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/- | |
Copyright (c) 2021 Martin Zinkevich. All rights reserved. | |
Released under Apache 2.0 license as described in the file LICENSE. | |
Authors: Johannes Hölzl, Martin Zinkevich, Rémy Degenne | |
-/ | |
import logic.encodable.lattice | |
import measure_theory.measurable_space_def | |
/-! | |
# Induction principles for measurable sets, related to π-systems and λ-systems. | |
## Main statements | |
* The main theorem of this file is Dynkin's π-λ theorem, which appears | |
here as an induction principle `induction_on_inter`. Suppose `s` is a | |
collection of subsets of `α` such that the intersection of two members | |
of `s` belongs to `s` whenever it is nonempty. Let `m` be the σ-algebra | |
generated by `s`. In order to check that a predicate `C` holds on every | |
member of `m`, it suffices to check that `C` holds on the members of `s` and | |
that `C` is preserved by complementation and *disjoint* countable | |
unions. | |
* The proof of this theorem relies on the notion of `is_pi_system`, i.e., a collection of sets | |
which is closed under binary non-empty intersections. Note that this is a small variation around | |
the usual notion in the literature, which often requires that a π-system is non-empty, and closed | |
also under disjoint intersections. This variation turns out to be convenient for the | |
formalization. | |
* The proof of Dynkin's π-λ theorem also requires the notion of `dynkin_system`, i.e., a collection | |
of sets which 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. | |
* `generate_pi_system g` gives the minimal π-system containing `g`. | |
This can be considered a Galois insertion into both measurable spaces and sets. | |
* `generate_from_generate_pi_system_eq` proves that if you start from a collection of sets `g`, | |
take the generated π-system, and then the generated σ-algebra, you get the same result as | |
the σ-algebra generated from `g`. This is useful because there are connections between | |
independent sets that are π-systems and the generated independent spaces. | |
* `mem_generate_pi_system_Union_elim` and `mem_generate_pi_system_Union_elim'` show that any | |
element of the π-system generated from the union of a set of π-systems can be | |
represented as the intersection of a finite number of elements from these sets. | |
* `pi_Union_Inter` defines a new π-system from a family of π-systems `π : ι → set (set α)` and a | |
set of finsets `S : set (finset α)`. `pi_Union_Inter π S` is the set of sets that can be written | |
as `⋂ x ∈ t, f x` for some `t ∈ S` and sets `f x ∈ π x`. If `S` is union-closed, then it is a | |
π-system. The π-systems used to prove Kolmogorov's 0-1 law will be defined using this mechanism | |
(TODO). | |
## Implementation details | |
* `is_pi_system` is a predicate, not a type. Thus, we don't explicitly define the galois | |
insertion, nor do we define a complete lattice. In theory, we could define a complete | |
lattice and galois insertion on the subtype corresponding to `is_pi_system`. | |
-/ | |
open measurable_space set | |
open_locale classical measure_theory | |
/-- 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. -/ | |
def is_pi_system {α} (C : set (set α)) : Prop := | |
∀ s t ∈ C, (s ∩ t : set α).nonempty → s ∩ t ∈ C | |
namespace measurable_space | |
lemma is_pi_system_measurable_set {α:Type*} [measurable_space α] : | |
is_pi_system {s : set α | measurable_set s} := | |
λ s hs t ht _, hs.inter ht | |
end measurable_space | |
lemma is_pi_system.singleton {α} (S : set α) : is_pi_system ({S} : set (set α)) := | |
begin | |
intros s h_s t h_t h_ne, | |
rw [set.mem_singleton_iff.1 h_s, set.mem_singleton_iff.1 h_t, set.inter_self, | |
set.mem_singleton_iff], | |
end | |
lemma is_pi_system.insert_empty {α} {S : set (set α)} (h_pi : is_pi_system S) : | |
is_pi_system (insert ∅ S) := | |
begin | |
intros s hs t ht hst, | |
cases hs, | |
{ simp [hs], }, | |
{ cases ht, | |
{ simp [ht], }, | |
{ exact set.mem_insert_of_mem _ (h_pi s hs t ht hst), }, }, | |
end | |
lemma is_pi_system.insert_univ {α} {S : set (set α)} (h_pi : is_pi_system S) : | |
is_pi_system (insert set.univ S) := | |
begin | |
intros s hs t ht hst, | |
cases hs, | |
{ cases ht; simp [hs, ht], }, | |
{ cases ht, | |
{ simp [hs, ht], }, | |
{ exact set.mem_insert_of_mem _ (h_pi s hs t ht hst), }, }, | |
end | |
lemma is_pi_system.comap {α β} {S : set (set β)} (h_pi : is_pi_system S) (f : α → β) : | |
is_pi_system {s : set α | ∃ t ∈ S, f ⁻¹' t = s} := | |
begin | |
rintros _ ⟨s, hs_mem, rfl⟩ _ ⟨t, ht_mem, rfl⟩ hst, | |
rw ← set.preimage_inter at hst ⊢, | |
refine ⟨s ∩ t, h_pi s hs_mem t ht_mem _, rfl⟩, | |
by_contra, | |
rw set.not_nonempty_iff_eq_empty at h, | |
rw h at hst, | |
simpa using hst, | |
end | |
section order | |
variables {α : Type*} {ι ι' : Sort*} [linear_order α] | |
lemma is_pi_system_image_Iio (s : set α) : is_pi_system (Iio '' s) := | |
begin | |
rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ -, | |
exact ⟨a ⊓ b, inf_ind a b ha hb, Iio_inter_Iio.symm⟩ | |
end | |
lemma is_pi_system_Iio : is_pi_system (range Iio : set (set α)) := | |
@image_univ α _ Iio ▸ is_pi_system_image_Iio univ | |
lemma is_pi_system_image_Ioi (s : set α) : is_pi_system (Ioi '' s) := | |
@is_pi_system_image_Iio αᵒᵈ _ s | |
lemma is_pi_system_Ioi : is_pi_system (range Ioi : set (set α)) := | |
@image_univ α _ Ioi ▸ is_pi_system_image_Ioi univ | |
lemma is_pi_system_Ixx_mem {Ixx : α → α → set α} {p : α → α → Prop} | |
(Hne : ∀ {a b}, (Ixx a b).nonempty → p a b) | |
(Hi : ∀ {a₁ b₁ a₂ b₂}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (max a₁ a₂) (min b₁ b₂)) (s t : set α) : | |
is_pi_system {S | ∃ (l ∈ s) (u ∈ t) (hlu : p l u), Ixx l u = S} := | |
begin | |
rintro _ ⟨l₁, hls₁, u₁, hut₁, hlu₁, rfl⟩ _ ⟨l₂, hls₂, u₂, hut₂, hlu₂, rfl⟩, | |
simp only [Hi, ← sup_eq_max, ← inf_eq_min], | |
exact λ H, ⟨l₁ ⊔ l₂, sup_ind l₁ l₂ hls₁ hls₂, u₁ ⊓ u₂, inf_ind u₁ u₂ hut₁ hut₂, Hne H, rfl⟩ | |
end | |
lemma is_pi_system_Ixx {Ixx : α → α → set α} {p : α → α → Prop} | |
(Hne : ∀ {a b}, (Ixx a b).nonempty → p a b) | |
(Hi : ∀ {a₁ b₁ a₂ b₂}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (max a₁ a₂) (min b₁ b₂)) | |
(f : ι → α) (g : ι' → α) : | |
@is_pi_system α ({S | ∃ i j (h : p (f i) (g j)), Ixx (f i) (g j) = S}) := | |
by simpa only [exists_range_iff] using is_pi_system_Ixx_mem @Hne @Hi (range f) (range g) | |
lemma is_pi_system_Ioo_mem (s t : set α) : | |
is_pi_system {S | ∃ (l ∈ s) (u ∈ t) (h : l < u), Ioo l u = S} := | |
is_pi_system_Ixx_mem (λ a b ⟨x, hax, hxb⟩, hax.trans hxb) (λ _ _ _ _, Ioo_inter_Ioo) s t | |
lemma is_pi_system_Ioo (f : ι → α) (g : ι' → α) : | |
@is_pi_system α {S | ∃ l u (h : f l < g u), Ioo (f l) (g u) = S} := | |
is_pi_system_Ixx (λ a b ⟨x, hax, hxb⟩, hax.trans hxb) (λ _ _ _ _, Ioo_inter_Ioo) f g | |
lemma is_pi_system_Ioc_mem (s t : set α) : | |
is_pi_system {S | ∃ (l ∈ s) (u ∈ t) (h : l < u), Ioc l u = S} := | |
is_pi_system_Ixx_mem (λ a b ⟨x, hax, hxb⟩, hax.trans_le hxb) (λ _ _ _ _, Ioc_inter_Ioc) s t | |
lemma is_pi_system_Ioc (f : ι → α) (g : ι' → α) : | |
@is_pi_system α {S | ∃ i j (h : f i < g j), Ioc (f i) (g j) = S} := | |
is_pi_system_Ixx (λ a b ⟨x, hax, hxb⟩, hax.trans_le hxb) (λ _ _ _ _, Ioc_inter_Ioc) f g | |
lemma is_pi_system_Ico_mem (s t : set α) : | |
is_pi_system {S | ∃ (l ∈ s) (u ∈ t) (h : l < u), Ico l u = S} := | |
is_pi_system_Ixx_mem (λ a b ⟨x, hax, hxb⟩, hax.trans_lt hxb) (λ _ _ _ _, Ico_inter_Ico) s t | |
lemma is_pi_system_Ico (f : ι → α) (g : ι' → α) : | |
@is_pi_system α {S | ∃ i j (h : f i < g j), Ico (f i) (g j) = S} := | |
is_pi_system_Ixx (λ a b ⟨x, hax, hxb⟩, hax.trans_lt hxb) (λ _ _ _ _, Ico_inter_Ico) f g | |
lemma is_pi_system_Icc_mem (s t : set α) : | |
is_pi_system {S | ∃ (l ∈ s) (u ∈ t) (h : l ≤ u), Icc l u = S} := | |
is_pi_system_Ixx_mem (λ a b, nonempty_Icc.1) (λ _ _ _ _, Icc_inter_Icc) s t | |
lemma is_pi_system_Icc (f : ι → α) (g : ι' → α) : | |
@is_pi_system α {S | ∃ i j (h : f i ≤ g j), Icc (f i) (g j) = S} := | |
is_pi_system_Ixx (λ a b, nonempty_Icc.1) (λ _ _ _ _, Icc_inter_Icc) f g | |
end order | |
/-- Given a collection `S` of subsets of `α`, then `generate_pi_system S` is the smallest | |
π-system containing `S`. -/ | |
inductive generate_pi_system {α} (S : set (set α)) : set (set α) | |
| base {s : set α} (h_s : s ∈ S) : generate_pi_system s | |
| inter {s t : set α} (h_s : generate_pi_system s) (h_t : generate_pi_system t) | |
(h_nonempty : (s ∩ t).nonempty) : generate_pi_system (s ∩ t) | |
lemma is_pi_system_generate_pi_system {α} (S : set (set α)) : | |
is_pi_system (generate_pi_system S) := | |
λ s h_s t h_t h_nonempty, generate_pi_system.inter h_s h_t h_nonempty | |
lemma subset_generate_pi_system_self {α} (S : set (set α)) : S ⊆ generate_pi_system S := | |
λ s, generate_pi_system.base | |
lemma generate_pi_system_subset_self {α} {S : set (set α)} (h_S : is_pi_system S) : | |
generate_pi_system S ⊆ S := | |
begin | |
intros x h, | |
induction h with s h_s s u h_gen_s h_gen_u h_nonempty h_s h_u, | |
{ exact h_s, }, | |
{ exact h_S _ h_s _ h_u h_nonempty, }, | |
end | |
lemma generate_pi_system_eq {α} {S : set (set α)} (h_pi : is_pi_system S) : | |
generate_pi_system S = S := | |
set.subset.antisymm (generate_pi_system_subset_self h_pi) (subset_generate_pi_system_self S) | |
lemma generate_pi_system_mono {α} {S T : set (set α)} (hST : S ⊆ T) : | |
generate_pi_system S ⊆ generate_pi_system T := | |
begin | |
intros t ht, | |
induction ht with s h_s s u h_gen_s h_gen_u h_nonempty h_s h_u, | |
{ exact generate_pi_system.base (set.mem_of_subset_of_mem hST h_s),}, | |
{ exact is_pi_system_generate_pi_system T _ h_s _ h_u h_nonempty, }, | |
end | |
lemma generate_pi_system_measurable_set {α} [M : measurable_space α] {S : set (set α)} | |
(h_meas_S : ∀ s ∈ S, measurable_set s) (t : set α) | |
(h_in_pi : t ∈ generate_pi_system S) : measurable_set t := | |
begin | |
induction h_in_pi with s h_s s u h_gen_s h_gen_u h_nonempty h_s h_u, | |
{ apply h_meas_S _ h_s, }, | |
{ apply measurable_set.inter h_s h_u, }, | |
end | |
lemma generate_from_measurable_set_of_generate_pi_system {α} {g : set (set α)} (t : set α) | |
(ht : t ∈ generate_pi_system g) : | |
measurable_set[generate_from g] t := | |
@generate_pi_system_measurable_set α (generate_from g) g | |
(λ s h_s_in_g, measurable_set_generate_from h_s_in_g) t ht | |
lemma generate_from_generate_pi_system_eq {α} {g : set (set α)} : | |
generate_from (generate_pi_system g) = generate_from g := | |
begin | |
apply le_antisymm; apply generate_from_le, | |
{ exact λ t h_t, generate_from_measurable_set_of_generate_pi_system t h_t, }, | |
{ exact λ t h_t, measurable_set_generate_from (generate_pi_system.base h_t), }, | |
end | |
/- Every element of the π-system generated by the union of a family of π-systems | |
is a finite intersection of elements from the π-systems. | |
For an indexed union version, see `mem_generate_pi_system_Union_elim'`. -/ | |
lemma mem_generate_pi_system_Union_elim {α β} {g : β → set (set α)} | |
(h_pi : ∀ b, is_pi_system (g b)) (t : set α) (h_t : t ∈ generate_pi_system (⋃ b, g b)) : | |
∃ (T : finset β) (f : β → set α), (t = ⋂ b ∈ T, f b) ∧ (∀ b ∈ T, f b ∈ g b) := | |
begin | |
induction h_t with s h_s s t' h_gen_s h_gen_t' h_nonempty h_s h_t', | |
{ rcases h_s with ⟨t', ⟨⟨b, rfl⟩, h_s_in_t'⟩⟩, | |
refine ⟨{b}, (λ _, s), _⟩, | |
simpa using h_s_in_t', }, | |
{ rcases h_t' with ⟨T_t', ⟨f_t', ⟨rfl, h_t'⟩⟩⟩, | |
rcases h_s with ⟨T_s, ⟨f_s, ⟨rfl, h_s⟩ ⟩ ⟩, | |
use [(T_s ∪ T_t'), (λ (b:β), | |
if (b ∈ T_s) then (if (b ∈ T_t') then (f_s b ∩ (f_t' b)) else (f_s b)) | |
else (if (b ∈ T_t') then (f_t' b) else (∅ : set α)))], | |
split, | |
{ ext a, | |
simp_rw [set.mem_inter_iff, set.mem_Inter, finset.mem_union, or_imp_distrib], | |
rw ← forall_and_distrib, | |
split; intros h1 b; by_cases hbs : b ∈ T_s; by_cases hbt : b ∈ T_t'; specialize h1 b; | |
simp only [hbs, hbt, if_true, if_false, true_implies_iff, and_self, false_implies_iff, | |
and_true, true_and] at h1 ⊢, | |
all_goals { exact h1, }, }, | |
intros b h_b, | |
split_ifs with hbs hbt hbt, | |
{ refine h_pi b (f_s b) (h_s b hbs) (f_t' b) (h_t' b hbt) (set.nonempty.mono _ h_nonempty), | |
exact set.inter_subset_inter (set.bInter_subset_of_mem hbs) (set.bInter_subset_of_mem hbt), }, | |
{ exact h_s b hbs, }, | |
{ exact h_t' b hbt, }, | |
{ rw finset.mem_union at h_b, | |
apply false.elim (h_b.elim hbs hbt), }, }, | |
end | |
/- Every element of the π-system generated by an indexed union of a family of π-systems | |
is a finite intersection of elements from the π-systems. | |
For a total union version, see `mem_generate_pi_system_Union_elim`. -/ | |
lemma mem_generate_pi_system_Union_elim' {α β} {g : β → set (set α)} {s : set β} | |
(h_pi : ∀ b ∈ s, is_pi_system (g b)) (t : set α) (h_t : t ∈ generate_pi_system (⋃ b ∈ s, g b)) : | |
∃ (T : finset β) (f : β → set α), (↑T ⊆ s) ∧ (t = ⋂ b ∈ T, f b) ∧ (∀ b ∈ T, f b ∈ g b) := | |
begin | |
have : t ∈ generate_pi_system (⋃ (b : subtype s), (g ∘ subtype.val) b), | |
{ suffices h1 : (⋃ (b : subtype s), (g ∘ subtype.val) b) = (⋃ b ∈ s, g b), by rwa h1, | |
ext x, | |
simp only [exists_prop, set.mem_Union, function.comp_app, subtype.exists, subtype.coe_mk], | |
refl }, | |
rcases @mem_generate_pi_system_Union_elim α (subtype s) (g ∘ subtype.val) | |
(λ b, h_pi b.val b.property) t this with ⟨T, ⟨f, ⟨rfl, h_t'⟩⟩⟩, | |
refine ⟨T.image subtype.val, function.extend subtype.val f (λ b : β, (∅ : set α)), by simp, _, _⟩, | |
{ ext a, split; | |
{ simp only [set.mem_Inter, subtype.forall, finset.set_bInter_finset_image], | |
intros h1 b h_b h_b_in_T, | |
have h2 := h1 b h_b h_b_in_T, | |
revert h2, | |
rw function.extend_apply subtype.val_injective, | |
apply id } }, | |
{ intros b h_b, | |
simp_rw [finset.mem_image, exists_prop, subtype.exists, | |
exists_and_distrib_right, exists_eq_right] at h_b, | |
cases h_b, | |
have h_b_alt : b = (subtype.mk b h_b_w).val := rfl, | |
rw [h_b_alt, function.extend_apply subtype.val_injective], | |
apply h_t', | |
apply h_b_h }, | |
end | |
section Union_Inter | |
/-! ### π-system generated by finite intersections of sets of a π-system family -/ | |
/-- From a set of finsets `S : set (finset ι)` and a family of sets of sets `π : ι → set (set α)`, | |
define the set of sets that can be written as `⋂ x ∈ t, f x` for some `t ∈ S` and sets `f x ∈ π x`. | |
If `S` is union-closed and `π` is a family of π-systems, then it is a π-system. | |
The π-systems used to prove Kolmogorov's 0-1 law are of that form. -/ | |
def pi_Union_Inter {α ι} (π : ι → set (set α)) (S : set (finset ι)) : set (set α) := | |
{s : set α | ∃ (t : finset ι) (htS : t ∈ S) (f : ι → set α) (hf : ∀ x, x ∈ t → f x ∈ π x), | |
s = ⋂ x ∈ t, f x} | |
/-- If `S` is union-closed and `π` is a family of π-systems, then `pi_Union_Inter π S` is a | |
π-system. -/ | |
lemma is_pi_system_pi_Union_Inter {α ι} (π : ι → set (set α)) | |
(hpi : ∀ x, is_pi_system (π x)) (S : set (finset ι)) (h_sup : sup_closed S) : | |
is_pi_system (pi_Union_Inter π S) := | |
begin | |
rintros t1 ⟨p1, hp1S, f1, hf1m, ht1_eq⟩ t2 ⟨p2, hp2S, f2, hf2m, ht2_eq⟩ h_nonempty, | |
simp_rw [pi_Union_Inter, set.mem_set_of_eq], | |
let g := λ n, (ite (n ∈ p1) (f1 n) set.univ) ∩ (ite (n ∈ p2) (f2 n) set.univ), | |
use [p1 ∪ p2, h_sup p1 p2 hp1S hp2S, g], | |
have h_inter_eq : t1 ∩ t2 = ⋂ i ∈ p1 ∪ p2, g i, | |
{ rw [ht1_eq, ht2_eq], | |
simp_rw [← set.inf_eq_inter, g], | |
ext1 x, | |
simp only [inf_eq_inter, mem_inter_eq, mem_Inter, finset.mem_union], | |
refine ⟨λ h i hi_mem_union, _, λ h, ⟨λ i hi1, _, λ i hi2, _⟩⟩, | |
{ split_ifs, | |
exacts [⟨h.1 i h_1, h.2 i h_2⟩, ⟨h.1 i h_1, set.mem_univ _⟩, | |
⟨set.mem_univ _, h.2 i h_2⟩, ⟨set.mem_univ _, set.mem_univ _⟩], }, | |
{ specialize h i (or.inl hi1), | |
rw if_pos hi1 at h, | |
exact h.1, }, | |
{ specialize h i (or.inr hi2), | |
rw if_pos hi2 at h, | |
exact h.2, }, }, | |
refine ⟨λ n hn, _, h_inter_eq⟩, | |
simp_rw g, | |
split_ifs with hn1 hn2, | |
{ refine hpi n (f1 n) (hf1m n hn1) (f2 n) (hf2m n hn2) (set.ne_empty_iff_nonempty.mp (λ h, _)), | |
rw h_inter_eq at h_nonempty, | |
suffices h_empty : (⋂ i ∈ p1 ∪ p2, g i) = ∅, | |
from (set.not_nonempty_iff_eq_empty.mpr h_empty) h_nonempty, | |
refine le_antisymm (set.Inter_subset_of_subset n _) (set.empty_subset _), | |
refine set.Inter_subset_of_subset hn _, | |
simp_rw [g, if_pos hn1, if_pos hn2], | |
exact h.subset, }, | |
{ simp [hf1m n hn1], }, | |
{ simp [hf2m n h], }, | |
{ exact absurd hn (by simp [hn1, h]), }, | |
end | |
lemma pi_Union_Inter_mono_left {α ι} {π π' : ι → set (set α)} (h_le : ∀ i, π i ⊆ π' i) | |
(S : set (finset ι)) : | |
pi_Union_Inter π S ⊆ pi_Union_Inter π' S := | |
begin | |
rintros s ⟨t, ht_mem, ft, hft_mem_pi, rfl⟩, | |
exact ⟨t, ht_mem, ft, λ x hxt, h_le x (hft_mem_pi x hxt), rfl⟩, | |
end | |
lemma generate_from_pi_Union_Inter_le {α ι} {m : measurable_space α} | |
(π : ι → set (set α)) (h : ∀ n, generate_from (π n) ≤ m) (S : set (finset ι)) : | |
generate_from (pi_Union_Inter π S) ≤ m := | |
begin | |
refine generate_from_le _, | |
rintros t ⟨ht_p, ht_p_mem, ft, hft_mem_pi, rfl⟩, | |
refine finset.measurable_set_bInter _ (λ x hx_mem, (h x) _ _), | |
exact measurable_set_generate_from (hft_mem_pi x hx_mem), | |
end | |
lemma subset_pi_Union_Inter {α ι} {π : ι → set (set α)} {S : set (finset ι)} | |
(h_univ : ∀ i, set.univ ∈ π i) {i : ι} {s : finset ι} (hsS : s ∈ S) (his : i ∈ s) : | |
π i ⊆ pi_Union_Inter π S := | |
begin | |
refine λ t ht_pii, ⟨s, hsS, (λ j, ite (j = i) t set.univ), ⟨λ m h_pm, _, _⟩⟩, | |
{ split_ifs, | |
{ rwa h, }, | |
{ exact h_univ m, }, }, | |
{ ext1 x, | |
simp_rw set.mem_Inter, | |
split; intro hx, | |
{ intros j h_p_j, | |
split_ifs, | |
{ exact hx, }, | |
{ exact set.mem_univ _, }, }, | |
{ simpa using hx i his, }, }, | |
end | |
lemma mem_pi_Union_Inter_of_measurable_set {α ι} (m : ι → measurable_space α) | |
{S : set (finset ι)} {i : ι} {t : finset ι} (htS : t ∈ S) (hit : i ∈ t) (s : set α) | |
(hs : measurable_set[m i] s) : | |
s ∈ pi_Union_Inter (λ n, {s | measurable_set[m n] s}) S := | |
subset_pi_Union_Inter (λ i, measurable_set.univ) htS hit hs | |
lemma le_generate_from_pi_Union_Inter {α ι} {π : ι → set (set α)} | |
(S : set (finset ι)) (h_univ : ∀ n, set.univ ∈ π n) {x : ι} | |
{t : finset ι} (htS : t ∈ S) (hxt : x ∈ t) : | |
generate_from (π x) ≤ generate_from (pi_Union_Inter π S) := | |
generate_from_mono (subset_pi_Union_Inter h_univ htS hxt) | |
lemma measurable_set_supr_of_mem_pi_Union_Inter {α ι} (m : ι → measurable_space α) | |
(S : set (finset ι)) (t : set α) (ht : t ∈ pi_Union_Inter (λ n, {s | measurable_set[m n] s}) S) : | |
measurable_set[⨆ i (hi : ∃ s ∈ S, i ∈ s), m i] t := | |
begin | |
rcases ht with ⟨pt, hpt, ft, ht_m, rfl⟩, | |
refine pt.measurable_set_bInter (λ i hi, _), | |
suffices h_le : m i ≤ (⨆ i (hi : ∃ s ∈ S, i ∈ s), m i), from h_le (ft i) (ht_m i hi), | |
have hi' : ∃ s ∈ S, i ∈ s, from ⟨pt, hpt, hi⟩, | |
exact le_supr₂ i hi', | |
end | |
lemma generate_from_pi_Union_Inter_measurable_space {α ι} (m : ι → measurable_space α) | |
(S : set (finset ι)) : | |
generate_from (pi_Union_Inter (λ n, {s | measurable_set[m n] s}) S) | |
= ⨆ i (hi : ∃ p ∈ S, i ∈ p), m i := | |
begin | |
refine le_antisymm _ _, | |
{ rw ← @generate_from_measurable_set α (⨆ i (hi : ∃ p ∈ S, i ∈ p), m i), | |
exact generate_from_mono (measurable_set_supr_of_mem_pi_Union_Inter m S), }, | |
{ refine supr₂_le (λ i hi, _), | |
rcases hi with ⟨p, hpS, hpi⟩, | |
rw ← @generate_from_measurable_set α (m i), | |
exact generate_from_mono (mem_pi_Union_Inter_of_measurable_set m hpS hpi), }, | |
end | |
end Union_Inter | |
namespace measurable_space | |
variable {α : Type*} | |
/-! ## Dynkin systems and Π-λ theorem -/ | |
/-- 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". | |
-/ | |
structure dynkin_system (α : Type*) := | |
(has : set α → Prop) | |
(has_empty : has ∅) | |
(has_compl : ∀ {a}, has a → has aᶜ) | |
(has_Union_nat : ∀ {f : ℕ → set α}, pairwise (disjoint on f) → (∀ i, has (f i)) → has (⋃ i, f i)) | |
namespace dynkin_system | |
@[ext] lemma ext : ∀ {d₁ d₂ : dynkin_system α}, (∀ s : set α, d₁.has s ↔ d₂.has s) → d₁ = d₂ | |
| ⟨s₁, _, _, _⟩ ⟨s₂, _, _, _⟩ h := have s₁ = s₂, from funext $ assume x, propext $ h x, | |
by subst this | |
variable (d : dynkin_system α) | |
lemma has_compl_iff {a} : d.has aᶜ ↔ d.has a := | |
⟨λ h, by simpa using d.has_compl h, λ h, d.has_compl h⟩ | |
lemma has_univ : d.has univ := | |
by simpa using d.has_compl d.has_empty | |
theorem has_Union {β} [encodable β] {f : β → set α} | |
(hd : pairwise (disjoint on f)) (h : ∀ i, d.has (f i)) : d.has (⋃ i, f i) := | |
by { rw ← encodable.Union_decode₂, exact | |
d.has_Union_nat (encodable.Union_decode₂_disjoint_on hd) | |
(λ n, encodable.Union_decode₂_cases d.has_empty h) } | |
theorem has_union {s₁ s₂ : set α} | |
(h₁ : d.has s₁) (h₂ : d.has s₂) (h : s₁ ∩ s₂ ⊆ ∅) : d.has (s₁ ∪ s₂) := | |
by { rw union_eq_Union, exact | |
d.has_Union (pairwise_disjoint_on_bool.2 h) (bool.forall_bool.2 ⟨h₂, h₁⟩) } | |
lemma has_diff {s₁ s₂ : set α} (h₁ : d.has s₁) (h₂ : d.has s₂) (h : s₂ ⊆ s₁) : d.has (s₁ \ s₂) := | |
begin | |
apply d.has_compl_iff.1, | |
simp [diff_eq, compl_inter], | |
exact d.has_union (d.has_compl h₁) h₂ (λ x ⟨h₁, h₂⟩, h₁ (h h₂)), | |
end | |
instance : has_le (dynkin_system α) := | |
{ le := λ m₁ m₂, m₁.has ≤ m₂.has } | |
lemma le_def {α} {a b : dynkin_system α} : a ≤ b ↔ a.has ≤ b.has := iff.rfl | |
instance : partial_order (dynkin_system α) := | |
{ le_refl := assume a b, le_rfl, | |
le_trans := assume a b c hab hbc, le_def.mpr (le_trans hab hbc), | |
le_antisymm := assume a b h₁ h₂, ext $ assume s, ⟨h₁ s, h₂ s⟩, | |
..dynkin_system.has_le } | |
/-- Every measurable space (σ-algebra) forms a Dynkin system -/ | |
def of_measurable_space (m : measurable_space α) : dynkin_system α := | |
{ has := m.measurable_set', | |
has_empty := m.measurable_set_empty, | |
has_compl := m.measurable_set_compl, | |
has_Union_nat := assume f _ hf, m.measurable_set_Union f hf } | |
lemma of_measurable_space_le_of_measurable_space_iff {m₁ m₂ : measurable_space α} : | |
of_measurable_space m₁ ≤ of_measurable_space m₂ ↔ m₁ ≤ m₂ := | |
iff.rfl | |
/-- The least Dynkin system containing a collection of basic sets. | |
This inductive type gives the underlying collection of sets. -/ | |
inductive generate_has (s : set (set α)) : set α → Prop | |
| basic : ∀ t ∈ s, generate_has t | |
| empty : generate_has ∅ | |
| compl : ∀ {a}, generate_has a → generate_has aᶜ | |
| Union : ∀ {f : ℕ → set α}, pairwise (disjoint on f) → | |
(∀ i, generate_has (f i)) → generate_has (⋃ i, f i) | |
lemma generate_has_compl {C : set (set α)} {s : set α} : generate_has C sᶜ ↔ generate_has C s := | |
by { refine ⟨_, generate_has.compl⟩, intro h, convert generate_has.compl h, simp } | |
/-- The least Dynkin system containing a collection of basic sets. -/ | |
def generate (s : set (set α)) : dynkin_system α := | |
{ has := generate_has s, | |
has_empty := generate_has.empty, | |
has_compl := assume a, generate_has.compl, | |
has_Union_nat := assume f, generate_has.Union } | |
lemma generate_has_def {C : set (set α)} : (generate C).has = generate_has C := rfl | |
instance : inhabited (dynkin_system α) := ⟨generate univ⟩ | |
/-- If a Dynkin system is closed under binary intersection, then it forms a `σ`-algebra. -/ | |
def to_measurable_space (h_inter : ∀ s₁ s₂, d.has s₁ → d.has s₂ → d.has (s₁ ∩ s₂)) := | |
{ measurable_space . | |
measurable_set' := d.has, | |
measurable_set_empty := d.has_empty, | |
measurable_set_compl := assume s h, d.has_compl h, | |
measurable_set_Union := λ f hf, | |
begin | |
rw ←Union_disjointed, | |
exact d.has_Union (disjoint_disjointed _) | |
(λ n, disjointed_rec (λ t i h, h_inter _ _ h $ d.has_compl $ hf i) (hf n)), | |
end } | |
lemma of_measurable_space_to_measurable_space | |
(h_inter : ∀ s₁ s₂, d.has s₁ → d.has s₂ → d.has (s₁ ∩ s₂)) : | |
of_measurable_space (d.to_measurable_space h_inter) = d := | |
ext $ assume s, iff.rfl | |
/-- If `s` is in a Dynkin system `d`, we can form the new Dynkin system `{s ∩ t | t ∈ d}`. -/ | |
def restrict_on {s : set α} (h : d.has s) : dynkin_system α := | |
{ has := λ t, d.has (t ∩ s), | |
has_empty := by simp [d.has_empty], | |
has_compl := assume t hts, | |
have tᶜ ∩ s = ((t ∩ s)ᶜ) \ sᶜ, | |
from set.ext $ assume x, by { by_cases x ∈ s; simp [h] }, | |
by { rw [this], exact d.has_diff (d.has_compl hts) (d.has_compl h) | |
(compl_subset_compl.mpr $ inter_subset_right _ _) }, | |
has_Union_nat := assume f hd hf, | |
begin | |
rw [inter_comm, inter_Union], | |
apply d.has_Union_nat, | |
{ exact λ i j h x ⟨⟨_, h₁⟩, _, h₂⟩, hd i j h ⟨h₁, h₂⟩ }, | |
{ simpa [inter_comm] using hf }, | |
end } | |
lemma generate_le {s : set (set α)} (h : ∀ t ∈ s, d.has t) : generate s ≤ d := | |
λ t ht, ht.rec_on h d.has_empty | |
(assume a _ h, d.has_compl h) | |
(assume f hd _ hf, d.has_Union hd hf) | |
lemma generate_has_subset_generate_measurable {C : set (set α)} {s : set α} | |
(hs : (generate C).has s) : measurable_set[generate_from C] s := | |
generate_le (of_measurable_space (generate_from C)) (λ t, measurable_set_generate_from) s hs | |
lemma generate_inter {s : set (set α)} | |
(hs : is_pi_system s) {t₁ t₂ : set α} | |
(ht₁ : (generate s).has t₁) (ht₂ : (generate s).has t₂) : (generate s).has (t₁ ∩ t₂) := | |
have generate s ≤ (generate s).restrict_on ht₂, | |
from generate_le _ $ assume s₁ hs₁, | |
have (generate s).has s₁, from generate_has.basic s₁ hs₁, | |
have generate s ≤ (generate s).restrict_on this, | |
from generate_le _ $ assume s₂ hs₂, | |
show (generate s).has (s₂ ∩ s₁), from | |
(s₂ ∩ s₁).eq_empty_or_nonempty.elim | |
(λ h, h.symm ▸ generate_has.empty) | |
(λ h, generate_has.basic _ $ hs _ hs₂ _ hs₁ h), | |
have (generate s).has (t₂ ∩ s₁), from this _ ht₂, | |
show (generate s).has (s₁ ∩ t₂), by rwa [inter_comm], | |
this _ ht₁ | |
/-- | |
**Dynkin's π-λ theorem**: | |
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). | |
-/ | |
lemma generate_from_eq {s : set (set α)} (hs : is_pi_system s) : | |
generate_from s = (generate s).to_measurable_space (λ t₁ t₂, generate_inter hs) := | |
le_antisymm | |
(generate_from_le $ assume t ht, generate_has.basic t ht) | |
(of_measurable_space_le_of_measurable_space_iff.mp $ | |
by { rw [of_measurable_space_to_measurable_space], | |
exact (generate_le _ $ assume t ht, measurable_set_generate_from ht) }) | |
end dynkin_system | |
theorem induction_on_inter {C : set α → Prop} {s : set (set α)} [m : measurable_space α] | |
(h_eq : m = generate_from s) (h_inter : is_pi_system s) | |
(h_empty : C ∅) (h_basic : ∀ t ∈ s, C t) (h_compl : ∀ t, measurable_set t → C t → C tᶜ) | |
(h_union : ∀ f : ℕ → set α, pairwise (disjoint on f) → | |
(∀ i, measurable_set (f i)) → (∀ i, C (f i)) → C (⋃ i, f i)) : | |
∀ ⦃t⦄, measurable_set t → C t := | |
have eq : measurable_set = dynkin_system.generate_has s, | |
by { rw [h_eq, dynkin_system.generate_from_eq h_inter], refl }, | |
assume t ht, | |
have dynkin_system.generate_has s t, by rwa [eq] at ht, | |
this.rec_on h_basic h_empty | |
(assume t ht, h_compl t $ by { rw [eq], exact ht }) | |
(assume f hf ht, h_union f hf $ assume i, by { rw [eq], exact ht _ }) | |
end measurable_space | |