The filter of small sets

This file defines the filter of small sets w.r.t. a filter f, which is the largest filter containing all powersets of members of f.

g converges to f.smallSets if for all s ∈ f, eventually we have g x ⊆ s.

An example usage is that if f : ι → E → ℝ is a family of nonnegative functions with integral 1, then saying that fun i ↦ support (f i) tendsto (𝓝 0).smallSets is a way of saying that f tends to the Dirac delta distribution.

def Filter.smallSets {α : Type u_1} (l : Filter α) : Filter (Set α)

The filter l.smallSets is the largest filter containing all powersets of members of l.

Equations

• Filter.smallSets l = Filter.lift' l Set.powerset

theorem Filter.smallSets_eq_generate {α : Type u_1} {f : Filter α} : Filter.smallSets f = Filter.generate (Set.powerset '' f.sets)

theorem Filter.HasBasis.smallSets {α : Type u_1} {ι : Sort u_3} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : Filter.HasBasis l p s) : Filter.HasBasis (Filter.smallSets l) p fun (i : ι) => 𝒫 s i

theorem Filter.hasBasis_smallSets {α : Type u_1} (l : Filter α) : Filter.HasBasis (Filter.smallSets l) (fun (t : Set α) => t ∈ l) Set.powerset

theorem Filter.tendsto_smallSets_iff {α : Type u_1} {β : Type u_2} {la : Filter α} {lb : Filter β} {f : α → Set β} : Filter.Tendsto f la (Filter.smallSets lb) ↔ ∀ t ∈ lb, ∀ᶠ (x : α) in la, f x ⊆ t

g converges to f.smallSets if for all s ∈ f, eventually we have g x ⊆ s.

theorem Filter.eventually_smallSets {α : Type u_1} {l : Filter α} {p : Set α → Prop} : (∀ᶠ (s : Set α) in Filter.smallSets l, p s) ↔ ∃ s ∈ l, ∀ t ⊆ s, p t

theorem Filter.eventually_smallSets' {α : Type u_1} {l : Filter α} {p : Set α → Prop} (hp : ∀ ⦃s t : Set α⦄, s ⊆ t → p t → p s) : (∀ᶠ (s : Set α) in Filter.smallSets l, p s) ↔ ∃ s ∈ l, p s

theorem Filter.frequently_smallSets {α : Type u_1} {l : Filter α} {p : Set α → Prop} : (∃ᶠ (s : Set α) in Filter.smallSets l, p s) ↔ ∀ t ∈ l, ∃ s ⊆ t, p s

theorem Filter.frequently_smallSets_mem {α : Type u_1} (l : Filter α) : ∃ᶠ (s : Set α) in Filter.smallSets l, s ∈ l

@[simp]

theorem Filter.tendsto_image_smallSets {α : Type u_1} {β : Type u_2} {la : Filter α} {lb : Filter β} {f : α → β} : Filter.Tendsto (fun (x : Set α) => f '' x) (Filter.smallSets la) (Filter.smallSets lb) ↔ Filter.Tendsto f la lb

theorem Filter.Tendsto.image_smallSets {α : Type u_1} {β : Type u_2} {la : Filter α} {lb : Filter β} {f : α → β} : Filter.Tendsto f la lb → Filter.Tendsto (fun (x : Set α) => f '' x) (Filter.smallSets la) (Filter.smallSets lb)

Alias of the reverse direction of Filter.tendsto_image_smallSets.

theorem Filter.HasAntitoneBasis.tendsto_smallSets {α : Type u_1} {l : Filter α} {ι : Type u_4} [Preorder ι] {s : ι → Set α} (hl : Filter.HasAntitoneBasis l s) : Filter.Tendsto s Filter.atTop (Filter.smallSets l)

theorem Filter.monotone_smallSets {α : Type u_1} : Monotone Filter.smallSets

@[simp]

theorem Filter.smallSets_bot {α : Type u_1} : Filter.smallSets ⊥ = pure ∅

@[simp]

theorem Filter.smallSets_top {α : Type u_1} : Filter.smallSets ⊤ = ⊤

@[simp]

theorem Filter.smallSets_principal {α : Type u_1} (s : Set α) : Filter.smallSets (Filter.principal s) = Filter.principal (𝒫 s)

theorem Filter.smallSets_comap {α : Type u_1} {β : Type u_2} (l : Filter β) (f : α → β) : Filter.smallSets (Filter.comap f l) = Filter.lift' l (Set.powerset ∘ Set.preimage f)

theorem Filter.comap_smallSets {α : Type u_1} {β : Type u_2} (l : Filter β) (f : α → Set β) : Filter.comap f (Filter.smallSets l) = Filter.lift' l (Set.preimage f ∘ Set.powerset)

theorem Filter.smallSets_iInf {α : Type u_1} {ι : Sort u_3} {f : ι → Filter α} : Filter.smallSets (iInf f) = ⨅ (i : ι), Filter.smallSets (f i)

theorem Filter.smallSets_inf {α : Type u_1} (l₁ : Filter α) (l₂ : Filter α) : Filter.smallSets (l₁ ⊓ l₂) = Filter.smallSets l₁ ⊓ Filter.smallSets l₂

instance Filter.smallSets_neBot {α : Type u_1} (l : Filter α) : Filter.NeBot (Filter.smallSets l)

Equations

• (_ : Filter.NeBot (Filter.smallSets l)) = (_ : Filter.NeBot (Filter.lift' l Set.powerset))

theorem Filter.Tendsto.smallSets_mono {α : Type u_1} {β : Type u_2} {la : Filter α} {lb : Filter β} {s : α → Set β} {t : α → Set β} (ht : Filter.Tendsto t la (Filter.smallSets lb)) (hst : ∀ᶠ (x : α) in la, s x ⊆ t x) : Filter.Tendsto s la (Filter.smallSets lb)

theorem Filter.Tendsto.of_smallSets {α : Type u_1} {β : Type u_2} {la : Filter α} {lb : Filter β} {s : α → Set β} {f : α → β} (hs : Filter.Tendsto s la (Filter.smallSets lb)) (hf : ∀ᶠ (x : α) in la, f x ∈ s x) : Filter.Tendsto f la lb

Generalized squeeze theorem (also known as sandwich theorem). If s : α → Set β is a family of sets that tends to Filter.smallSets lb along la and f : α → β is a function such that f x ∈ s x eventually along la, then f tends to lb along la.

If s x is the closed interval [g x, h x] for some functions g, h that tend to the same limit 𝓝 y, then we obtain the standard squeeze theorem, see tendsto_of_tendsto_of_tendsto_of_le_of_le'.

@[simp]

theorem Filter.eventually_smallSets_eventually {α : Type u_1} {l : Filter α} {l' : Filter α} {p : α → Prop} : (∀ᶠ (s : Set α) in Filter.smallSets l, ∀ᶠ (x : α) in l', x ∈ s → p x) ↔ ∀ᶠ (x : α) in l ⊓ l', p x

@[simp]

theorem Filter.eventually_smallSets_forall {α : Type u_1} {l : Filter α} {p : α → Prop} : (∀ᶠ (s : Set α) in Filter.smallSets l, ∀ x ∈ s, p x) ↔ ∀ᶠ (x : α) in l, p x

theorem Filter.Eventually.smallSets {α : Type u_1} {l : Filter α} {p : α → Prop} : (∀ᶠ (x : α) in l, p x) → ∀ᶠ (s : Set α) in Filter.smallSets l, ∀ x ∈ s, p x

Alias of the reverse direction of Filter.eventually_smallSets_forall.

theorem Filter.Eventually.of_smallSets {α : Type u_1} {l : Filter α} {p : α → Prop} : (∀ᶠ (s : Set α) in Filter.smallSets l, ∀ x ∈ s, p x) → ∀ᶠ (x : α) in l, p x

Alias of the forward direction of Filter.eventually_smallSets_forall.

@[simp]

theorem Filter.eventually_smallSets_subset {α : Type u_1} {l : Filter α} {s : Set α} : (∀ᶠ (t : Set α) in Filter.smallSets l, t ⊆ s) ↔ s ∈ l