File: Mathlib/MeasureTheory/OuterMeasure/Operations.lean

full_name: MeasureTheory.OuterMeasure.instZero
code: instance instZero : Zero (OuterMeasure α) 
goal: Zero (MeasureTheory.OuterMeasure α)

full_name: MeasureTheory.OuterMeasure.coe_zero
code: @[simp]
theorem coe_zero : ⇑(0 : OuterMeasure α) = 0 
goal: ⇑0 = 0

full_name: MeasureTheory.OuterMeasure.instInhabited
code: instance instInhabited : Inhabited (OuterMeasure α) 
goal: Inhabited (MeasureTheory.OuterMeasure α)

full_name: MeasureTheory.OuterMeasure.instAdd
code: instance instAdd : Add (OuterMeasure α) 
goal: Add (MeasureTheory.OuterMeasure α)

full_name: MeasureTheory.OuterMeasure.coe_add
code: @[simp]
theorem coe_add (m₁ m₂ : OuterMeasure α) : ⇑(m₁ + m₂) = m₁ + m₂ 
goal: ⇑(m₁ + m₂) = ⇑m₁ + ⇑m₂

full_name: MeasureTheory.OuterMeasure.add_apply
code: theorem add_apply (m₁ m₂ : OuterMeasure α) (s : Set α) : (m₁ + m₂) s = m₁ s + m₂ s 
goal: (m₁ + m₂) s = m₁ s + m₂ s

full_name: MeasureTheory.OuterMeasure.instSMul
code: instance instSMul : SMul R (OuterMeasure α) 
goal: SMul R (MeasureTheory.OuterMeasure α)

full_name: MeasureTheory.OuterMeasure.coe_smul
code: @[simp]
theorem coe_smul (c : R) (m : OuterMeasure α) : ⇑(c • m) = c • ⇑m 
goal: ⇑(c • m) = c • ⇑m

full_name: MeasureTheory.OuterMeasure.smul_apply
code: theorem smul_apply (c : R) (m : OuterMeasure α) (s : Set α) : (c • m) s = c • m s 
goal: (c • m) s = c • m s

full_name: MeasureTheory.OuterMeasure.instSMulCommClass
code: instance instSMulCommClass [SMulCommClass R R' ℝ≥0∞] : SMulCommClass R R' (OuterMeasure α) 
goal: SMulCommClass R R' (MeasureTheory.OuterMeasure α)

full_name: MeasureTheory.OuterMeasure.instIsScalarTower
code: instance instIsScalarTower [SMul R R'] [IsScalarTower R R' ℝ≥0∞] :
    IsScalarTower R R' (OuterMeasure α) 
goal: IsScalarTower R R' (MeasureTheory.OuterMeasure α)

full_name: MeasureTheory.OuterMeasure.instIsCentralScalar
code: instance instIsCentralScalar [SMul Rᵐᵒᵖ ℝ≥0∞] [IsCentralScalar R ℝ≥0∞] :
    IsCentralScalar R (OuterMeasure α) 
goal: IsCentralScalar R (MeasureTheory.OuterMeasure α)

full_name: MeasureTheory.OuterMeasure.instMulAction
code: instance instMulAction {R : Type*} [Monoid R] [MulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] :
    MulAction R (OuterMeasure α) 
goal: MulAction R (MeasureTheory.OuterMeasure α)

full_name: MeasureTheory.OuterMeasure.addCommMonoid
code: instance addCommMonoid : AddCommMonoid (OuterMeasure α) 
goal: AddCommMonoid (MeasureTheory.OuterMeasure α)

full_name: MeasureTheory.OuterMeasure.coeFnAddMonoidHom_apply
code: /-- `(⇑)` as an `AddMonoidHom`. -/
@[simps]
def coeFnAddMonoidHom : OuterMeasure α →+ Set α → ℝ≥0∞ 
goal: ∀ (a : MeasureTheory.OuterMeasure α) (a_1 : Set α), MeasureTheory.OuterMeasure.coeFnAddMonoidHom a a_1 = a a_1

full_name: MeasureTheory.OuterMeasure.coeFnAddMonoidHom
code: /-- `(⇑)` as an `AddMonoidHom`. -/
@[simps]
def coeFnAddMonoidHom : OuterMeasure α →+ Set α → ℝ≥0∞ 
goal: MeasureTheory.OuterMeasure α →+ Set α → ℝ≥0∞

full_name: MeasureTheory.OuterMeasure.instDistribMulAction
code: instance instDistribMulAction {R : Type*} [Monoid R] [DistribMulAction R ℝ≥0∞]
    [IsScalarTower R ℝ≥0∞ ℝ≥0∞] :
    DistribMulAction R (OuterMeasure α) 
goal: DistribMulAction R (MeasureTheory.OuterMeasure α)

full_name: MeasureTheory.OuterMeasure.instModule
code: instance instModule {R : Type*} [Semiring R] [Module R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] :
    Module R (OuterMeasure α) 
goal: Module R (MeasureTheory.OuterMeasure α)

full_name: MeasureTheory.OuterMeasure.instBot
code: instance instBot : Bot (OuterMeasure α) 
goal: Bot (MeasureTheory.OuterMeasure α)

full_name: MeasureTheory.OuterMeasure.coe_bot
code: @[simp]
theorem coe_bot : (⊥ : OuterMeasure α) = 0 
goal: ⊥ = 0

full_name: MeasureTheory.OuterMeasure.instPartialOrder
code: instance instPartialOrder : PartialOrder (OuterMeasure α) 
goal: PartialOrder (MeasureTheory.OuterMeasure α)

full_name: MeasureTheory.OuterMeasure.orderBot
code: instance orderBot : OrderBot (OuterMeasure α) 
goal: OrderBot (MeasureTheory.OuterMeasure α)

full_name: MeasureTheory.OuterMeasure.univ_eq_zero_iff
code: theorem univ_eq_zero_iff (m : OuterMeasure α) : m univ = 0 ↔ m = 0 
goal: m Set.univ = 0 ↔ m = 0

full_name: MeasureTheory.OuterMeasure.instSupSet
code: instance instSupSet : SupSet (OuterMeasure α) 
goal: SupSet (MeasureTheory.OuterMeasure α)

full_name: MeasureTheory.OuterMeasure.instCompleteLattice
code: instance instCompleteLattice : CompleteLattice (OuterMeasure α) 
goal: CompleteLattice (MeasureTheory.OuterMeasure α)

full_name: MeasureTheory.OuterMeasure.sSup_apply
code: @[simp]
theorem sSup_apply (ms : Set (OuterMeasure α)) (s : Set α) :
    (sSup ms) s = ⨆ m ∈ ms, (m : OuterMeasure α) s 
goal: (sSup ms) s = ⨆ m ∈ ms, m s

full_name: MeasureTheory.OuterMeasure.iSup_apply
code: @[simp]
theorem iSup_apply {ι} (f : ι → OuterMeasure α) (s : Set α) : (⨆ i : ι, f i) s = ⨆ i, f i s 
goal: (⨆ i, f i) s = ⨆ i, (f i) s

full_name: MeasureTheory.OuterMeasure.coe_iSup
code: @[norm_cast]
theorem coe_iSup {ι} (f : ι → OuterMeasure α) : ⇑(⨆ i, f i) = ⨆ i, ⇑(f i) 
goal: ⇑(⨆ i, f i) = ⨆ i, ⇑(f i)

full_name: MeasureTheory.OuterMeasure.sup_apply
code: @[simp]
theorem sup_apply (m₁ m₂ : OuterMeasure α) (s : Set α) : (m₁ ⊔ m₂) s = m₁ s ⊔ m₂ s 
goal: (m₁ ⊔ m₂) s = m₁ s ⊔ m₂ s

full_name: MeasureTheory.OuterMeasure.smul_iSup
code: theorem smul_iSup {R : Type*} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞]
    {ι : Sort*} (f : ι → OuterMeasure α) (c : R) :
    (c • ⨆ i, f i) = ⨆ i, c • f i 
goal: c • ⨆ i, f i = ⨆ i, c • f i

full_name: MeasureTheory.OuterMeasure.mono''
code: @[mono, gcongr]
theorem mono'' {m₁ m₂ : OuterMeasure α} {s₁ s₂ : Set α} (hm : m₁ ≤ m₂) (hs : s₁ ⊆ s₂) :
    m₁ s₁ ≤ m₂ s₂ 
goal: m₁ s₁ ≤ m₂ s₂

full_name: MeasureTheory.OuterMeasure.map
code: /-- The pushforward of `m` along `f`. The outer measure on `s` is defined to be `m (f ⁻¹' s)`. -/
def map {β} (f : α → β) : OuterMeasure α →ₗ[ℝ≥0∞] OuterMeasure β 
goal: MeasureTheory.OuterMeasure α →ₗ[ℝ≥0∞] MeasureTheory.OuterMeasure β

full_name: MeasureTheory.OuterMeasure.map_apply
code: @[simp]
theorem map_apply {β} (f : α → β) (m : OuterMeasure α) (s : Set β) : map f m s = m (f ⁻¹' s) 
goal: ((MeasureTheory.OuterMeasure.map f) m) s = m (f ⁻¹' s)

full_name: MeasureTheory.OuterMeasure.map_id
code: @[simp]
theorem map_id (m : OuterMeasure α) : map id m = m 
goal: (MeasureTheory.OuterMeasure.map id) m = m

full_name: MeasureTheory.OuterMeasure.map_map
code: @[simp]
theorem map_map {β γ} (f : α → β) (g : β → γ) (m : OuterMeasure α) :
    map g (map f m) = map (g ∘ f) m 
goal: (MeasureTheory.OuterMeasure.map g) ((MeasureTheory.OuterMeasure.map f) m) = (MeasureTheory.OuterMeasure.map (g ∘ f)) m

full_name: MeasureTheory.OuterMeasure.map_mono
code: @[mono]
theorem map_mono {β} (f : α → β) : Monotone (map f) 
goal: Monotone ⇑(MeasureTheory.OuterMeasure.map f)

full_name: MeasureTheory.OuterMeasure.map_sup
code: @[simp]
theorem map_sup {β} (f : α → β) (m m' : OuterMeasure α) : map f (m ⊔ m') = map f m ⊔ map f m' 
goal: (MeasureTheory.OuterMeasure.map f) (m ⊔ m') =
  (MeasureTheory.OuterMeasure.map f) m ⊔ (MeasureTheory.OuterMeasure.map f) m'

full_name: MeasureTheory.OuterMeasure.map_iSup
code: @[simp]
theorem map_iSup {β ι} (f : α → β) (m : ι → OuterMeasure α) : map f (⨆ i, m i) = ⨆ i, map f (m i) 
goal: (MeasureTheory.OuterMeasure.map f) (⨆ i, m i) = ⨆ i, (MeasureTheory.OuterMeasure.map f) (m i)

full_name: MeasureTheory.OuterMeasure.instFunctor
code: instance instFunctor : Functor OuterMeasure 
goal: Functor MeasureTheory.OuterMeasure

full_name: MeasureTheory.OuterMeasure.instLawfulFunctor
code: instance instLawfulFunctor : LawfulFunctor OuterMeasure 
goal: LawfulFunctor MeasureTheory.OuterMeasure

full_name: MeasureTheory.OuterMeasure.dirac
code: /-- The dirac outer measure. -/
def dirac (a : α) : OuterMeasure α 
goal: MeasureTheory.OuterMeasure α

full_name: MeasureTheory.OuterMeasure.dirac_apply
code: @[simp]
theorem dirac_apply (a : α) (s : Set α) : dirac a s = indicator s (fun _ => 1) a 
goal: (MeasureTheory.OuterMeasure.dirac a) s = s.indicator (fun x => 1) a

full_name: MeasureTheory.OuterMeasure.sum
code: /-- The sum of an (arbitrary) collection of outer measures. -/
def sum {ι} (f : ι → OuterMeasure α) : OuterMeasure α 
goal: MeasureTheory.OuterMeasure α

full_name: MeasureTheory.OuterMeasure.sum_apply
code: @[simp]
theorem sum_apply {ι} (f : ι → OuterMeasure α) (s : Set α) : sum f s = ∑' i, f i s 
goal: (MeasureTheory.OuterMeasure.sum f) s = ∑' (i : ι), (f i) s

full_name: MeasureTheory.OuterMeasure.smul_dirac_apply
code: theorem smul_dirac_apply (a : ℝ≥0∞) (b : α) (s : Set α) :
    (a • dirac b) s = indicator s (fun _ => a) b 
goal: (a • MeasureTheory.OuterMeasure.dirac b) s = s.indicator (fun x => a) b

full_name: MeasureTheory.OuterMeasure.comap
code: /-- Pullback of an `OuterMeasure`: `comap f μ s = μ (f '' s)`. -/
def comap {β} (f : α → β) : OuterMeasure β →ₗ[ℝ≥0∞] OuterMeasure α 
goal: MeasureTheory.OuterMeasure β →ₗ[ℝ≥0∞] MeasureTheory.OuterMeasure α

full_name: MeasureTheory.OuterMeasure.comap_apply
code: @[simp]
theorem comap_apply {β} (f : α → β) (m : OuterMeasure β) (s : Set α) : comap f m s = m (f '' s) 
goal: ((MeasureTheory.OuterMeasure.comap f) m) s = m (f '' s)

full_name: MeasureTheory.OuterMeasure.comap_mono
code: @[mono]
theorem comap_mono {β} (f : α → β) : Monotone (comap f) 
goal: Monotone ⇑(MeasureTheory.OuterMeasure.comap f)

full_name: MeasureTheory.OuterMeasure.comap_iSup
code: @[simp]
theorem comap_iSup {β ι} (f : α → β) (m : ι → OuterMeasure β) :
    comap f (⨆ i, m i) = ⨆ i, comap f (m i) 
goal: (MeasureTheory.OuterMeasure.comap f) (⨆ i, m i) = ⨆ i, (MeasureTheory.OuterMeasure.comap f) (m i)

full_name: MeasureTheory.OuterMeasure.restrict
code: /-- Restrict an `OuterMeasure` to a set. -/
def restrict (s : Set α) : OuterMeasure α →ₗ[ℝ≥0∞] OuterMeasure α 
goal: MeasureTheory.OuterMeasure α →ₗ[ℝ≥0∞] MeasureTheory.OuterMeasure α

full_name: MeasureTheory.OuterMeasure.restrict_apply
code: @[simp]
theorem restrict_apply (s t : Set α) (m : OuterMeasure α) : restrict s m t = m (t ∩ s) 
goal: ((MeasureTheory.OuterMeasure.restrict s) m) t = m (t ∩ s)

full_name: MeasureTheory.OuterMeasure.restrict_mono
code: @[mono]
theorem restrict_mono {s t : Set α} (h : s ⊆ t) {m m' : OuterMeasure α} (hm : m ≤ m') :
    restrict s m ≤ restrict t m' 
goal: (MeasureTheory.OuterMeasure.restrict s) m ≤ (MeasureTheory.OuterMeasure.restrict t) m'

full_name: MeasureTheory.OuterMeasure.restrict_univ
code: @[simp]
theorem restrict_univ (m : OuterMeasure α) : restrict univ m = m 
goal: (MeasureTheory.OuterMeasure.restrict Set.univ) m = m

full_name: MeasureTheory.OuterMeasure.restrict_empty
code: @[simp]
theorem restrict_empty (m : OuterMeasure α) : restrict ∅ m = 0 
goal: (MeasureTheory.OuterMeasure.restrict ∅) m = 0

full_name: MeasureTheory.OuterMeasure.restrict_iSup
code: @[simp]
theorem restrict_iSup {ι} (s : Set α) (m : ι → OuterMeasure α) :
    restrict s (⨆ i, m i) = ⨆ i, restrict s (m i) 
goal: (MeasureTheory.OuterMeasure.restrict s) (⨆ i, m i) = ⨆ i, (MeasureTheory.OuterMeasure.restrict s) (m i)

full_name: MeasureTheory.OuterMeasure.map_comap
code: theorem map_comap {β} (f : α → β) (m : OuterMeasure β) : map f (comap f m) = restrict (range f) m 
goal: (MeasureTheory.OuterMeasure.map f) ((MeasureTheory.OuterMeasure.comap f) m) =
  (MeasureTheory.OuterMeasure.restrict (Set.range f)) m

full_name: MeasureTheory.OuterMeasure.map_comap_le
code: theorem map_comap_le {β} (f : α → β) (m : OuterMeasure β) : map f (comap f m) ≤ m 
goal: (MeasureTheory.OuterMeasure.map f) ((MeasureTheory.OuterMeasure.comap f) m) ≤ m

full_name: MeasureTheory.OuterMeasure.restrict_le_self
code: theorem restrict_le_self (m : OuterMeasure α) (s : Set α) : restrict s m ≤ m 
goal: (MeasureTheory.OuterMeasure.restrict s) m ≤ m

full_name: MeasureTheory.OuterMeasure.map_le_restrict_range
code: @[simp]
theorem map_le_restrict_range {β} {ma : OuterMeasure α} {mb : OuterMeasure β} {f : α → β} :
    map f ma ≤ restrict (range f) mb ↔ map f ma ≤ mb 
goal: (MeasureTheory.OuterMeasure.map f) ma ≤ (MeasureTheory.OuterMeasure.restrict (Set.range f)) mb ↔
  (MeasureTheory.OuterMeasure.map f) ma ≤ mb

full_name: MeasureTheory.OuterMeasure.map_comap_of_surjective
code: theorem map_comap_of_surjective {β} {f : α → β} (hf : Surjective f) (m : OuterMeasure β) :
    map f (comap f m) = m 
goal: (MeasureTheory.OuterMeasure.map f) ((MeasureTheory.OuterMeasure.comap f) m) = m

full_name: MeasureTheory.OuterMeasure.le_comap_map
code: theorem le_comap_map {β} (f : α → β) (m : OuterMeasure α) : m ≤ comap f (map f m) 
goal: m ≤ (MeasureTheory.OuterMeasure.comap f) ((MeasureTheory.OuterMeasure.map f) m)

full_name: MeasureTheory.OuterMeasure.comap_map
code: theorem comap_map {β} {f : α → β} (hf : Injective f) (m : OuterMeasure α) : comap f (map f m) = m 
goal: (MeasureTheory.OuterMeasure.comap f) ((MeasureTheory.OuterMeasure.map f) m) = m

full_name: MeasureTheory.OuterMeasure.top_apply
code: @[simp]
theorem top_apply {s : Set α} (h : s.Nonempty) : (⊤ : OuterMeasure α) s = ∞ 
goal: ⊤ s = ⊤

full_name: MeasureTheory.OuterMeasure.top_apply'
code: theorem top_apply' (s : Set α) : (⊤ : OuterMeasure α) s = ⨅ h : s = ∅, 0 
goal: ⊤ s = ⨅ (_ : s = ∅), 0

full_name: MeasureTheory.OuterMeasure.comap_top
code: @[simp]
theorem comap_top (f : α → β) : comap f ⊤ = ⊤ 
goal: (MeasureTheory.OuterMeasure.comap f) ⊤ = ⊤

full_name: MeasureTheory.OuterMeasure.map_top
code: theorem map_top (f : α → β) : map f ⊤ = restrict (range f) ⊤ 
goal: (MeasureTheory.OuterMeasure.map f) ⊤ = (MeasureTheory.OuterMeasure.restrict (Set.range f)) ⊤

full_name: MeasureTheory.OuterMeasure.map_top_of_surjective
code: @[simp]
theorem map_top_of_surjective (f : α → β) (hf : Surjective f) : map f ⊤ = ⊤ 
goal: (MeasureTheory.OuterMeasure.map f) ⊤ = ⊤