ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.condexp_add
theorem condexp_add (hf : Integrable f μ) (hg : Integrable g μ) :
    μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m] 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → F'} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: μ[f + g|m] =ᵐ[μ] μ[f|m] + μ[g|m]
-- MeasureTheory.condexp_finset_sum
theorem condexp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → F'}
    (hf : ∀ i ∈ s, Integrable (f i) μ) : μ[∑ i ∈ s, f i|m] =ᵐ[μ] ∑ i ∈ s, μ[f i|m] 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_5} {s : Finset ι} {f : ι → α → F'} (hf : ∀ i ∈ s, MeasureTheory.Integrable (f i) μ)
Goal: μ[∑ i ∈ s, f i|m] =ᵐ[μ] ∑ i ∈ s, μ[f i|m]
-- MeasureTheory.condexp_smul
theorem condexp_smul (c : 𝕜) (f : α → F') : μ[c • f|m] =ᵐ[μ] c • μ[f|m] 
Arguments: {α : Type u_1} {F' : Type u_3} {𝕜 : Type u_4} [RCLike 𝕜] [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (c : 𝕜) (f : α → F')
Goal: μ[c • f|m] =ᵐ[μ] c • μ[f|m]
-- MeasureTheory.condexp_neg
theorem condexp_neg (f : α → F') : μ[-f|m] =ᵐ[μ] -μ[f|m] 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → F')
Goal: μ[-f|m] =ᵐ[μ] -μ[f|m]
-- MeasureTheory.condexp_sub
theorem condexp_sub (hf : Integrable f μ) (hg : Integrable g μ) :
    μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m] 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → F'} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: μ[f - g|m] =ᵐ[μ] μ[f|m] - μ[g|m]
-- MeasureTheory.condexp_condexp_of_le
theorem condexp_condexp_of_le {m₁ m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm₁₂ : m₁ ≤ m₂)
    (hm₂ : m₂ ≤ m0) [SigmaFinite (μ.trim hm₂)] : μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁] 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {f : α → F'} {m₁ m₂ m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm₁₂ : m₁ ≤ m₂) (hm₂ : m₂ ≤ m0) [MeasureTheory.SigmaFinite (μ.trim hm₂)]
Goal: μ[μ[f|m₂]|m₁] =ᵐ[μ] μ[f|m₁]
-- MeasureTheory.condexp_mono
theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
    [OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
    μ[f|m] ≤ᵐ[μ] μ[g|m] 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] [OrderedSMul ℝ E] {f g : α → E} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ) (hfg : f ≤ᵐ[μ] g)
Goal: μ[f|m] ≤ᵐ[μ] μ[g|m]
-- MeasureTheory.condexp_nonneg
theorem condexp_nonneg {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
    [OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m] 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] [OrderedSMul ℝ E] {f : α → E} (hf : 0 ≤ᵐ[μ] f)
Goal: 0 ≤ᵐ[μ] μ[f|m]
-- MeasureTheory.condexp_nonpos
theorem condexp_nonpos {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
    [OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_5} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] [OrderedSMul ℝ E] {f : α → E} (hf : f ≤ᵐ[μ] 0)
Goal: μ[f|m] ≤ᵐ[μ] 0
-- MeasureTheory.tendsto_condexpL1_of_dominated_convergence
theorem tendsto_condexpL1_of_dominated_convergence (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
    {fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ)
    (hfs_meas : ∀ n, AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : Integrable bound_fs μ)
    (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
    (hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) :
    Tendsto (fun n => condexpL1 hm μ (fs n)) atTop (𝓝 (condexpL1 hm μ f)) 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) [MeasureTheory.SigmaFinite (μ.trim hm)] {fs : ℕ → α → F'} {f : α → F'} (bound_fs : α → ℝ) (hfs_meas : ∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (fs n) μ) (h_int_bound_fs : MeasureTheory.Integrable bound_fs μ) (hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x) (hfs : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => fs n x) Filter.atTop (𝓝 (f x)))
Goal: Filter.Tendsto (fun n => MeasureTheory.condexpL1 hm μ (fs n)) Filter.atTop (𝓝 (MeasureTheory.condexpL1 hm μ f))
-- MeasureTheory.tendsto_condexp_unique
theorem tendsto_condexp_unique (fs gs : ℕ → α → F') (f g : α → F')
    (hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
    (hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
    (hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
    (h_int_bound_fs : Integrable bound_fs μ) (bound_gs : α → ℝ)
    (h_int_bound_gs : Integrable bound_gs μ) (hfs_bound : ∀ n, ∀ᵐ x ∂μ, ‖fs n x‖ ≤ bound_fs x)
    (hgs_bound : ∀ n, ∀ᵐ x ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ n, μ[fs n|m] =ᵐ[μ] μ[gs n|m]) :
    μ[f|m] =ᵐ[μ] μ[g|m] 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (fs gs : ℕ → α → F') (f g : α → F') (hfs_int : ∀ (n : ℕ), MeasureTheory.Integrable (fs n) μ) (hgs_int : ∀ (n : ℕ), MeasureTheory.Integrable (gs n) μ) (hfs : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => fs n x) Filter.atTop (𝓝 (f x))) (hgs : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => gs n x) Filter.atTop (𝓝 (g x))) (bound_fs : α → ℝ) (h_int_bound_fs : MeasureTheory.Integrable bound_fs μ) (bound_gs : α → ℝ) (h_int_bound_gs : MeasureTheory.Integrable bound_gs μ) (hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x) (hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x) (hfg : ∀ (n : ℕ), μ[fs n|m] =ᵐ[μ] μ[gs n|m])
Goal: μ[f|m] =ᵐ[μ] μ[g|m]
-- MeasurableSpace.measurableSet_iUnion
@[class] structure MeasurableSpace (α : Type*) where
    MeasurableSet' : Set α → Prop
    measurableSet_empty : MeasurableSet' ∅
    measurableSet_compl : ∀ s, MeasurableSet' s → MeasurableSet' sᶜ
    measurableSet_iUnion : ∀ f : ℕ → Set α, (∀ i, MeasurableSet' (f i)) → MeasurableSet' (⋃ i, f i)
Arguments: {α : Type u_7} (self : MeasurableSpace α) (f : ℕ → Set α)
Goal: (∀ (i : ℕ), self.MeasurableSet' (f i)) → self.MeasurableSet' (⋃ i, f i)
-- MeasurableSpace.measurableSet_compl
@[class] structure MeasurableSpace (α : Type*) where
    MeasurableSet' : Set α → Prop
    measurableSet_empty : MeasurableSet' ∅
    measurableSet_compl : ∀ s, MeasurableSet' s → MeasurableSet' sᶜ
    measurableSet_iUnion : ∀ f : ℕ → Set α, (∀ i, MeasurableSet' (f i)) → MeasurableSet' (⋃ i, f i)
Arguments: {α : Type u_7} (self : MeasurableSpace α) (s : Set α)
Goal: self.MeasurableSet' s → self.MeasurableSet' sᶜ
-- MeasurableSpace.measurableSet_empty
@[class] structure MeasurableSpace (α : Type*) where
    MeasurableSet' : Set α → Prop
    measurableSet_empty : MeasurableSet' ∅
    measurableSet_compl : ∀ s, MeasurableSet' s → MeasurableSet' sᶜ
    measurableSet_iUnion : ∀ f : ℕ → Set α, (∀ i, MeasurableSet' (f i)) → MeasurableSet' (⋃ i, f i)
Arguments: {α : Type u_7} (self : MeasurableSpace α)
Goal: self.MeasurableSet' ∅
-- MeasurableSet.empty
@[simp, measurability]
theorem MeasurableSet.empty [MeasurableSpace α] : MeasurableSet (∅ : Set α) 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: MeasurableSet ∅
-- MeasurableSet.compl
@[measurability]
protected theorem MeasurableSet.compl : MeasurableSet s → MeasurableSet sᶜ 
Arguments: {α : Type u_1} {s : Set α} {m : MeasurableSpace α}
Goal: MeasurableSet s → MeasurableSet sᶜ
-- MeasurableSet.of_compl
protected theorem MeasurableSet.of_compl (h : MeasurableSet sᶜ) : MeasurableSet s 
Arguments: {α : Type u_1} {s : Set α} {m : MeasurableSpace α} (h : MeasurableSet sᶜ)
Goal: MeasurableSet s
-- MeasurableSet.compl_iff
@[simp]
theorem MeasurableSet.compl_iff : MeasurableSet sᶜ ↔ MeasurableSet s 
Arguments: {α : Type u_1} {s : Set α} {m : MeasurableSpace α}
Goal: MeasurableSet sᶜ ↔ MeasurableSet s
-- MeasurableSet.univ
@[simp, measurability]
protected theorem MeasurableSet.univ : MeasurableSet (univ : Set α) 
Arguments: {α : Type u_1} {m : MeasurableSpace α}
Goal: MeasurableSet Set.univ
-- Subsingleton.measurableSet
@[nontriviality, measurability]
theorem Subsingleton.measurableSet [Subsingleton α] {s : Set α} : MeasurableSet s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} [Subsingleton α] {s : Set α}
Goal: MeasurableSet s
-- MeasurableSet.congr
theorem MeasurableSet.congr {s t : Set α} (hs : MeasurableSet s) (h : s = t) : MeasurableSet t 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s t : Set α} (hs : MeasurableSet s) (h : s = t)
Goal: MeasurableSet t
-- MeasurableSet.iUnion
@[measurability]
protected theorem MeasurableSet.iUnion [Countable ι] ⦃f : ι → Set α⦄
    (h : ∀ b, MeasurableSet (f b)) : MeasurableSet (⋃ b, f b) 
Arguments: {α : Type u_1} {ι : Sort u_6} {m : MeasurableSpace α} [Countable ι] ⦃f : ι → Set α⦄ (h : ∀ (b : ι), MeasurableSet (f b))
Goal: MeasurableSet (⋃ b, f b)
-- MeasurableSet.biUnion
protected theorem MeasurableSet.biUnion {f : β → Set α} {s : Set β} (hs : s.Countable)
    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋃ b ∈ s, f b) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {f : β → Set α} {s : Set β} (hs : s.Countable) (h : ∀ b ∈ s, MeasurableSet (f b))
Goal: MeasurableSet (⋃ b ∈ s, f b)
-- Set.Finite.measurableSet_biUnion
theorem Set.Finite.measurableSet_biUnion {f : β → Set α} {s : Set β} (hs : s.Finite)
    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋃ b ∈ s, f b) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {f : β → Set α} {s : Set β} (hs : s.Finite) (h : ∀ b ∈ s, MeasurableSet (f b))
Goal: MeasurableSet (⋃ b ∈ s, f b)
-- Finset.measurableSet_biUnion
theorem Finset.measurableSet_biUnion {f : β → Set α} (s : Finset β)
    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋃ b ∈ s, f b) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {f : β → Set α} (s : Finset β) (h : ∀ b ∈ s, MeasurableSet (f b))
Goal: MeasurableSet (⋃ b ∈ s, f b)
-- MeasurableSet.sUnion
protected theorem MeasurableSet.sUnion {s : Set (Set α)} (hs : s.Countable)
    (h : ∀ t ∈ s, MeasurableSet t) : MeasurableSet (⋃₀ s) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s : Set (Set α)} (hs : s.Countable) (h : ∀ t ∈ s, MeasurableSet t)
Goal: MeasurableSet (⋃₀ s)
-- Set.Finite.measurableSet_sUnion
theorem Set.Finite.measurableSet_sUnion {s : Set (Set α)} (hs : s.Finite)
    (h : ∀ t ∈ s, MeasurableSet t) : MeasurableSet (⋃₀ s) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s : Set (Set α)} (hs : s.Finite) (h : ∀ t ∈ s, MeasurableSet t)
Goal: MeasurableSet (⋃₀ s)
-- MeasurableSet.iInter
@[measurability]
theorem MeasurableSet.iInter [Countable ι] {f : ι → Set α} (h : ∀ b, MeasurableSet (f b)) :
    MeasurableSet (⋂ b, f b) 
Arguments: {α : Type u_1} {ι : Sort u_6} {m : MeasurableSpace α} [Countable ι] {f : ι → Set α} (h : ∀ (b : ι), MeasurableSet (f b))
Goal: MeasurableSet (⋂ b, f b)
-- MeasurableSet.biInter
theorem MeasurableSet.biInter {f : β → Set α} {s : Set β} (hs : s.Countable)
    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋂ b ∈ s, f b) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {f : β → Set α} {s : Set β} (hs : s.Countable) (h : ∀ b ∈ s, MeasurableSet (f b))
Goal: MeasurableSet (⋂ b ∈ s, f b)
-- Set.Finite.measurableSet_biInter
theorem Set.Finite.measurableSet_biInter {f : β → Set α} {s : Set β} (hs : s.Finite)
    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋂ b ∈ s, f b) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {f : β → Set α} {s : Set β} (hs : s.Finite) (h : ∀ b ∈ s, MeasurableSet (f b))
Goal: MeasurableSet (⋂ b ∈ s, f b)
-- Finset.measurableSet_biInter
theorem Finset.measurableSet_biInter {f : β → Set α} (s : Finset β)
    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋂ b ∈ s, f b) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {f : β → Set α} (s : Finset β) (h : ∀ b ∈ s, MeasurableSet (f b))
Goal: MeasurableSet (⋂ b ∈ s, f b)
-- MeasurableSet.sInter
theorem MeasurableSet.sInter {s : Set (Set α)} (hs : s.Countable) (h : ∀ t ∈ s, MeasurableSet t) :
    MeasurableSet (⋂₀ s) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s : Set (Set α)} (hs : s.Countable) (h : ∀ t ∈ s, MeasurableSet t)
Goal: MeasurableSet (⋂₀ s)
-- Set.Finite.measurableSet_sInter
theorem Set.Finite.measurableSet_sInter {s : Set (Set α)} (hs : s.Finite)
    (h : ∀ t ∈ s, MeasurableSet t) : MeasurableSet (⋂₀ s) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s : Set (Set α)} (hs : s.Finite) (h : ∀ t ∈ s, MeasurableSet t)
Goal: MeasurableSet (⋂₀ s)
-- MeasurableSet.union
@[simp, measurability]
protected theorem MeasurableSet.union {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁)
    (h₂ : MeasurableSet s₂) : MeasurableSet (s₁ ∪ s₂) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁) (h₂ : MeasurableSet s₂)
Goal: MeasurableSet (s₁ ∪ s₂)
-- MeasurableSet.inter
@[simp, measurability]
protected theorem MeasurableSet.inter {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁)
    (h₂ : MeasurableSet s₂) : MeasurableSet (s₁ ∩ s₂) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁) (h₂ : MeasurableSet s₂)
Goal: MeasurableSet (s₁ ∩ s₂)
-- MeasurableSet.diff
@[simp, measurability]
protected theorem MeasurableSet.diff {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁)
    (h₂ : MeasurableSet s₂) : MeasurableSet (s₁ \ s₂) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁) (h₂ : MeasurableSet s₂)
Goal: MeasurableSet (s₁ \ s₂)
-- MeasurableSet.himp
@[simp, measurability]
protected lemma MeasurableSet.himp {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁) (h₂ : MeasurableSet s₂) :
    MeasurableSet (s₁ ⇨ s₂) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁) (h₂ : MeasurableSet s₂)
Goal: MeasurableSet (s₁ ⇨ s₂)
-- MeasurableSet.symmDiff
@[simp, measurability]
protected theorem MeasurableSet.symmDiff {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁)
    (h₂ : MeasurableSet s₂) : MeasurableSet (s₁ ∆ s₂) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁) (h₂ : MeasurableSet s₂)
Goal: MeasurableSet (symmDiff s₁ s₂)
-- MeasurableSet.bihimp
@[simp, measurability]
protected lemma MeasurableSet.bihimp {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁)
    (h₂ : MeasurableSet s₂) : MeasurableSet (s₁ ⇔ s₂) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁) (h₂ : MeasurableSet s₂)
Goal: MeasurableSet (bihimp s₁ s₂)
-- MeasurableSet.ite
@[simp, measurability]
protected theorem MeasurableSet.ite {t s₁ s₂ : Set α} (ht : MeasurableSet t)
    (h₁ : MeasurableSet s₁) (h₂ : MeasurableSet s₂) : MeasurableSet (t.ite s₁ s₂) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {t s₁ s₂ : Set α} (ht : MeasurableSet t) (h₁ : MeasurableSet s₁) (h₂ : MeasurableSet s₂)
Goal: MeasurableSet (t.ite s₁ s₂)
-- MeasurableSet.ite'
theorem MeasurableSet.ite' {s t : Set α} {p : Prop} (hs : p → MeasurableSet s)
    (ht : ¬p → MeasurableSet t) : MeasurableSet (ite p s t) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s t : Set α} {p : Prop} (hs : p → MeasurableSet s) (ht : ¬p → MeasurableSet t)
Goal: MeasurableSet (if p then s else t)
-- MeasurableSet.cond
@[simp, measurability]
protected theorem MeasurableSet.cond {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁)
    (h₂ : MeasurableSet s₂) {i : Bool} : MeasurableSet (cond i s₁ s₂) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁) (h₂ : MeasurableSet s₂) {i : Bool}
Goal: MeasurableSet (bif i then s₁ else s₂)
-- MeasurableSet.disjointed
@[simp, measurability]
protected theorem MeasurableSet.disjointed {f : ℕ → Set α} (h : ∀ i, MeasurableSet (f i)) (n) :
    MeasurableSet (disjointed f n) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : ℕ → Set α} (h : ∀ (i : ℕ), MeasurableSet (f i)) (n : ℕ)
Goal: MeasurableSet (disjointed f n)
-- MeasurableSet.const
protected theorem MeasurableSet.const (p : Prop) : MeasurableSet { _a : α | p } 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (p : Prop)
Goal: MeasurableSet {_a | p}
-- nonempty_measurable_superset
theorem nonempty_measurable_superset (s : Set α) : Nonempty { t // s ⊆ t ∧ MeasurableSet t } 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s : Set α)
Goal: Nonempty { t // s ⊆ t ∧ MeasurableSet t }
-- MeasurableSpace.measurableSet_injective
theorem MeasurableSpace.measurableSet_injective : Injective (@MeasurableSet α)
  
Arguments: {α : Type u_1}
Goal: Function.Injective (@MeasurableSet α)
-- MeasurableSpace.ext
@[ext]
theorem MeasurableSpace.ext {m₁ m₂ : MeasurableSpace α}
    (h : ∀ s : Set α, MeasurableSet[m₁] s ↔ MeasurableSet[m₂] s) : m₁ = m₂ 
Arguments: {α : Type u_1} {m₁ m₂ : MeasurableSpace α} (h : ∀ (s : Set α), MeasurableSet s ↔ MeasurableSet s)
Goal: m₁ = m₂
-- MeasurableSpace.ext_iff
theorem MeasurableSpace.ext_iff {m₁ m₂ : MeasurableSpace α} :
    m₁ = m₂ ↔ ∀ s : Set α, MeasurableSet[m₁] s ↔ MeasurableSet[m₂] s 
Arguments: {α : Type u_1} {m₁ m₂ : MeasurableSpace α}
Goal: m₁ = m₂ ↔ ∀ (s : Set α), MeasurableSet s ↔ MeasurableSet s
-- MeasurableSingletonClass.measurableSet_singleton
class MeasurableSingletonClass (α : Type*) [MeasurableSpace α] : Prop where
    measurableSet_singleton : ∀ x, MeasurableSet ({x} : Set α)
Arguments: {α : Type u_7} [MeasurableSpace α] [self : MeasurableSingletonClass α] (x : α)
Goal: MeasurableSet {x}
-- MeasurableSet.singleton
@[simp]
lemma MeasurableSet.singleton [MeasurableSpace α] [MeasurableSingletonClass α] (a : α) :
    MeasurableSet {a} 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (a : α)
Goal: MeasurableSet {a}
-- measurableSet_eq
@[measurability]
theorem measurableSet_eq {a : α} : MeasurableSet { x | x = a } 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] {a : α}
Goal: MeasurableSet {x | x = a}
-- MeasurableSet.insert
@[measurability]
protected theorem MeasurableSet.insert {s : Set α} (hs : MeasurableSet s) (a : α) :
    MeasurableSet (insert a s) 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] {s : Set α} (hs : MeasurableSet s) (a : α)
Goal: MeasurableSet (insert a s)
-- measurableSet_insert
@[simp]
theorem measurableSet_insert {a : α} {s : Set α} : MeasurableSet (insert a s) ↔ MeasurableSet s 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] {a : α} {s : Set α}
Goal: MeasurableSet (insert a s) ↔ MeasurableSet s
-- Set.Subsingleton.measurableSet
theorem Set.Subsingleton.measurableSet {s : Set α} (hs : s.Subsingleton) : MeasurableSet s 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] {s : Set α} (hs : s.Subsingleton)
Goal: MeasurableSet s
-- Set.Finite.measurableSet
theorem Set.Finite.measurableSet {s : Set α} (hs : s.Finite) : MeasurableSet s 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] {s : Set α} (hs : s.Finite)
Goal: MeasurableSet s
-- Finset.measurableSet
@[measurability]
protected theorem Finset.measurableSet (s : Finset α) : MeasurableSet (↑s : Set α) 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (s : Finset α)
Goal: MeasurableSet ↑s
-- Set.Countable.measurableSet
theorem Set.Countable.measurableSet {s : Set α} (hs : s.Countable) : MeasurableSet s 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] {s : Set α} (hs : s.Countable)
Goal: MeasurableSet s
-- MeasurableSpace.measurableSet_copy
lemma measurableSet_copy {m : MeasurableSpace α} {p : Set α → Prop}
    (h : ∀ s, p s ↔ MeasurableSet[m] s) {s} : MeasurableSet[.copy m p h] s ↔ p s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {p : Set α → Prop} (h : ∀ (s : Set α), p s ↔ MeasurableSet s) {s : Set α}
Goal: MeasurableSet s ↔ p s
-- MeasurableSpace.copy_eq
lemma copy_eq {m : MeasurableSpace α} {p : Set α → Prop} (h : ∀ s, p s ↔ MeasurableSet[m] s) :
    m.copy p h = m 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {p : Set α → Prop} (h : ∀ (s : Set α), p s ↔ MeasurableSet s)
Goal: m.copy p h = m
-- MeasurableSpace.le_def
theorem le_def {α} {a b : MeasurableSpace α} : a ≤ b ↔ a.MeasurableSet' ≤ b.MeasurableSet' 
Arguments: {α : Type u_7} {a b : MeasurableSpace α}
Goal: a ≤ b ↔ a.MeasurableSet' ≤ b.MeasurableSet'
-- MeasurableSpace.measurableSet_generateFrom
theorem measurableSet_generateFrom {s : Set (Set α)} {t : Set α} (ht : t ∈ s) :
    MeasurableSet[generateFrom s] t 
Arguments: {α : Type u_1} {s : Set (Set α)} {t : Set α} (ht : t ∈ s)
Goal: MeasurableSet t
-- MeasurableSpace.generateFrom_induction
@[elab_as_elim]
theorem generateFrom_induction (p : Set α → Prop) (C : Set (Set α)) (hC : ∀ t ∈ C, p t)
    (h_empty : p ∅) (h_compl : ∀ t, p t → p tᶜ)
    (h_Union : ∀ f : ℕ → Set α, (∀ n, p (f n)) → p (⋃ i, f i)) {s : Set α}
    (hs : MeasurableSet[generateFrom C] s) : p s 
Arguments: {α : Type u_1} (p : Set α → Prop) (C : Set (Set α)) (hC : ∀ t ∈ C, p t) (h_empty : p ∅) (h_compl : ∀ (t : Set α), p t → p tᶜ) (h_Union : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n)) → p (⋃ i, f i)) {s : Set α} (hs : MeasurableSet s)
Goal: p s
-- MeasurableSpace.generateFrom_le
theorem generateFrom_le {s : Set (Set α)} {m : MeasurableSpace α}
    (h : ∀ t ∈ s, MeasurableSet[m] t) : generateFrom s ≤ m 
Arguments: {α : Type u_1} {s : Set (Set α)} {m : MeasurableSpace α} (h : ∀ t ∈ s, MeasurableSet t)
Goal: MeasurableSpace.generateFrom s ≤ m
-- MeasurableSpace.generateFrom_le_iff
theorem generateFrom_le_iff {s : Set (Set α)} (m : MeasurableSpace α) :
    generateFrom s ≤ m ↔ s ⊆ { t | MeasurableSet[m] t } 
Arguments: {α : Type u_1} {s : Set (Set α)} (m : MeasurableSpace α)
Goal: MeasurableSpace.generateFrom s ≤ m ↔ s ⊆ {t | MeasurableSet t}
-- MeasurableSpace.generateFrom_measurableSet
@[simp]
theorem generateFrom_measurableSet [MeasurableSpace α] :
    generateFrom { s : Set α | MeasurableSet s } = ‹_› 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: MeasurableSpace.generateFrom {s | MeasurableSet s} = inst✝
-- MeasurableSpace.forall_generateFrom_mem_iff_mem_iff
theorem forall_generateFrom_mem_iff_mem_iff {S : Set (Set α)} {x y : α} :
    (∀ s, MeasurableSet[generateFrom S] s → (x ∈ s ↔ y ∈ s)) ↔ (∀ s ∈ S, x ∈ s ↔ y ∈ s) 
Arguments: {α : Type u_1} {S : Set (Set α)} {x y : α}
Goal: (∀ (s : Set α), MeasurableSet s → (x ∈ s ↔ y ∈ s)) ↔ ∀ s ∈ S, x ∈ s ↔ y ∈ s
-- MeasurableSpace.mkOfClosure_sets
theorem mkOfClosure_sets {s : Set (Set α)} {hs : { t | MeasurableSet[generateFrom s] t } = s} :
    MeasurableSpace.mkOfClosure s hs = generateFrom s 
Arguments: {α : Type u_1} {s : Set (Set α)} {hs : {t | MeasurableSet t} = s}
Goal: MeasurableSpace.mkOfClosure s hs = MeasurableSpace.generateFrom s
-- MeasurableSpace.generateFrom_mono
@[mono]
theorem generateFrom_mono {s t : Set (Set α)} (h : s ⊆ t) : generateFrom s ≤ generateFrom t 
Arguments: {α : Type u_1} {s t : Set (Set α)} (h : s ⊆ t)
Goal: MeasurableSpace.generateFrom s ≤ MeasurableSpace.generateFrom t
-- MeasurableSpace.generateFrom_sup_generateFrom
theorem generateFrom_sup_generateFrom {s t : Set (Set α)} :
    generateFrom s ⊔ generateFrom t = generateFrom (s ∪ t) 
Arguments: {α : Type u_1} {s t : Set (Set α)}
Goal: MeasurableSpace.generateFrom s ⊔ MeasurableSpace.generateFrom t = MeasurableSpace.generateFrom (s ∪ t)
-- MeasurableSpace.iSup_generateFrom
lemma iSup_generateFrom (s : ι → Set (Set α)) :
    ⨆ i, generateFrom (s i) = generateFrom (⋃ i, s i) 
Arguments: {α : Type u_1} {ι : Sort u_6} (s : ι → Set (Set α))
Goal: ⨆ i, MeasurableSpace.generateFrom (s i) = MeasurableSpace.generateFrom (⋃ i, s i)