₁ ∈ p₁, t₂ ∈ s₂, then μ (t₁ ∩ t₂) = μ (t₁) * μ (t₂)

Equations
def probability_theory.Indep {Ω : Type u_1} {ι : Type u_2} (m : ι → measurable_space Ω) [measurable_space Ω] (μ : measure_theory.measure Ω . "volume_tac") :
Prop

A family of measurable space structures (i.e. of σ-algebras) is independent with respect to a measure μ (typically defined on a finer σ-algebra) if the family of sets of measurable sets they define is independent. m : ι → measurable_space Ω is independent with respect to measure μ if for any finite set of indices s = {i_1, ..., i_n}, for any sets f i_1 ∈ m i_1, ..., f i_n ∈ m i_n, then μ (⋂ i in s, f i) = ∏ i in s, μ (f i).

Equations
def probability_theory.indep {Ω : Type u_1} (m₁ m₂ : measurable_space Ω) [measurable_space Ω] (μ : measure_theory.measure Ω . "volume_tac") :
Prop

Two measurable space structures (or σ-algebras) m₁, m₂ are independent with respect to a measure μ (defined on a third σ-algebra) if for any sets t₁ ∈ m₁, t₂ ∈ m₂, μ (t₁ ∩ t₂) = μ (t₁) * μ (t₂)

Equations
def probability_theory.Indep_set {Ω : Type u_1} {ι : Type u_2} [measurable_space Ω] (s : ι → set Ω) (μ : measure_theory.measure Ω . "volume_tac") :
Prop

A family of sets is independent if the family of measurable space structures they generate is independent. For a set s, the generated measurable space has measurable sets ∅, s, sᶜ, univ.

Equations
def probability_theory.indep_set {Ω : Type u_1} [measurable_space Ω] (s t : set Ω) (μ : measure_theory.measure Ω . "volume_tac") :
Prop

Two sets are independent if the two measurable space structures they generate are independent. For a set s, the generated measurable space structure has measurable sets ∅, s, sᶜ, univ.

Equations
def probability_theory.Indep_fun {Ω : Type u_1} {ι : Type u_2} [measurable_space Ω] {β : ι → Type u_3} (m : Π (x : ι), measurable_space (β x)) (f : Π (x : ι), Ω → β x) (μ : measure_theory.measure Ω . "volume_tac") :
Prop

A family of functions defined on the same space Ω and taking values in possibly different spaces, each with a measurable space structure, is independent if the family of measurable space structures they generate on Ω is independent. For a function g with codomain having measurable space structure m, the generated measurable space structure is measurable_space.comap g m.

Equations
def probability_theory.indep_fun {Ω : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space Ω] [mβ : measurable_space β] [mγ : measurable_space γ] (f : Ω → β) (g : Ω → γ) (μ : measure_theory.measure Ω . "volume_tac") :
Prop

Two functions are independent if the two measurable space structures they generate are independent. For a function f with codomain having measurable space structure m, the generated measurable space structure is measurable_space.comap f m.

Equations
theorem probability_theory.indep_sets.symm {Ω : Type u_1} {s₁ s₂ : set (set Ω)} [measurable_space Ω] {μ : measure_theory.measure Ω} (h : probability_theory.indep_sets s₁ s₂ μ) :
theorem probability_theory.indep.symm {Ω : Type u_1} {m₁ m₂ : measurable_space Ω} [measurable_space Ω] {μ : measure_theory.measure Ω} (h : probability_theory.indep m₁ m₂ μ) :
theorem probability_theory.indep_sets_of_indep_sets_of_le_left {Ω : Type u_1} {s₁ s₂ s₃ : set (set Ω)} [measurable_space Ω] {μ : measure_theory.measure Ω} (h_indep : probability_theory.indep_sets s₁ s₂ μ) (h31 : s₃ s₁) :
theorem probability_theory.indep_sets_of_indep_sets_of_le_right {Ω : Type u_1} {s₁ s₂ s₃ : set (set Ω)} [measurable_space Ω] {μ : measure_theory.measure Ω} (h_indep : probability_theory.indep_sets s₁ s₂ μ) (h32 : s₃ s₂) :
theorem probability_theory.indep_of_indep_of_le_left {Ω : Type u_1} {m₁ m₂ m₃ : measurable_space Ω} [measurable_space Ω] {μ : measure_theory.measure Ω} (h_indep : probability_theory.indep m₁ m₂ μ) (h31 : m₃ m₁) :
theorem probability_theory.indep_of_indep_of_le_right {Ω : Type u_1} {m₁ m₂ m₃ : measurable_space Ω} [measurable_space Ω] {μ : measure_theory.measure Ω} (h_indep : probability_theory.indep m₁ m₂ μ) (h32 : m₃ m₂) :
theorem probability_theory.indep_sets.union {Ω : Type u_1} [measurable_space Ω] {s₁ s₂ s' : set (set Ω)} {μ : measure_theory.measure Ω} (h₁ : probability_theory.indep_sets s₁ s' μ) (h₂ : probability_theory.indep_sets s₂ s' μ) :
@[simp]
theorem probability_theory.indep_sets.Union {Ω : Type u_1} {ι : Type u_2} [measurable_space Ω] {s : ι → set (set Ω)} {s' : set (set Ω)} {μ : measure_theory.measure Ω} (hyp : ∀ (n : ι), probability_theory.indep_sets (s n) s' μ) :
probability_theory.indep_sets (⋃ (n : ι), s n) s' μ
theorem probability_theory.indep_sets.inter {Ω : Type u_1} [measurable_space Ω] {s₁ s' : set (set Ω)} (s₂ : set (set Ω)) {μ : measure_theory.measure Ω} (h₁ : probability_theory.indep_sets s₁ s' μ) :
theorem probability_theory.indep_sets.Inter {Ω : Type u_1} {ι : Type u_2} [measurable_space Ω] {s : ι → set (set Ω)} {s' : set (set Ω)} {μ : measure_theory.measure Ω} (h : ∃ (n : ι), probability_theory.indep_sets (s n) s' μ) :
probability_theory.indep_sets (⋂ (n : ι), s n) s' μ
theorem probability_theory.indep_sets_singleton_iff {Ω : Type u_1} [measurable_space Ω] {s t : set Ω} {μ : measure_theory.measure Ω} :
probability_theory.indep_sets {s} {t} μ μ (s t) = μ s * μ t

Deducing indep from Indep #

theorem probability_theory.Indep_sets.indep_sets {Ω : Type u_1} {ι : Type u_2} {s : ι → set (set Ω)} [measurable_space Ω] {μ : measure_theory.measure Ω} (h_indep : probability_theory.Indep_sets s μ) {i j : ι} (hij : i j) :
theorem probability_theory.Indep.indep {Ω : Type u_1} {ι : Type u_2} {m : ι → measurable_space Ω} [measurable_space Ω] {μ : measure_theory.measure Ω} (h_indep : probability_theory.Indep m μ) {i j : ι} (hij : i j) :
theorem probability_theory.Indep_fun.indep_fun {Ω : Type u_1} {ι : Type u_2} {m₀ : measurable_space Ω} {μ : measure_theory.measure Ω} {β : ι → Type u_3} {m : Π (x : ι), measurable_space (β x)} {f : Π (i : ι), Ω → β i} (hf_Indep : probability_theory.Indep_fun m f μ) {i j : ι} (hij : i j) :

π-system lemma #

Independence of measurable spaces is equivalent to independence of generating π-systems.

Independence of measurable space structures implies independence of generating π-systems #

theorem probability_theory.Indep.Indep_sets {Ω : Type u_1} {ι : Type u_2} [measurable_space Ω] {μ : measure_theory.measure Ω} {m : ι → measurable_space Ω} {s : ι → set (set Ω)} (hms : ∀ (n : ι), m n = measurable_space.generate_from (s n)) (h_indep : probability_theory.Indep m μ) :

Independence of generating π-systems implies independence of measurable space structures #

theorem probability_theory.indep_sets.indep {Ω : Type u_1} {m1 m2 m : measurable_space Ω} {μ : measure_theory.measure Ω} [measure_theory.is_probability_measure μ] {p1 p2 : set (set Ω)} (h1 : m1 m) (h2 : m2 m) (hp1 : is_pi_system p1) (hp2 : is_pi_system p2) (hpm1 : m1 = measurable_space.generate_from p1) (hpm2 : m2 = measurable_space.generate_from p2) (hyp : probability_theory.indep_sets p1 p2 μ) :
theorem probability_theory.Indep_sets.pi_Union_Inter_singleton {Ω : Type u_1} {ι : Type u_2} {m0 : measurable_space Ω} {μ : measure_theory.measure Ω} {π : ι → set (set Ω)} {a : ι} {S : finset ι} (hp_ind : probability_theory.Indep_sets π μ) (haS : a S) :
theorem probability_theory.Indep_sets.Indep_aux {Ω : Type u_1} {ι : Type u_2} {m0 : measurable_space Ω} {μ : measure_theory.measure Ω} [measure_theory.is_probability_measure μ] (m : ι → measurable_space Ω) (h_le : ∀ (i : ι), m i m0) (π : ι → set (set Ω)) (h_pi : ∀ (n : ι), is_pi_system (π n)) (hp_univ : ∀ (i : ι), set.univ π i) (h_generate : ∀ (i : ι), m i = measurable_space.generate_from (π i)) (h_ind : probability_theory.Indep_sets π μ) :

Auxiliary lemma for Indep_sets.Indep.

theorem probability_theory.Indep_sets.Indep {Ω : Type u_1} {ι : Type u_2} {m0 : measurable_space Ω} {μ : measure_theory.measure Ω} [measure_theory.is_probability_measure μ] (m : ι → measurable_space Ω) (h_le : ∀ (i : ι), m i m0) (π : ι → set (set Ω)) (h_pi : ∀ (n : ι), is_pi_system (π n)) (h_generate : ∀ (i : ι), m i = measurable_space.generate_from (π i)) (h_ind : probability_theory.Indep_sets π μ) :

The measurable space structures generated by independent pi-systems are independent.

Independence of measurable sets #

We prove the following equivalences on indep_set, for measurable sets s, t.

theorem probability_theory.indep_sets.indep_set_of_mem {Ω : Type u_1} [measurable_space Ω] {s t : set Ω} (S T : set (set Ω)) (hs : s S) (ht : t T) (hs_meas : measurable_set s) (ht_meas : measurable_set t) (μ : measure_theory.measure Ω . "volume_tac") [measure_theory.is_probability_measure μ] (h_indep : probability_theory.indep_sets S T μ) :

Independence of random variables #

theorem probability_theory.indep_fun_iff_measure_inter_preimage_eq_mul {Ω : Type u_1} {β : Type u_3} {β' : Type u_4} {mΩ : measurable_space Ω} {μ : measure_theory.measure Ω} {f : Ω → β} {g : Ω → β'} {mβ : measurable_space β} {mβ' : measurable_space β'} :
probability_theory.indep_fun f g μ ∀ (s : set β) (t : set β'), measurable_set smeasurable_set tμ (f ⁻¹' s g ⁻¹' t) = μ (f ⁻¹' s) * μ (g ⁻¹' t)
theorem probability_theory.Indep_fun_iff_measure_inter_preimage_eq_mul {Ω : Type u_1} {mΩ : measurable_space Ω} {μ : measure_theory.measure Ω} {ι : Type u_2} {β : ι → Type u_3} (m : Π (x : ι), measurable_space (β x)) (f : Π (i : ι), Ω → β i) :
probability_theory.Indep_fun m f μ ∀ (S : finset ι) {sets : Π (i : ι), set (β i)}, (∀ (i : ι), i Smeasurable_set (sets i))μ (⋂ (i : ι) (H : i S), f i ⁻¹' sets i) = S.prod (λ (i : ι), μ (f i ⁻¹' sets i))
theorem probability_theory.indep_fun_iff_indep_set_preimage {Ω : Type u_1} {β : Type u_3} {β' : Type u_4} {mΩ : measurable_space Ω} {μ : measure_theory.measure Ω} {f : Ω → β} {g : Ω → β'} {mβ : measurable_space β} {mβ' : measurable_space β'} [measure_theory.is_probability_measure μ] (hf : measurable f) (hg : measurable g) :
theorem probability_theory.indep_fun.symm {Ω : Type u_1} {β : Type u_3} {mΩ : measurable_space Ω} {μ : measure_theory.measure Ω} {mβ : measurable_space β} {f g : Ω → β} (hfg : probability_theory.indep_fun f g μ) :
theorem probability_theory.indep_fun.ae_eq {Ω : Type u_1} {β : Type u_3} {mΩ : measurable_space Ω} {μ : measure_theory.measure Ω} {mβ : measurable_space β} {f g f' g' : Ω → β} (hfg : probability_theory.indep_fun f g μ) (hf : f =ᵐ[μ] f') (hg : g =ᵐ[μ] g') :
theorem probability_theory.indep_fun.comp {Ω : Type u_1} {β : Type u_3} {β' : Type u_4} {γ : Type u_5} {γ' : Type u_6} {mΩ : measurable_space Ω} {μ : measure_theory.measure Ω} {f : Ω → β} {g : Ω → β'} {mβ : measurable_space β} {mβ' : measurable_space β'} {mγ : measurable_space γ} {mγ' : measurable_space γ'} {φ : β → γ} {ψ : β' → γ'} (hfg : probability_theory.indep_fun f g μ) (hφ : measurable φ) (hψ : measurable ψ) :
theorem probability_theory.Indep_fun.indep_fun_finset {Ω : Type u_1} {mΩ : measurable_space Ω} {μ : measure_theory.measure Ω} [measure_theory.is_probability_measure μ] {ι : Type u_2} {β : ι → Type u_3} {m : Π (i : ι), measurable_space (β i)} {f : Π (i : ι), Ω → β i} (S T : finset ι) (hST : disjoint S T) (hf_Indep : probability_theory.Indep_fun m f μ) (hf_meas : ∀ (i : ι), measurable (f i)) :
probability_theory.indep_fun (λ (a : Ω) (i : S), f i a) (λ (a : Ω) (i : T), f i a) μ

If f is a family of mutually independent random variables (Indep_fun m f μ) and S, T are two disjoint finite index sets, then the tuple formed by f i for i ∈ S is independent of the tuple (f i)_i for i ∈ T.

theorem probability_theory.Indep_fun.indep_fun_prod {Ω : Type u_1} {mΩ : measurable_space Ω} {μ : measure_theory.measure Ω} [measure_theory.is_probability_measure μ] {ι : Type u_2} {β : ι → Type u_3} {m : Π (i : ι), measurable_space (β i)} {f : Π (i : ι), Ω → β i} (hf_Indep : probability_theory.Indep_fun m f μ) (hf_meas : ∀ (i : ι), measurable (f i)) (i j k : ι) (hik : i k) (hjk : j k) :
probability_theory.indep_fun (λ (a : Ω), (f i a, f j a)) (f k) μ
theorem probability_theory.Indep_fun.mul {Ω : Type u_1} {mΩ : measurable_space Ω} {μ : measure_theory.measure Ω} [measure_theory.is_probability_measure μ] {ι : Type u_2} {β : Type u_3} {m : measurable_space β} [has_mul β] [has_measurable_mul₂ β] {f : ι → Ω → β} (hf_Indep : probability_theory.Indep_fun (λ (_x : ι), m) f μ) (hf_meas : ∀ (i : ι), measurable (f i)) (i j k : ι) (hik : i k) (hjk : j k) :
probability_theory.indep_fun (f i * f j) (f k) μ
theorem probability_theory.Indep_fun.add {Ω : Type u_1} {mΩ : measurable_space Ω} {μ : measure_theory.measure Ω} [measure_theory.is_probability_measure μ] {ι : Type u_2} {β : Type u_3} {m : measurable_space β} [has_add β] [has_measurable_add₂ β] {f : ι → Ω → β} (hf_Indep : probability_theory.Indep_fun (λ (_x : ι), m) f μ) (hf_meas : ∀ (i : ι), measurable (f i)) (i j k : ι) (hik : i k) (hjk : j k) :
probability_theory.indep_fun (f i + f j) (f k) μ
theorem probability_theory.Indep_fun.indep_fun_finset_sum_of_not_mem {Ω : Type u_1} {mΩ : measurable_space Ω} {μ : measure_theory.measure Ω} [measure_theory.is_probability_measure μ] {ι : Type u_2} {β : Type u_3} {m : measurable_space β} [add_comm_monoid β] [has_measurable_add₂ β] {f : ι → Ω → β} (hf_Indep : probability_theory.Indep_fun (λ (_x : ι), m) f μ) (hf_meas : ∀ (i : ι), measurable (f i)) {s : finset ι} {i : ι} (hi : i s) :
probability_theory.indep_fun (s.sum (λ (j : ι), f j)) (f i) μ
theorem probability_theory.Indep_fun.indep_fun_finset_prod_of_not_mem {Ω : Type u_1} {mΩ : measurable_space Ω} {μ : measure_theory.measure Ω} [measure_theory.is_probability_measure μ] {ι : Type u_2} {β : Type u_3} {m : measurable_space β} [comm_monoid β] [has_measurable_mul₂ β] {f : ι → Ω → β} (hf_Indep : probability_theory.Indep_fun (λ (_x : ι), m) f μ) (hf_meas : ∀ (i : ι), measurable (f i)) {s : finset ι} {i : ι} (hi : i s) :
probability_theory.indep_fun (s.prod (λ (j : ι), f j)) (f i) μ
theorem probability_theory.Indep_fun.indep_fun_prod_range_succ {Ω : Type u_1} {mΩ : measurable_space Ω} {μ : measure_theory.measure Ω} [measure_theory.is_probability_measure μ] {β : Type u_2} {m : measurable_space β} [comm_monoid β] [has_measurable_mul₂ β] {f : Ω → β} (hf_Indep : probability_theory.Indep_fun (λ (_x : ), m) f μ) (hf_meas : ∀ (i : ), measurable (f i)) (n : ) :
probability_theory.indep_fun ((finset.range n).prod (λ (j : ), f j)) (f n) μ
theorem probability_theory.Indep_fun.indep_fun_sum_range_succ {Ω : Type u_1} {mΩ : measurable_space Ω} {μ : measure_theory.measure Ω} [measure_theory.is_probability_measure μ] {β : Type u_2} {m : measurable_space β} [add_comm_monoid β] [has_measurable_add₂ β] {f : Ω → β} (hf_Indep : probability_theory.Indep_fun (λ (_x : ), m) f μ) (hf_meas : ∀ (i : ), measurable (f i)) (n : ) :
probability_theory.indep_fun ((finset.range n).sum (λ (j : ), f j)) (f n) μ