mathlib documentation

measure_theory.pi_system

Induction principles for measurable sets, related to π-systems and λ-systems. #

Main statements #

Implementation details #

def is_pi_system {α : Type u_1} (C : set (set α)) :
Prop

A π-system is a collection of subsets of α that is closed under binary intersection of non-disjoint sets. Usually it is also required that the collection is nonempty, but we don't do that here.

Equations
theorem is_pi_system.singleton {α : Type u_1} (S : set α) :
inductive generate_pi_system {α : Type u_1} (S : set (set α)) :
set (set α)

Given a collection S of subsets of α, then generate_pi_system S is the smallest π-system containing S.

theorem is_pi_system_generate_pi_system {α : Type u_1} (S : set (set α)) :
theorem subset_generate_pi_system_self {α : Type u_1} (S : set (set α)) :
theorem generate_pi_system_subset_self {α : Type u_1} {S : set (set α)} (h_S : is_pi_system S) :
theorem generate_pi_system_eq {α : Type u_1} {S : set (set α)} (h_pi : is_pi_system S) :
theorem generate_pi_system_mono {α : Type u_1} {S T : set (set α)} (hST : S T) :
theorem generate_pi_system_measurable_set {α : Type u_1} [M : measurable_space α] {S : set (set α)} (h_meas_S : ∀ (s : set α), s Smeasurable_set s) (t : set α) (h_in_pi : t generate_pi_system S) :
theorem mem_generate_pi_system_Union_elim {α : Type u_1} {β : Type u_2} {g : β → set (set α)} (h_pi : ∀ (b : β), is_pi_system (g b)) (t : set α) (h_t : t generate_pi_system (⋃ (b : β), g b)) :
∃ (T : finset β) (f : β → set α), (t = ⋂ (b : β) (H : b T), f b) ∀ (b : β), b Tf b g b
theorem mem_generate_pi_system_Union_elim' {α : Type u_1} {β : Type u_2} {g : β → set (set α)} {s : set β} (h_pi : ∀ (b : β), b sis_pi_system (g b)) (t : set α) (h_t : t generate_pi_system (⋃ (b : β) (H : b s), g b)) :
∃ (T : finset β) (f : β → set α), T s (t = ⋂ (b : β) (H : b T), f b) ∀ (b : β), b Tf b g b
structure measurable_space.dynkin_system (α : Type u_2) :
Type u_2

A Dynkin system is a collection of subsets of a type α that contains the empty set, is closed under complementation and under countable union of pairwise disjoint sets. The disjointness condition is the only difference with σ-algebras.

The main purpose of Dynkin systems is to provide a powerful induction rule for σ-algebras generated by a collection of sets which is stable under intersection.

A Dynkin system is also known as a "λ-system" or a "d-system".

@[ext]
theorem measurable_space.dynkin_system.ext {α : Type u_1} {d₁ d₂ : measurable_space.dynkin_system α} :
(∀ (s : set α), d₁.has s d₂.has s)d₁ = d₂
theorem measurable_space.dynkin_system.has_Union {α : Type u_1} (d : measurable_space.dynkin_system α) {β : Type u_2} [encodable β] {f : β → set α} (hd : pairwise (disjoint on f)) (h : ∀ (i : β), d.has (f i)) :
d.has (⋃ (i : β), f i)
theorem measurable_space.dynkin_system.has_union {α : Type u_1} (d : measurable_space.dynkin_system α) {s₁ s₂ : set α} (h₁ : d.has s₁) (h₂ : d.has s₂) (h : s₁ s₂ ) :
d.has (s₁ s₂)
theorem measurable_space.dynkin_system.has_diff {α : Type u_1} (d : measurable_space.dynkin_system α) {s₁ s₂ : set α} (h₁ : d.has s₁) (h₂ : d.has s₂) (h : s₂ s₁) :
d.has (s₁ \ s₂)
@[instance]
Equations

Every measurable space (σ-algebra) forms a Dynkin system

Equations
inductive measurable_space.dynkin_system.generate_has {α : Type u_1} (s : set (set α)) :
set α → Prop

The least Dynkin system containing a collection of basic sets. This inductive type gives the underlying collection of sets.

The least Dynkin system containing a collection of basic sets.

Equations
def measurable_space.dynkin_system.to_measurable_space {α : Type u_1} (d : measurable_space.dynkin_system α) (h_inter : ∀ (s₁ s₂ : set α), d.has s₁d.has s₂d.has (s₁ s₂)) :

If a Dynkin system is closed under binary intersection, then it forms a σ-algebra.

Equations
theorem measurable_space.dynkin_system.of_measurable_space_to_measurable_space {α : Type u_1} (d : measurable_space.dynkin_system α) (h_inter : ∀ (s₁ s₂ : set α), d.has s₁d.has s₂d.has (s₁ s₂)) :

If s is in a Dynkin system d, we can form the new Dynkin system {s ∩ t | t ∈ d}.

Equations
theorem measurable_space.dynkin_system.generate_le {α : Type u_1} (d : measurable_space.dynkin_system α) {s : set (set α)} (h : ∀ (t : set α), t sd.has t) :

Given a collection of sets closed under binary intersections, then the Dynkin system it generates is equal to the σ-algebra it generates. This result is known as the π-λ theorem. A collection of sets closed under binary intersection is called a π-system (often requiring additionnally that is is non-empty, but we drop this condition in the formalization).

theorem measurable_space.induction_on_inter {α : Type u_1} {C : set α → Prop} {s : set (set α)} [m : measurable_space α] (h_eq : m = measurable_space.generate_from s) (h_inter : is_pi_system s) (h_empty : C ) (h_basic : ∀ (t : set α), t sC t) (h_compl : ∀ (t : set α), measurable_set tC tC t) (h_union : ∀ (f : set α), pairwise (disjoint on f)(∀ (i : ), measurable_set (f i))(∀ (i : ), C (f i))C (⋃ (i : ), f i)) ⦃t : set α⦄ :
measurable_set tC t