ADEInequality.A stringlengths 2 203 | Mathlib.NumberTheory.ADEInequality stringlengths 6 90 | ℕ+ → Multiset ℕ+ stringlengths 1 26.8k |
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ADEInequality.A' | Mathlib.NumberTheory.ADEInequality | ℕ+ → ℕ+ → Multiset ℕ+ |
ADEInequality.A'.eq_1 | Mathlib.NumberTheory.ADEInequality | ∀ (q r : ℕ+), ADEInequality.A' q r = {1, q, r} |
ADEInequality.Admissible | Mathlib.NumberTheory.ADEInequality | Multiset ℕ+ → Prop |
ADEInequality.Admissible.eq_1 | Mathlib.NumberTheory.ADEInequality | ∀ (pqr : Multiset ℕ+), ADEInequality.Admissible pqr = ((∃ q r, ADEInequality.A' q r = pqr) ∨ (∃ r, ADEInequality.D' r = pqr) ∨ ADEInequality.E' 3 = pqr ∨ ADEInequality.E' 4 = pqr ∨ ADEInequality.E' 5 = pqr) |
ADEInequality.Admissible.one_lt_sumInv | Mathlib.NumberTheory.ADEInequality | ∀ {pqr : Multiset ℕ+}, ADEInequality.Admissible pqr → 1 < ADEInequality.sumInv pqr |
ADEInequality.D' | Mathlib.NumberTheory.ADEInequality | ℕ+ → Multiset ℕ+ |
ADEInequality.D'.eq_1 | Mathlib.NumberTheory.ADEInequality | ∀ (r : ℕ+), ADEInequality.D' r = {2, 2, r} |
ADEInequality.E' | Mathlib.NumberTheory.ADEInequality | ℕ+ → Multiset ℕ+ |
ADEInequality.E'.eq_1 | Mathlib.NumberTheory.ADEInequality | ∀ (r : ℕ+), ADEInequality.E' r = {2, 3, r} |
ADEInequality.E6 | Mathlib.NumberTheory.ADEInequality | Multiset ℕ+ |
ADEInequality.E7 | Mathlib.NumberTheory.ADEInequality | Multiset ℕ+ |
ADEInequality.E8 | Mathlib.NumberTheory.ADEInequality | Multiset ℕ+ |
ADEInequality.admissible_A' | Mathlib.NumberTheory.ADEInequality | ∀ (q r : ℕ+), ADEInequality.Admissible (ADEInequality.A' q r) |
ADEInequality.admissible_D' | Mathlib.NumberTheory.ADEInequality | ∀ (n : ℕ+), ADEInequality.Admissible (ADEInequality.D' n) |
ADEInequality.admissible_E'3 | Mathlib.NumberTheory.ADEInequality | ADEInequality.Admissible (ADEInequality.E' 3) |
ADEInequality.admissible_E'4 | Mathlib.NumberTheory.ADEInequality | ADEInequality.Admissible (ADEInequality.E' 4) |
ADEInequality.admissible_E'5 | Mathlib.NumberTheory.ADEInequality | ADEInequality.Admissible (ADEInequality.E' 5) |
ADEInequality.admissible_E6 | Mathlib.NumberTheory.ADEInequality | ADEInequality.Admissible ADEInequality.E6 |
ADEInequality.admissible_E7 | Mathlib.NumberTheory.ADEInequality | ADEInequality.Admissible ADEInequality.E7 |
ADEInequality.admissible_E8 | Mathlib.NumberTheory.ADEInequality | ADEInequality.Admissible ADEInequality.E8 |
ADEInequality.admissible_of_one_lt_sumInv | Mathlib.NumberTheory.ADEInequality | ∀ {p q r : ℕ+}, 1 < ADEInequality.sumInv {p, q, r} → ADEInequality.Admissible {p, q, r} |
ADEInequality.admissible_of_one_lt_sumInv_aux | Mathlib.NumberTheory.ADEInequality | ∀ {pqr : List ℕ+}, pqr.SortedLE → pqr.length = 3 → 1 < ADEInequality.sumInv ↑pqr → ADEInequality.Admissible ↑pqr |
ADEInequality.admissible_of_one_lt_sumInv_aux' | Mathlib.NumberTheory.ADEInequality | ∀ {p q r : ℕ+}, p ≤ q → q ≤ r → 1 < ADEInequality.sumInv {p, q, r} → ADEInequality.Admissible {p, q, r} |
ADEInequality.classification | Mathlib.NumberTheory.ADEInequality | ∀ (p q r : ℕ+), 1 < ADEInequality.sumInv {p, q, r} ↔ ADEInequality.Admissible {p, q, r} |
ADEInequality.lt_four | Mathlib.NumberTheory.ADEInequality | ∀ {q r : ℕ+}, q ≤ r → 1 < ADEInequality.sumInv {2, q, r} → q < 4 |
ADEInequality.lt_six | Mathlib.NumberTheory.ADEInequality | ∀ {r : ℕ+}, 1 < ADEInequality.sumInv {2, 3, r} → r < 6 |
ADEInequality.lt_three | Mathlib.NumberTheory.ADEInequality | ∀ {p q r : ℕ+}, p ≤ q → q ≤ r → 1 < ADEInequality.sumInv {p, q, r} → p < 3 |
ADEInequality.sumInv | Mathlib.NumberTheory.ADEInequality | Multiset ℕ+ → ℚ |
ADEInequality.sumInv.eq_1 | Mathlib.NumberTheory.ADEInequality | ∀ (pqr : Multiset ℕ+), ADEInequality.sumInv pqr = (Multiset.map (fun x => (↑↑x)⁻¹) pqr).sum |
ADEInequality.sumInv_pqr | Mathlib.NumberTheory.ADEInequality | ∀ (p q r : ℕ+), ADEInequality.sumInv {p, q, r} = (↑↑p)⁻¹ + (↑↑q)⁻¹ + (↑↑r)⁻¹ |
AEMeasurable | Mathlib.MeasureTheory.Measure.MeasureSpaceDef | {α : Type u_1} → {β : Type u_2} → [MeasurableSpace β] → {_m : MeasurableSpace α} → (α → β) → autoParam (MeasureTheory.Measure α) AEMeasurable._auto_1 → Prop |
AEMeasurable.abs | Mathlib.MeasureTheory.Order.Group.Lattice | ∀ {α : Type u_1} {β : Type u_2} [inst : Lattice α] [inst_1 : AddGroup α] [inst_2 : MeasurableSpace α] [inst_3 : MeasurableSpace β] {f : β → α} [MeasurableNeg α] [MeasurableSup₂ α] {μ : MeasureTheory.Measure β}, AEMeasurable f μ → AEMeasurable (fun x => |f x|) μ |
AEMeasurable.add | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {M : Type u_2} {α : Type u_3} [inst : MeasurableSpace M] [inst_1 : Add M] {m : MeasurableSpace α} {f g : α → M} {μ : MeasureTheory.Measure α} [MeasurableAdd₂ M], AEMeasurable f μ → AEMeasurable g μ → AEMeasurable (fun a => f a + g a) μ |
AEMeasurable.add' | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {M : Type u_2} {α : Type u_3} [inst : MeasurableSpace M] [inst_1 : Add M] {m : MeasurableSpace α} {f g : α → M} {μ : MeasureTheory.Measure α} [MeasurableAdd₂ M], AEMeasurable f μ → AEMeasurable g μ → AEMeasurable (f + g) μ |
AEMeasurable.add_const | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {M : Type u_2} {α : Type u_3} [inst : MeasurableSpace M] [inst_1 : Add M] {m : MeasurableSpace α} {f : α → M} {μ : MeasureTheory.Measure α} [MeasurableAdd M], AEMeasurable f μ → ∀ (c : M), AEMeasurable (fun x => f x + c) μ |
AEMeasurable.add_iff_left | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {α : Type u_1} {G : Type u_2} [inst : MeasurableSpace G] [inst_1 : MeasurableSpace α] [inst_2 : AddCommGroup G] [MeasurableAdd₂ G] [MeasurableNeg G] {μ : MeasureTheory.Measure α} {f g : α → G}, AEMeasurable f μ → (AEMeasurable (g + f) μ ↔ AEMeasurable g μ) |
AEMeasurable.add_iff_right | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {α : Type u_1} {G : Type u_2} [inst : MeasurableSpace G] [inst_1 : MeasurableSpace α] [inst_2 : AddCommGroup G] [MeasurableAdd₂ G] [MeasurableNeg G] {μ : MeasureTheory.Measure α} {f g : α → G}, AEMeasurable f μ → (AEMeasurable (f + g) μ ↔ AEMeasurable g μ) |
AEMeasurable.add_measure | Mathlib.MeasureTheory.Measure.AEMeasurable | ∀ {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {μ ν : MeasureTheory.Measure α} {f : α → β}, AEMeasurable f μ → AEMeasurable f ν → AEMeasurable f (μ + ν) |
AEMeasurable.ae_eq_mk | Mathlib.MeasureTheory.Measure.MeasureSpaceDef | ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [inst : MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} (h : AEMeasurable f μ), f =ᵐ[μ] AEMeasurable.mk f h |
AEMeasurable.ae_inf_principal_eq_mk | Mathlib.MeasureTheory.Measure.AEMeasurable | ∀ {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} {s : Set α} (h : AEMeasurable f (μ.restrict s)), f =ᶠ[MeasureTheory.ae μ ⊓ Filter.principal s] AEMeasurable.mk f h |
AEMeasurable.ae_mem_imp_eq_mk | Mathlib.MeasureTheory.Measure.AEMeasurable | ∀ {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} {s : Set α} (h : AEMeasurable f (μ.restrict s)), ∀ᵐ (x : α) ∂μ, x ∈ s → f x = AEMeasurable.mk f h x |
AEMeasurable.ae_of_bind | Mathlib.MeasureTheory.Measure.GiryMonad | ∀ {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {γ : Type u_3} {x : MeasurableSpace γ} {m : MeasureTheory.Measure α} {f : α → MeasureTheory.Measure β} {g : β → γ}, AEMeasurable f m → AEMeasurable g (m.bind f) → ∀ᵐ (a : α) ∂m, AEMeasurable g (f a) |
AEMeasurable.ae_of_join | Mathlib.MeasureTheory.Measure.GiryMonad | ∀ {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {m : MeasureTheory.Measure (MeasureTheory.Measure α)} {f : α → β}, AEMeasurable f m.join → ∀ᵐ (μ : MeasureTheory.Measure α) ∂m, AEMeasurable f μ |
AEMeasurable.aestronglyMeasurable | Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | ∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace β] {m₀ : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → β} [inst_1 : MeasurableSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [OpensMeasurableSpace β] [SecondCountableTopology β], AEMeasurable f μ → MeasureTheory.AEStronglyMeasurable f μ |
AEMeasurable.apply_continuousLinearMap | Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap | ∀ {α : Type u_1} [inst : MeasurableSpace α] {𝕜 : Type u_2} [inst_1 : NontriviallyNormedField 𝕜] {E : Type u_3} [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace 𝕜 E] [inst_4 : MeasurableSpace E] [BorelSpace E] {F : Type u_4} [inst_6 : NormedAddCommGroup F] [inst_7 : NormedSpace 𝕜 F] {φ : α → F →L[𝕜] E} {μ : Me... |
AEMeasurable.biInf | Mathlib.MeasureTheory.Constructions.BorelSpace.Order | ∀ {α : Type u_1} {δ : Type u_4} [inst : TopologicalSpace α] {mα : MeasurableSpace α} [BorelSpace α] {mδ : MeasurableSpace δ} [inst_2 : ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Type u_5} {μ : MeasureTheory.Measure δ} (s : Set ι) {f : ι → δ → α}, s.Countable → (∀ i ∈ s, AEMea... |
AEMeasurable.biSup | Mathlib.MeasureTheory.Constructions.BorelSpace.Order | ∀ {α : Type u_1} {δ : Type u_4} [inst : TopologicalSpace α] {mα : MeasurableSpace α} [BorelSpace α] {mδ : MeasurableSpace δ} [inst_2 : ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Type u_5} {μ : MeasureTheory.Measure δ} (s : Set ι) {f : ι → δ → α}, s.Countable → (∀ i ∈ s, AEMea... |
AEMeasurable.carg | Mathlib.MeasureTheory.Function.SpecialFunctions.Basic | ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℂ}, AEMeasurable f μ → AEMeasurable (fun x => (f x).arg) μ |
AEMeasurable.ccos | Mathlib.MeasureTheory.Function.SpecialFunctions.Basic | ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℂ}, AEMeasurable f μ → AEMeasurable (fun x => Complex.cos (f x)) μ |
AEMeasurable.ccosh | Mathlib.MeasureTheory.Function.SpecialFunctions.Basic | ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℂ}, AEMeasurable f μ → AEMeasurable (fun x => Complex.cosh (f x)) μ |
AEMeasurable.cexp | Mathlib.MeasureTheory.Function.SpecialFunctions.Basic | ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℂ}, AEMeasurable f μ → AEMeasurable (fun x => Complex.exp (f x)) μ |
AEMeasurable.clog | Mathlib.MeasureTheory.Function.SpecialFunctions.Basic | ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℂ}, AEMeasurable f μ → AEMeasurable (fun x => Complex.log (f x)) μ |
AEMeasurable.coe_ereal_ennreal | Mathlib.MeasureTheory.Constructions.BorelSpace.Real | ∀ {α : Type u_1} {mα : MeasurableSpace α} {f : α → ENNReal} {μ : MeasureTheory.Measure α}, AEMeasurable f μ → AEMeasurable (fun x => ↑(f x)) μ |
AEMeasurable.coe_nnreal_ennreal | Mathlib.MeasureTheory.Constructions.BorelSpace.Real | ∀ {α : Type u_1} {mα : MeasurableSpace α} {f : α → NNReal} {μ : MeasureTheory.Measure α}, AEMeasurable f μ → AEMeasurable (fun x => ↑(f x)) μ |
AEMeasurable.coe_nnreal_real | Mathlib.MeasureTheory.Constructions.BorelSpace.Real | ∀ {α : Type u_1} {mα : MeasurableSpace α} {f : α → NNReal} {μ : MeasureTheory.Measure α}, AEMeasurable f μ → AEMeasurable (fun x => ↑(f x)) μ |
AEMeasurable.coe_real_ereal | Mathlib.MeasureTheory.Constructions.BorelSpace.Real | ∀ {α : Type u_1} {mα : MeasurableSpace α} {f : α → ℝ} {μ : MeasureTheory.Measure α}, AEMeasurable f μ → AEMeasurable (fun x => ↑(f x)) μ |
AEMeasurable.comp_aemeasurable | Mathlib.MeasureTheory.Measure.AEMeasurable | ∀ {α : Type u_2} {β : Type u_3} {δ : Type u_5} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] [inst_1 : MeasurableSpace δ] {μ : MeasureTheory.Measure α} {f : α → δ} {g : δ → β}, AEMeasurable g (MeasureTheory.Measure.map f μ) → AEMeasurable f μ → AEMeasurable (g ∘ f) μ |
AEMeasurable.comp_aemeasurable' | Mathlib.MeasureTheory.Measure.AEMeasurable | ∀ {α : Type u_2} {β : Type u_3} {δ : Type u_5} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] [inst_1 : MeasurableSpace δ] {μ : MeasureTheory.Measure α} {f : α → δ} {g : δ → β}, AEMeasurable g (MeasureTheory.Measure.map f μ) → AEMeasurable f μ → AEMeasurable (fun x => g (f x)) μ |
AEMeasurable.comp_fst | Mathlib.MeasureTheory.Measure.Prod | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ}, AEMeasurable f μ → AEMeasurable (fun z => f z.1) (μ.prod ν) |
AEMeasurable.comp_measurable | Mathlib.MeasureTheory.Measure.AEMeasurable | ∀ {α : Type u_2} {β : Type u_3} {δ : Type u_5} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] [inst_1 : MeasurableSpace δ] {μ : MeasureTheory.Measure α} {f : α → δ} {g : δ → β}, AEMeasurable g (MeasureTheory.Measure.map f μ) → Measurable f → AEMeasurable (g ∘ f) μ |
AEMeasurable.comp_quasiMeasurePreserving | Mathlib.MeasureTheory.Measure.AEMeasurable | ∀ {α : Type u_2} {β : Type u_3} {δ : Type u_5} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] [inst_1 : MeasurableSpace δ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure δ} {f : α → δ} {g : δ → β}, AEMeasurable g ν → MeasureTheory.Measure.QuasiMeasurePreserving f μ ν → AEMeasurable (g ∘ f) μ |
AEMeasurable.comp_snd | Mathlib.MeasureTheory.Measure.Prod | ∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : β → γ}, AEMeasurable f ν → AEMeasurable (fun z => f z.2) (μ.prod ν) |
AEMeasurable.complex_ofReal | Mathlib.MeasureTheory.Function.SpecialFunctions.Basic | ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ}, AEMeasurable f μ → AEMeasurable (fun x => ↑(f x)) μ |
AEMeasurable.congr | Mathlib.MeasureTheory.Measure.MeasureSpaceDef | ∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [inst : MeasurableSpace β] {f g : α → β} {μ : MeasureTheory.Measure α}, AEMeasurable f μ → f =ᵐ[μ] g → AEMeasurable g μ |
AEMeasurable.const_add | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {M : Type u_2} {α : Type u_3} [inst : MeasurableSpace M] [inst_1 : Add M] {m : MeasurableSpace α} {f : α → M} {μ : MeasureTheory.Measure α} [MeasurableAdd M], AEMeasurable f μ → ∀ (c : M), AEMeasurable (fun x => c + f x) μ |
AEMeasurable.const_div | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {G : Type u_2} {α : Type u_3} [inst : MeasurableSpace G] [inst_1 : Div G] {m : MeasurableSpace α} {f : α → G} {μ : MeasureTheory.Measure α} [MeasurableDiv G], AEMeasurable f μ → ∀ (c : G), AEMeasurable (fun x => c / f x) μ |
AEMeasurable.const_inf | Mathlib.MeasureTheory.Order.Lattice | ∀ {M : Type u_1} [inst : MeasurableSpace M] {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → M} [inst_1 : Min M] [MeasurableInf M], AEMeasurable f μ → ∀ (c : M), AEMeasurable (fun x => c ⊓ f x) μ |
AEMeasurable.const_inner | Mathlib.MeasureTheory.Function.SpecialFunctions.Inner | ∀ {α : Type u_1} {𝕜 : Type u_2} {E : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E] {m : MeasurableSpace α} [inst_3 : MeasurableSpace E] [OpensMeasurableSpace E] [SecondCountableTopology E] {μ : MeasureTheory.Measure α} {f : α → E} {c : E}, AEMeasurable f μ → AEMeasurabl... |
AEMeasurable.const_mul | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {M : Type u_2} {α : Type u_3} [inst : MeasurableSpace M] [inst_1 : Mul M] {m : MeasurableSpace α} {f : α → M} {μ : MeasureTheory.Measure α} [MeasurableMul M], AEMeasurable f μ → ∀ (c : M), AEMeasurable (fun x => c * f x) μ |
AEMeasurable.const_pow | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {β : Type u_2} {γ : Type u_3} {α : Type u_4} [inst : MeasurableSpace β] [inst_1 : MeasurableSpace γ] [inst_2 : Pow β γ] [MeasurablePow β γ] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {g : α → γ}, AEMeasurable g μ → ∀ (c : β), AEMeasurable (fun x => c ^ g x) μ |
AEMeasurable.const_smul | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {M : Type u_2} {X : Type u_3} {α : Type u_4} [inst : MeasurableSpace X] [inst_1 : SMul M X] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {g : α → X} [MeasurableConstSMul M X], AEMeasurable g μ → ∀ (c : M), AEMeasurable (c • g) μ |
AEMeasurable.const_sub | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {G : Type u_2} {α : Type u_3} [inst : MeasurableSpace G] [inst_1 : Sub G] {m : MeasurableSpace α} {f : α → G} {μ : MeasureTheory.Measure α} [MeasurableSub G], AEMeasurable f μ → ∀ (c : G), AEMeasurable (fun x => c - f x) μ |
AEMeasurable.const_sup | Mathlib.MeasureTheory.Order.Lattice | ∀ {M : Type u_1} [inst : MeasurableSpace M] {α : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → M} [inst_1 : Max M] [MeasurableSup M], AEMeasurable f μ → ∀ (c : M), AEMeasurable (fun x => c ⊔ f x) μ |
AEMeasurable.const_vadd | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {M : Type u_2} {X : Type u_3} {α : Type u_4} [inst : MeasurableSpace X] [inst_1 : VAdd M X] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {g : α → X} [MeasurableConstVAdd M X], AEMeasurable g μ → ∀ (c : M), AEMeasurable (c +ᵥ g) μ |
AEMeasurable.cos | Mathlib.MeasureTheory.Function.SpecialFunctions.Basic | ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ}, AEMeasurable f μ → AEMeasurable (fun x => Real.cos (f x)) μ |
AEMeasurable.cosh | Mathlib.MeasureTheory.Function.SpecialFunctions.Basic | ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ}, AEMeasurable f μ → AEMeasurable (fun x => Real.cosh (f x)) μ |
AEMeasurable.csin | Mathlib.MeasureTheory.Function.SpecialFunctions.Basic | ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℂ}, AEMeasurable f μ → AEMeasurable (fun x => Complex.sin (f x)) μ |
AEMeasurable.csinh | Mathlib.MeasureTheory.Function.SpecialFunctions.Basic | ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℂ}, AEMeasurable f μ → AEMeasurable (fun x => Complex.sinh (f x)) μ |
AEMeasurable.dist | Mathlib.MeasureTheory.Constructions.BorelSpace.Metric | ∀ {α : Type u_1} {β : Type u_2} [inst : PseudoMetricSpace α] [inst_1 : MeasurableSpace α] [OpensMeasurableSpace α] [inst_3 : MeasurableSpace β] [SecondCountableTopology α] {f g : β → α} {μ : MeasureTheory.Measure β}, AEMeasurable f μ → AEMeasurable g μ → AEMeasurable (fun b => dist (f b) (g b)) μ |
AEMeasurable.div | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {G : Type u_2} {α : Type u_3} [inst : MeasurableSpace G] [inst_1 : Div G] {m : MeasurableSpace α} {f g : α → G} {μ : MeasureTheory.Measure α} [MeasurableDiv₂ G], AEMeasurable f μ → AEMeasurable g μ → AEMeasurable (fun a => f a / g a) μ |
AEMeasurable.div' | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {G : Type u_2} {α : Type u_3} [inst : MeasurableSpace G] [inst_1 : Div G] {m : MeasurableSpace α} {f g : α → G} {μ : MeasureTheory.Measure α} [MeasurableDiv₂ G], AEMeasurable f μ → AEMeasurable g μ → AEMeasurable (f / g) μ |
AEMeasurable.div_const | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {G : Type u_2} {α : Type u_3} [inst : MeasurableSpace G] [inst_1 : Div G] {m : MeasurableSpace α} {f : α → G} {μ : MeasureTheory.Measure α} [MeasurableDiv G], AEMeasurable f μ → ∀ (c : G), AEMeasurable (fun x => f x / c) μ |
AEMeasurable.edist | Mathlib.MeasureTheory.Constructions.BorelSpace.Metric | ∀ {α : Type u_1} {β : Type u_2} [inst : PseudoEMetricSpace α] [inst_1 : MeasurableSpace α] [OpensMeasurableSpace α] [inst_3 : MeasurableSpace β] [SecondCountableTopology α] {f g : β → α} {μ : MeasureTheory.Measure β}, AEMeasurable f μ → AEMeasurable g μ → AEMeasurable (fun a => edist (f a) (g a)) μ |
AEMeasurable.ennreal_ofReal | Mathlib.MeasureTheory.Constructions.BorelSpace.Real | ∀ {α : Type u_1} {mα : MeasurableSpace α} {f : α → ℝ} {μ : MeasureTheory.Measure α}, AEMeasurable f μ → AEMeasurable (fun x => ENNReal.ofReal (f x)) μ |
AEMeasurable.ennreal_toNNReal | Mathlib.MeasureTheory.Constructions.BorelSpace.Real | ∀ {α : Type u_1} {mα : MeasurableSpace α} {f : α → ENNReal} {μ : MeasureTheory.Measure α}, AEMeasurable f μ → AEMeasurable (fun x => (f x).toNNReal) μ |
AEMeasurable.ennreal_toReal | Mathlib.MeasureTheory.Constructions.BorelSpace.Real | ∀ {α : Type u_1} {mα : MeasurableSpace α} {f : α → ENNReal} {μ : MeasureTheory.Measure α}, AEMeasurable f μ → AEMeasurable (fun x => (f x).toReal) μ |
AEMeasurable.ennreal_tsum | Mathlib.MeasureTheory.Constructions.BorelSpace.Real | ∀ {α : Type u_1} {mα : MeasurableSpace α} {ι : Type u_5} [Countable ι] {f : ι → α → ENNReal} {μ : MeasureTheory.Measure α}, (∀ (i : ι), AEMeasurable (f i) μ) → AEMeasurable (fun x => ∑' (i : ι), f i x) μ |
AEMeasurable.enorm | Mathlib.MeasureTheory.Constructions.BorelSpace.Metric | ∀ {β : Type u_2} {ε : Type u_5} [inst : MeasurableSpace ε] [inst_1 : TopologicalSpace ε] [inst_2 : ContinuousENorm ε] [OpensMeasurableSpace ε] [inst_4 : MeasurableSpace β] {f : β → ε} {μ : MeasureTheory.Measure β}, AEMeasurable f μ → AEMeasurable (fun x => ‖f x‖ₑ) μ |
AEMeasurable.eq_1 | Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | ∀ {α : Type u_1} {β : Type u_2} [inst : MeasurableSpace β] {_m : MeasurableSpace α} (f : α → β) (μ : MeasureTheory.Measure α), AEMeasurable f μ = ∃ g, Measurable g ∧ f =ᵐ[μ] g |
AEMeasurable.ereal_toENNReal | Mathlib.MeasureTheory.Constructions.BorelSpace.Real | ∀ {α : Type u_1} {mα : MeasurableSpace α} {f : α → EReal} {μ : MeasureTheory.Measure α}, AEMeasurable f μ → AEMeasurable (fun x => (f x).toENNReal) μ |
AEMeasurable.ereal_toReal | Mathlib.MeasureTheory.Constructions.BorelSpace.Real | ∀ {α : Type u_1} {mα : MeasurableSpace α} {f : α → EReal} {μ : MeasureTheory.Measure α}, AEMeasurable f μ → AEMeasurable (fun x => (f x).toReal) μ |
AEMeasurable.eval | Mathlib.MeasureTheory.Measure.MeasureSpaceDef | ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {δ : Type u_6} {X : δ → Type u_7} {mX : (a : δ) → MeasurableSpace (X a)} {g : α → (a : δ) → X a}, AEMeasurable g μ → ∀ (a : δ), AEMeasurable (fun x => g x a) μ |
AEMeasurable.exists_ae_eq_range_subset | Mathlib.MeasureTheory.Measure.AEMeasurable | ∀ {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α}, AEMeasurable f μ → ∀ {t : Set β}, (∀ᵐ (x : α) ∂μ, f x ∈ t) → t.Nonempty → ∃ g, Measurable g ∧ Set.range g ⊆ t ∧ f =ᵐ[μ] g |
AEMeasurable.exists_measurable_nonneg | Mathlib.MeasureTheory.Measure.AEMeasurable | ∀ {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [inst : Preorder β] [inst_1 : Zero β] {mβ : MeasurableSpace β} {f : α → β}, AEMeasurable f μ → (∀ᵐ (t : α) ∂μ, 0 ≤ f t) → ∃ g, Measurable g ∧ 0 ≤ g ∧ f =ᵐ[μ] g |
AEMeasurable.exp | Mathlib.MeasureTheory.Function.SpecialFunctions.Basic | ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ}, AEMeasurable f μ → AEMeasurable (fun x => Real.exp (f x)) μ |
AEMeasurable.fst | Mathlib.MeasureTheory.Measure.AEMeasurable | ∀ {α : Type u_2} {β : Type u_3} {γ : Type u_4} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] [inst_1 : MeasurableSpace γ] {μ : MeasureTheory.Measure α} {f : α → β × γ}, AEMeasurable f μ → AEMeasurable (fun x => (f x).1) μ |
AEMeasurable.fun_const_smul | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {M : Type u_2} {X : Type u_3} {α : Type u_4} [inst : MeasurableSpace X] [inst_1 : SMul M X] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {g : α → X} [MeasurableConstSMul M X], AEMeasurable g μ → ∀ (c : M), AEMeasurable (fun x => c • g x) μ |
AEMeasurable.fun_const_vadd | Mathlib.MeasureTheory.Group.Arithmetic | ∀ {M : Type u_2} {X : Type u_3} {α : Type u_4} [inst : MeasurableSpace X] [inst_1 : VAdd M X] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {g : α → X} [MeasurableConstVAdd M X], AEMeasurable g μ → ∀ (c : M), AEMeasurable (fun x => c +ᵥ g x) μ |
AEMeasurable.iInf | Mathlib.MeasureTheory.Constructions.BorelSpace.Order | ∀ {α : Type u_1} {δ : Type u_4} [inst : TopologicalSpace α] {mα : MeasurableSpace α} [BorelSpace α] {mδ : MeasurableSpace δ} [inst_2 : ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Sort u_5} {μ : MeasureTheory.Measure δ} [Countable ι] {f : ι → δ → α}, (∀ (i : ι), AEMeasurable (f... |
AEMeasurable.iSup | Mathlib.MeasureTheory.Constructions.BorelSpace.Order | ∀ {α : Type u_1} {δ : Type u_4} [inst : TopologicalSpace α] {mα : MeasurableSpace α} [BorelSpace α] {mδ : MeasurableSpace δ} [inst_2 : ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Sort u_5} {μ : MeasureTheory.Measure δ} [Countable ι] {f : ι → δ → α}, (∀ (i : ι), AEMeasurable (f... |
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