File: Mathlib/MeasureTheory/Order/Lattice.lean

full_name: MeasurableSup.measurable_const_sup
code: /-- We say that a type has `MeasurableSup` if `(c ⊔ ·)` and `(· ⊔ c)` are measurable functions.
For a typeclass assuming measurability of `uncurry (· ⊔ ·)` see `MeasurableSup₂`. -/
class MeasurableSup (M : Type*) [MeasurableSpace M] [Sup M] : Prop where
  measurable_const_sup : ∀ c : M, Measurable (c ⊔ ·)
  measurable_sup_const : ∀ c : M, Measurable (· ⊔ c)
goal: Measurable fun x => c ⊔ x

full_name: MeasurableSup
code: /-- We say that a type has `MeasurableSup` if `(c ⊔ ·)` and `(· ⊔ c)` are measurable functions.
For a typeclass assuming measurability of `uncurry (· ⊔ ·)` see `MeasurableSup₂`. -/
class MeasurableSup (M : Type*) [MeasurableSpace M] [Sup M] : Prop where
  measurable_const_sup : ∀ c : M, Measurable (c ⊔ ·)
  measurable_sup_const : ∀ c : M, Measurable (· ⊔ c)
goal: Prop

full_name: MeasurableSup.measurable_sup_const
code: /-- We say that a type has `MeasurableSup` if `(c ⊔ ·)` and `(· ⊔ c)` are measurable functions.
For a typeclass assuming measurability of `uncurry (· ⊔ ·)` see `MeasurableSup₂`. -/
class MeasurableSup (M : Type*) [MeasurableSpace M] [Sup M] : Prop where
  measurable_const_sup : ∀ c : M, Measurable (c ⊔ ·)
  measurable_sup_const : ∀ c : M, Measurable (· ⊔ c)
goal: Measurable fun x => x ⊔ c

full_name: MeasurableSup₂.measurable_sup
code: /-- We say that a type has `MeasurableSup₂` if `uncurry (· ⊔ ·)` is a measurable functions.
For a typeclass assuming measurability of `(c ⊔ ·)` and `(· ⊔ c)` see `MeasurableSup`. -/
class MeasurableSup₂ (M : Type*) [MeasurableSpace M] [Sup M] : Prop where
  measurable_sup : Measurable fun p : M × M => p.1 ⊔ p.2
goal: Measurable fun p => p.1 ⊔ p.2

full_name: MeasurableSup₂
code: /-- We say that a type has `MeasurableSup₂` if `uncurry (· ⊔ ·)` is a measurable functions.
For a typeclass assuming measurability of `(c ⊔ ·)` and `(· ⊔ c)` see `MeasurableSup`. -/
class MeasurableSup₂ (M : Type*) [MeasurableSpace M] [Sup M] : Prop where
  measurable_sup : Measurable fun p : M × M => p.1 ⊔ p.2
goal: Prop

full_name: MeasurableInf.measurable_const_inf
code: /-- We say that a type has `MeasurableInf` if `(c ⊓ ·)` and `(· ⊓ c)` are measurable functions.
For a typeclass assuming measurability of `uncurry (· ⊓ ·)` see `MeasurableInf₂`. -/
class MeasurableInf (M : Type*) [MeasurableSpace M] [Inf M] : Prop where
  measurable_const_inf : ∀ c : M, Measurable (c ⊓ ·)
  measurable_inf_const : ∀ c : M, Measurable (· ⊓ c)
goal: Measurable fun x => c ⊓ x

full_name: MeasurableInf.measurable_inf_const
code: /-- We say that a type has `MeasurableInf` if `(c ⊓ ·)` and `(· ⊓ c)` are measurable functions.
For a typeclass assuming measurability of `uncurry (· ⊓ ·)` see `MeasurableInf₂`. -/
class MeasurableInf (M : Type*) [MeasurableSpace M] [Inf M] : Prop where
  measurable_const_inf : ∀ c : M, Measurable (c ⊓ ·)
  measurable_inf_const : ∀ c : M, Measurable (· ⊓ c)
goal: Measurable fun x => x ⊓ c

full_name: MeasurableInf
code: /-- We say that a type has `MeasurableInf` if `(c ⊓ ·)` and `(· ⊓ c)` are measurable functions.
For a typeclass assuming measurability of `uncurry (· ⊓ ·)` see `MeasurableInf₂`. -/
class MeasurableInf (M : Type*) [MeasurableSpace M] [Inf M] : Prop where
  measurable_const_inf : ∀ c : M, Measurable (c ⊓ ·)
  measurable_inf_const : ∀ c : M, Measurable (· ⊓ c)
goal: Prop

full_name: MeasurableInf₂.measurable_inf
code: /-- We say that a type has `MeasurableInf₂` if `uncurry (· ⊓ ·)` is a measurable functions.
For a typeclass assuming measurability of `(c ⊓ ·)` and `(· ⊓ c)` see `MeasurableInf`. -/
class MeasurableInf₂ (M : Type*) [MeasurableSpace M] [Inf M] : Prop where
  measurable_inf : Measurable fun p : M × M => p.1 ⊓ p.2
goal: Measurable fun p => p.1 ⊓ p.2

full_name: MeasurableInf₂
code: /-- We say that a type has `MeasurableInf₂` if `uncurry (· ⊓ ·)` is a measurable functions.
For a typeclass assuming measurability of `(c ⊓ ·)` and `(· ⊓ c)` see `MeasurableInf`. -/
class MeasurableInf₂ (M : Type*) [MeasurableSpace M] [Inf M] : Prop where
  measurable_inf : Measurable fun p : M × M => p.1 ⊓ p.2
goal: Prop

full_name: OrderDual.instMeasurableSup
code: instance (priority := 100) OrderDual.instMeasurableSup [Inf M] [MeasurableInf M] :
    MeasurableSup Mᵒᵈ 
goal: MeasurableSup Mᵒᵈ

full_name: OrderDual.instMeasurableInf
code: instance (priority := 100) OrderDual.instMeasurableInf [Sup M] [MeasurableSup M] :
    MeasurableInf Mᵒᵈ 
goal: MeasurableInf Mᵒᵈ

full_name: OrderDual.instMeasurableSup₂
code: instance (priority := 100) OrderDual.instMeasurableSup₂ [Inf M] [MeasurableInf₂ M] :
    MeasurableSup₂ Mᵒᵈ 
goal: MeasurableSup₂ Mᵒᵈ

full_name: OrderDual.instMeasurableInf₂
code: instance (priority := 100) OrderDual.instMeasurableInf₂ [Sup M] [MeasurableSup₂ M] :
    MeasurableInf₂ Mᵒᵈ 
goal: MeasurableInf₂ Mᵒᵈ

full_name: Measurable.const_sup
code: @[measurability]
theorem Measurable.const_sup (hf : Measurable f) (c : M) : Measurable fun x => c ⊔ f x 
goal: Measurable fun x => c ⊔ f x

full_name: AEMeasurable.const_sup
code: @[measurability]
theorem AEMeasurable.const_sup (hf : AEMeasurable f μ) (c : M) :
    AEMeasurable (fun x => c ⊔ f x) μ 
goal: AEMeasurable (fun x => c ⊔ f x) μ

full_name: Measurable.sup_const
code: @[measurability]
theorem Measurable.sup_const (hf : Measurable f) (c : M) : Measurable fun x => f x ⊔ c 
goal: Measurable fun x => f x ⊔ c

full_name: AEMeasurable.sup_const
code: @[measurability]
theorem AEMeasurable.sup_const (hf : AEMeasurable f μ) (c : M) :
    AEMeasurable (fun x => f x ⊔ c) μ 
goal: AEMeasurable (fun x => f x ⊔ c) μ

full_name: Measurable.sup'
code: @[measurability]
theorem Measurable.sup' (hf : Measurable f) (hg : Measurable g) : Measurable (f ⊔ g) 
goal: Measurable (f ⊔ g)

full_name: Measurable.sup
code: @[measurability]
theorem Measurable.sup (hf : Measurable f) (hg : Measurable g) : Measurable fun a => f a ⊔ g a 
goal: Measurable fun a => f a ⊔ g a

full_name: AEMeasurable.sup'
code: @[measurability]
theorem AEMeasurable.sup' (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (f ⊔ g) μ 
goal: AEMeasurable (f ⊔ g) μ

full_name: AEMeasurable.sup
code: @[measurability]
theorem AEMeasurable.sup (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (fun a => f a ⊔ g a) μ 
goal: AEMeasurable (fun a => f a ⊔ g a) μ

full_name: MeasurableSup₂.toMeasurableSup
code: instance (priority := 100) MeasurableSup₂.toMeasurableSup : MeasurableSup M 
goal: MeasurableSup M

full_name: Measurable.const_inf
code: @[measurability]
theorem Measurable.const_inf (hf : Measurable f) (c : M) : Measurable fun x => c ⊓ f x 
goal: Measurable fun x => c ⊓ f x

full_name: AEMeasurable.const_inf
code: @[measurability]
theorem AEMeasurable.const_inf (hf : AEMeasurable f μ) (c : M) :
    AEMeasurable (fun x => c ⊓ f x) μ 
goal: AEMeasurable (fun x => c ⊓ f x) μ

full_name: Measurable.inf_const
code: @[measurability]
theorem Measurable.inf_const (hf : Measurable f) (c : M) : Measurable fun x => f x ⊓ c 
goal: Measurable fun x => f x ⊓ c

full_name: AEMeasurable.inf_const
code: @[measurability]
theorem AEMeasurable.inf_const (hf : AEMeasurable f μ) (c : M) :
    AEMeasurable (fun x => f x ⊓ c) μ 
goal: AEMeasurable (fun x => f x ⊓ c) μ

full_name: Measurable.inf'
code: @[measurability]
theorem Measurable.inf' (hf : Measurable f) (hg : Measurable g) : Measurable (f ⊓ g) 
goal: Measurable (f ⊓ g)

full_name: Measurable.inf
code: @[measurability]
theorem Measurable.inf (hf : Measurable f) (hg : Measurable g) : Measurable fun a => f a ⊓ g a 
goal: Measurable fun a => f a ⊓ g a

full_name: AEMeasurable.inf'
code: @[measurability]
theorem AEMeasurable.inf' (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (f ⊓ g) μ 
goal: AEMeasurable (f ⊓ g) μ

full_name: AEMeasurable.inf
code: @[measurability]
theorem AEMeasurable.inf (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (fun a => f a ⊓ g a) μ 
goal: AEMeasurable (fun a => f a ⊓ g a) μ

full_name: MeasurableInf₂.to_hasMeasurableInf
code: instance (priority := 100) MeasurableInf₂.to_hasMeasurableInf : MeasurableInf M 
goal: MeasurableInf M

full_name: Finset.measurable_sup'
code: @[measurability]
theorem Finset.measurable_sup' {ι : Type*} {s : Finset ι} (hs : s.Nonempty) {f : ι → δ → α}
    (hf : ∀ n ∈ s, Measurable (f n)) : Measurable (s.sup' hs f) 
goal: Measurable (s.sup' hs f)

full_name: Finset.measurable_range_sup'
code: @[measurability]
theorem Finset.measurable_range_sup' {f : ℕ → δ → α} {n : ℕ} (hf : ∀ k ≤ n, Measurable (f k)) :
    Measurable ((range (n + 1)).sup' nonempty_range_succ f) 
goal: Measurable ((Finset.range (n + 1)).sup' ⋯ f)

full_name: Finset.measurable_range_sup''
code: @[measurability]
theorem Finset.measurable_range_sup'' {f : ℕ → δ → α} {n : ℕ} (hf : ∀ k ≤ n, Measurable (f k)) :
    Measurable fun x => (range (n + 1)).sup' nonempty_range_succ fun k => f k x 
goal: Measurable fun x => (Finset.range (n + 1)).sup' ⋯ fun k => f k x