mathlib documentation

probability_theory.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 {α} {m₁ m₂ : measurable_space α} [measurable_space α] {μ : measure α} ... . 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 α], and not m₁ or m₂.

References #

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

A family of sets of sets π : ι → set (set α) 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 {α : Type u_1} [measurable_space α] (s1 s2 : set (set α)) (μ : measure_theory.measure α . "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 {α : Type u_1} {ι : Type u_2} (m : ι → measurable_space α) [measurable_space α] (μ : measure_theory.measure α . "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 α 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 {α : Type u_1} (m₁ m₂ : measurable_space α) [measurable_space α] (μ : measure_theory.measure α . "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 {α : Type u_1} {ι : Type u_2} [measurable_space α] (s : ι → set α) (μ : measure_theory.measure α . "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 {α : Type u_1} [measurable_space α] (s t : set α) (μ : measure_theory.measure α . "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 {α : Type u_1} {ι : Type u_2} [measurable_space α] {β : ι → Type u_3} (m : Π (x : ι), measurable_space (β x)) (f : Π (x : ι), α → β x) (μ : measure_theory.measure α . "volume_tac") :
Prop

A family of functions defined on the same space α 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 α 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 {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α] (mβ : measurable_space β) (mγ : measurable_space γ) (f : α → β) (g : α → γ) (μ : measure_theory.measure α . "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 {α : Type u_1} {s₁ s₂ : set (set α)} [measurable_space α] {μ : measure_theory.measure α} (h : probability_theory.indep_sets s₁ s₂ μ) :
theorem probability_theory.indep.symm {α : Type u_1} {m₁ m₂ : measurable_space α} [measurable_space α] {μ : measure_theory.measure α} (h : probability_theory.indep m₁ m₂ μ) :
theorem probability_theory.indep_sets_of_indep_sets_of_le_left {α : Type u_1} {s₁ s₂ s₃ : set (set α)} [measurable_space α] {μ : measure_theory.measure α} (h_indep : probability_theory.indep_sets s₁ s₂ μ) (h31 : s₃ s₁) :
theorem probability_theory.indep_sets_of_indep_sets_of_le_right {α : Type u_1} {s₁ s₂ s₃ : set (set α)} [measurable_space α] {μ : measure_theory.measure α} (h_indep : probability_theory.indep_sets s₁ s₂ μ) (h32 : s₃ s₂) :
theorem probability_theory.indep_of_indep_of_le_left {α : Type u_1} {m₁ m₂ m₃ : measurable_space α} [measurable_space α] {μ : measure_theory.measure α} (h_indep : probability_theory.indep m₁ m₂ μ) (h31 : m₃ m₁) :
theorem probability_theory.indep_of_indep_of_le_right {α : Type u_1} {m₁ m₂ m₃ : measurable_space α} [measurable_space α] {μ : measure_theory.measure α} (h_indep : probability_theory.indep m₁ m₂ μ) (h32 : m₃ m₂) :
theorem probability_theory.indep_sets.union {α : Type u_1} [measurable_space α] {s₁ s₂ s' : set (set α)} {μ : measure_theory.measure α} (h₁ : probability_theory.indep_sets s₁ s' μ) (h₂ : probability_theory.indep_sets s₂ s' μ) :
@[simp]
theorem probability_theory.indep_sets.Union {α : Type u_1} {ι : Sort u_2} [measurable_space α] {s : ι → set (set α)} {s' : set (set α)} {μ : measure_theory.measure α} (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 {α : Type u_1} [measurable_space α] {s₁ s' : set (set α)} (s₂ : set (set α)) {μ : measure_theory.measure α} (h₁ : probability_theory.indep_sets s₁ s' μ) :
theorem probability_theory.indep_sets.Inter {α : Type u_1} {ι : Sort u_2} [measurable_space α] {s : ι → set (set α)} {s' : set (set α)} {μ : measure_theory.measure α} (h : ∃ (n : ι), probability_theory.indep_sets (s n) s' μ) :
probability_theory.indep_sets (⋂ (n : ι), s n) s' μ

Deducing indep from Indep #

theorem probability_theory.Indep_sets.indep_sets {α : Type u_1} {ι : Type u_2} {s : ι → set (set α)} [measurable_space α] {μ : measure_theory.measure α} (h_indep : probability_theory.Indep_sets s μ) {i j : ι} (hij : i j) :
theorem probability_theory.Indep.indep {α : Type u_1} {ι : Type u_2} {m : ι → measurable_space α} [measurable_space α] {μ : measure_theory.measure α} (h_indep : probability_theory.Indep m 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 {α : Type u_1} {ι : Type u_2} [measurable_space α] {μ : measure_theory.measure α} {m : ι → measurable_space α} {s : ι → set (set α)} (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 {α : Type u_1} {m1 m2 m : measurable_space α} {μ : measure_theory.measure α} [measure_theory.probability_measure μ] {p1 p2 : set (set α)} (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) :