Loogle!
Result
Found 78 definitions mentioning AEMeasurable and MeasureTheory.Measure.map.
- MeasureTheory.Measure.tendsto_ae_map Mathlib.MeasureTheory.Measure.MeasureSpace
∀ {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β}, AEMeasurable f μ → Filter.Tendsto f μ.ae (MeasureTheory.Measure.map f μ).ae - MeasureTheory.Measure.map_of_not_aemeasurable Mathlib.MeasureTheory.Measure.MeasureSpace
∀ {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α}, ¬AEMeasurable f μ → MeasureTheory.Measure.map f μ = 0 - MeasureTheory.ae_of_ae_map Mathlib.MeasureTheory.Measure.MeasureSpace
∀ {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β}, AEMeasurable f μ → ∀ {p : β → Prop}, (∀ᵐ (y : β) ∂MeasureTheory.Measure.map f μ, p y) → ∀ᵐ (x : α) ∂μ, p (f x) - MeasureTheory.ae_map_iff Mathlib.MeasureTheory.Measure.MeasureSpace
∀ {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β}, AEMeasurable f μ → ∀ {p : β → Prop}, MeasurableSet {x | p x} → ((∀ᵐ (y : β) ∂MeasureTheory.Measure.map f μ, p y) ↔ ∀ᵐ (x : α) ∂μ, p (f x)) - MeasureTheory.mem_ae_of_mem_ae_map Mathlib.MeasureTheory.Measure.MeasureSpace
∀ {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β}, AEMeasurable f μ → ∀ {s : Set β}, s ∈ (MeasureTheory.Measure.map f μ).ae → f ⁻¹' s ∈ μ.ae - MeasureTheory.mem_ae_map_iff Mathlib.MeasureTheory.Measure.MeasureSpace
∀ {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β}, AEMeasurable f μ → ∀ {s : Set β}, MeasurableSet s → (s ∈ (MeasureTheory.Measure.map f μ).ae ↔ f ⁻¹' s ∈ μ.ae) - MeasureTheory.Measure.map_eq_zero_iff Mathlib.MeasureTheory.Measure.MeasureSpace
∀ {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β}, AEMeasurable f μ → (MeasureTheory.Measure.map f μ = 0 ↔ μ = 0) - MeasureTheory.Measure.map_ne_zero_iff Mathlib.MeasureTheory.Measure.MeasureSpace
∀ {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β}, AEMeasurable f μ → (MeasureTheory.Measure.map f μ ≠ 0 ↔ μ ≠ 0) - MeasureTheory.Measure.map_apply_of_aemeasurable Mathlib.MeasureTheory.Measure.MeasureSpace
∀ {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β}, AEMeasurable f μ → ∀ {s : Set β}, MeasurableSet s → (MeasureTheory.Measure.map f μ) s = μ (f ⁻¹' s) - MeasureTheory.Measure.map_apply₀ Mathlib.MeasureTheory.Measure.MeasureSpace
∀ {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β}, AEMeasurable f μ → ∀ {s : Set β}, MeasureTheory.NullMeasurableSet s (MeasureTheory.Measure.map f μ) → (MeasureTheory.Measure.map f μ) s = μ (f ⁻¹' s) - MeasureTheory.Measure.le_map_apply Mathlib.MeasureTheory.Measure.MeasureSpace
∀ {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β}, AEMeasurable f μ → ∀ (s : Set β), μ (f ⁻¹' s) ≤ (MeasureTheory.Measure.map f μ) s - MeasureTheory.Measure.le_map_apply_image Mathlib.MeasureTheory.Measure.MeasureSpace
∀ {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β}, AEMeasurable f μ → ∀ (s : Set α), μ s ≤ (MeasureTheory.Measure.map f μ) (f '' s) - MeasureTheory.Measure.preimage_null_of_map_null Mathlib.MeasureTheory.Measure.MeasureSpace
∀ {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β}, AEMeasurable f μ → ∀ {s : Set β}, (MeasureTheory.Measure.map f μ) s = 0 → μ (f ⁻¹' s) = 0 - MeasureTheory.Measure.map_def Mathlib.MeasureTheory.Measure.MeasureSpace
∀ {α : Type u_8} {β : Type u_9} [inst : MeasurableSpace β] [inst_1 : MeasurableSpace α] (f : α → β) (μ : MeasureTheory.Measure α), MeasureTheory.Measure.map f μ = if hf : AEMeasurable f μ then (MeasureTheory.Measure.mapₗ (AEMeasurable.mk f hf)) μ else 0 - MeasureTheory.Measure.map_toOuterMeasure Mathlib.MeasureTheory.Measure.MeasureSpace
∀ {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β}, AEMeasurable f μ → (MeasureTheory.Measure.map f μ).toOuterMeasure = ((MeasureTheory.OuterMeasure.map f) μ.toOuterMeasure).trim - MeasureTheory.Measure.mapₗ_mk_apply_of_aemeasurable Mathlib.MeasureTheory.Measure.MeasureSpace
∀ {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β} (hf : AEMeasurable f μ), (MeasureTheory.Measure.mapₗ (AEMeasurable.mk f hf)) μ = MeasureTheory.Measure.map f μ - MeasureTheory.ae_eq_comp Mathlib.MeasureTheory.Measure.Restrict
∀ {α : Type u_2} {β : Type u_3} {δ : Type u_4} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β} {g g' : β → δ}, AEMeasurable f μ → (MeasureTheory.Measure.map f μ).ae.EventuallyEq g g' → μ.ae.EventuallyEq (g ∘ f) (g' ∘ f) - MeasureTheory.ae_eq_comp' Mathlib.MeasureTheory.Measure.Restrict
∀ {α : Type u_2} {β : Type u_3} {δ : Type u_4} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → β} {g g' : β → δ}, AEMeasurable f μ → ν.ae.EventuallyEq g g' → (MeasureTheory.Measure.map f μ).AbsolutelyContinuous ν → μ.ae.EventuallyEq (g ∘ f) (g' ∘ f) - MeasureTheory.isProbabilityMeasure_map Mathlib.MeasureTheory.Measure.Typeclasses
∀ {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} [inst_1 : MeasureTheory.IsProbabilityMeasure μ] {f : α → β}, AEMeasurable f μ → MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.map f μ) - MeasureTheory.SigmaFinite.of_map Mathlib.MeasureTheory.Measure.Typeclasses
∀ {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] (μ : MeasureTheory.Measure α) {f : α → β}, AEMeasurable f μ → MeasureTheory.SigmaFinite (MeasureTheory.Measure.map f μ) → MeasureTheory.SigmaFinite μ - aemeasurable_of_map_neZero Mathlib.MeasureTheory.Measure.AEMeasurable
∀ {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {f : α → β}, NeZero (MeasureTheory.Measure.map f μ) → AEMeasurable f μ - 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_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 - MeasurableEmbedding.aemeasurable_map_iff 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] {f : α → β} {μ : MeasureTheory.Measure α} {g : β → γ}, MeasurableEmbedding f → (AEMeasurable g (MeasureTheory.Measure.map f μ) ↔ AEMeasurable (g ∘ f) μ) - MeasureTheory.Measure.map_sum Mathlib.MeasureTheory.Measure.AEMeasurable
∀ {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [inst : MeasurableSpace β] {ι : Type u_7} {m : ι → MeasureTheory.Measure α} {f : α → β}, AEMeasurable f (MeasureTheory.Measure.sum m) → MeasureTheory.Measure.map f (MeasureTheory.Measure.sum m) = MeasureTheory.Measure.sum fun i => MeasureTheory.Measure.map f (m i) - MeasureTheory.Measure.restrict_map_of_aemeasurable Mathlib.MeasureTheory.Measure.AEMeasurable
∀ {α : Type u_2} {δ : Type u_5} {m0 : MeasurableSpace α} [inst : MeasurableSpace δ] {μ : MeasureTheory.Measure α} {f : α → δ}, AEMeasurable f μ → ∀ {s : Set δ}, MeasurableSet s → (MeasureTheory.Measure.map f μ).restrict s = MeasureTheory.Measure.map f (μ.restrict (f ⁻¹' s)) - AEMeasurable.map_map_of_aemeasurable 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} {g : β → γ} {f : α → β}, AEMeasurable g (MeasureTheory.Measure.map f μ) → AEMeasurable f μ → MeasureTheory.Measure.map g (MeasureTheory.Measure.map f μ) = MeasureTheory.Measure.map (g ∘ f) μ - MeasureTheory.Measure.map_mono_of_aemeasurable Mathlib.MeasureTheory.Measure.AEMeasurable
∀ {α : Type u_2} {δ : Type u_5} {m0 : MeasurableSpace α} [inst : MeasurableSpace δ] {μ ν : MeasureTheory.Measure α} {f : α → δ}, μ ≤ ν → AEMeasurable f ν → MeasureTheory.Measure.map f μ ≤ MeasureTheory.Measure.map f ν - aemeasurable_map_equiv_iff 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} (e : α ≃ᵐ β) {f : β → γ}, AEMeasurable f (MeasureTheory.Measure.map (⇑e) μ) ↔ AEMeasurable (f ∘ ⇑e) μ - MeasureTheory.Measure.InnerRegularWRT.map Mathlib.MeasureTheory.Measure.Regular
∀ {α : Type u_2} {β : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] {μ : MeasureTheory.Measure α} {pa qa : Set α → Prop}, μ.InnerRegularWRT pa qa → ∀ {f : α → β}, AEMeasurable f μ → ∀ {pb qb : Set β → Prop}, (∀ (U : Set β), qb U → qa (f ⁻¹' U)) → (∀ (K : Set α), pa K → pb (f '' K)) → (∀ (U : Set β), qb U → MeasurableSet U) → (MeasureTheory.Measure.map f μ).InnerRegularWRT pb qb - essSup_comp_le_essSup_map_measure Mathlib.MeasureTheory.Function.EssSup
∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : CompleteLattice β] {γ : Type u_3} {mγ : MeasurableSpace γ} {f : α → γ} {g : γ → β}, AEMeasurable f μ → essSup (g ∘ f) μ ≤ essSup g (MeasureTheory.Measure.map f μ) - essSup_map_measure_of_measurable Mathlib.MeasureTheory.Function.EssSup
∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : CompleteLattice β] {γ : Type u_3} {mγ : MeasurableSpace γ} {f : α → γ} {g : γ → β} [inst_1 : MeasurableSpace β] [inst_2 : TopologicalSpace β] [inst_3 : SecondCountableTopology β] [inst_4 : OrderClosedTopology β] [inst_5 : OpensMeasurableSpace β], Measurable g → AEMeasurable f μ → essSup g (MeasureTheory.Measure.map f μ) = essSup (g ∘ f) μ - essSup_map_measure Mathlib.MeasureTheory.Function.EssSup
∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : CompleteLattice β] {γ : Type u_3} {mγ : MeasurableSpace γ} {f : α → γ} {g : γ → β} [inst_1 : MeasurableSpace β] [inst_2 : TopologicalSpace β] [inst_3 : SecondCountableTopology β] [inst_4 : OrderClosedTopology β] [inst_5 : OpensMeasurableSpace β], AEMeasurable g (MeasureTheory.Measure.map f μ) → AEMeasurable f μ → essSup g (MeasureTheory.Measure.map f μ) = essSup (g ∘ f) μ - MeasureTheory.lintegral_map' Mathlib.MeasureTheory.Integral.Lebesgue
∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {mβ : MeasurableSpace β} {f : β → ENNReal} {g : α → β}, AEMeasurable f (MeasureTheory.Measure.map g μ) → AEMeasurable g μ → ∫⁻ (a : β), f a ∂MeasureTheory.Measure.map g μ = ∫⁻ (a : α), f (g a) ∂μ - MeasureTheory.AEStronglyMeasurable.comp_aemeasurable Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace β] {g : α → β} {γ : Type u_5} {x : MeasurableSpace γ} {x_1 : MeasurableSpace α} {f : γ → α} {μ : MeasureTheory.Measure γ}, MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map f μ) → AEMeasurable f μ → MeasureTheory.AEStronglyMeasurable (g ∘ f) μ - MeasureTheory.Memℒp.comp_of_map Mathlib.MeasureTheory.Function.LpSeminorm.Basic
∀ {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [inst : NormedAddCommGroup E] {β : Type u_5} {mβ : MeasurableSpace β} {f : α → β} {g : β → E}, MeasureTheory.Memℒp g p (MeasureTheory.Measure.map f μ) → AEMeasurable f μ → MeasureTheory.Memℒp (g ∘ f) p μ - MeasureTheory.snormEssSup_map_measure Mathlib.MeasureTheory.Function.LpSeminorm.Basic
∀ {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : NormedAddCommGroup E] {β : Type u_5} {mβ : MeasurableSpace β} {f : α → β} {g : β → E}, MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map f μ) → AEMeasurable f μ → MeasureTheory.snormEssSup g (MeasureTheory.Measure.map f μ) = MeasureTheory.snormEssSup (g ∘ f) μ - MeasureTheory.memℒp_map_measure_iff Mathlib.MeasureTheory.Function.LpSeminorm.Basic
∀ {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [inst : NormedAddCommGroup E] {β : Type u_5} {mβ : MeasurableSpace β} {f : α → β} {g : β → E}, MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map f μ) → AEMeasurable f μ → (MeasureTheory.Memℒp g p (MeasureTheory.Measure.map f μ) ↔ MeasureTheory.Memℒp (g ∘ f) p μ) - MeasureTheory.snorm_map_measure Mathlib.MeasureTheory.Function.LpSeminorm.Basic
∀ {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [inst : NormedAddCommGroup E] {β : Type u_5} {mβ : MeasurableSpace β} {f : α → β} {g : β → E}, MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map f μ) → AEMeasurable f μ → MeasureTheory.snorm g p (MeasureTheory.Measure.map f μ) = MeasureTheory.snorm (g ∘ f) p μ - MeasureTheory.Integrable.comp_aemeasurable Mathlib.MeasureTheory.Function.L1Space
∀ {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasurableSpace δ] [inst_1 : NormedAddCommGroup β] {f : α → δ} {g : δ → β}, MeasureTheory.Integrable g (MeasureTheory.Measure.map f μ) → AEMeasurable f μ → MeasureTheory.Integrable (g ∘ f) μ - MeasureTheory.integrable_map_measure Mathlib.MeasureTheory.Function.L1Space
∀ {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasurableSpace δ] [inst_1 : NormedAddCommGroup β] {f : α → δ} {g : δ → β}, MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map f μ) → AEMeasurable f μ → (MeasureTheory.Integrable g (MeasureTheory.Measure.map f μ) ↔ MeasureTheory.Integrable (g ∘ f) μ) - MeasureTheory.integral_map Mathlib.MeasureTheory.Integral.Bochner
∀ {α : Type u_1} {G : Type u_5} [inst : NormedAddCommGroup G] [inst_1 : NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [inst_2 : MeasurableSpace β] {φ : α → β}, AEMeasurable φ μ → ∀ {f : β → G}, MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map φ μ) → ∫ (y : β), f y ∂MeasureTheory.Measure.map φ μ = ∫ (x : α), f (φ x) ∂μ - MeasureTheory.setIntegral_map Mathlib.MeasureTheory.Integral.SetIntegral
∀ {X : Type u_1} {E : Type u_3} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace ℝ E] {μ : MeasureTheory.Measure X} {Y : Type u_5} [inst_3 : MeasurableSpace Y] {g : X → Y} {f : Y → E} {s : Set Y}, MeasurableSet s → MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map g μ) → AEMeasurable g μ → ∫ (y : Y) in s, f y ∂MeasureTheory.Measure.map g μ = ∫ (x : X) in g ⁻¹' s, f (g x) ∂μ - MeasureTheory.set_integral_map Mathlib.MeasureTheory.Integral.SetIntegral
∀ {X : Type u_1} {E : Type u_3} [inst : MeasurableSpace X] [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace ℝ E] {μ : MeasureTheory.Measure X} {Y : Type u_5} [inst_3 : MeasurableSpace Y] {g : X → Y} {f : Y → E} {s : Set Y}, MeasurableSet s → MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map g μ) → AEMeasurable g μ → ∫ (y : Y) in s, f y ∂MeasureTheory.Measure.map g μ = ∫ (x : X) in g ⁻¹' s, f (g x) ∂μ - MeasureTheory.Measure.fst_map_prod_mk₀ Mathlib.MeasureTheory.Constructions.Prod.Basic
∀ {α : Type u_1} {β : Type u_3} {γ : Type u_5} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] {X : α → β} {Y : α → γ} {μ : MeasureTheory.Measure α}, AEMeasurable Y μ → (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ).fst = MeasureTheory.Measure.map X μ - MeasureTheory.Measure.snd_map_prod_mk₀ Mathlib.MeasureTheory.Constructions.Prod.Basic
∀ {α : Type u_1} {β : Type u_3} {γ : Type u_5} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] {X : α → β} {Y : α → γ} {μ : MeasureTheory.Measure α}, AEMeasurable X μ → (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ).snd = MeasureTheory.Measure.map Y μ - essSup_comp_quotientAddGroup_mk Mathlib.MeasureTheory.Measure.Haar.Quotient
∀ {G : Type u_1} [inst : AddGroup G] [inst_1 : MeasurableSpace G] [inst_2 : TopologicalSpace G] [inst_3 : TopologicalAddGroup G] [inst_4 : BorelSpace G] {μ : MeasureTheory.Measure G} {Γ : AddSubgroup G} {𝓕 : Set G}, MeasureTheory.IsAddFundamentalDomain (↥Γ.op) 𝓕 μ → ∀ [inst_5 : Countable ↥Γ] [inst_6 : MeasurableSpace (G ⧸ Γ)] [inst_7 : BorelSpace (G ⧸ Γ)] [inst_8 : μ.IsAddRightInvariant] {g : G ⧸ Γ → ENNReal}, AEMeasurable g (MeasureTheory.Measure.map QuotientAddGroup.mk (μ.restrict 𝓕)) → essSup g (MeasureTheory.Measure.map QuotientAddGroup.mk (μ.restrict 𝓕)) = essSup (fun x => g ↑x) μ - essSup_comp_quotientGroup_mk Mathlib.MeasureTheory.Measure.Haar.Quotient
∀ {G : Type u_1} [inst : Group G] [inst_1 : MeasurableSpace G] [inst_2 : TopologicalSpace G] [inst_3 : TopologicalGroup G] [inst_4 : BorelSpace G] {μ : MeasureTheory.Measure G} {Γ : Subgroup G} {𝓕 : Set G}, MeasureTheory.IsFundamentalDomain (↥Γ.op) 𝓕 μ → ∀ [inst_5 : Countable ↥Γ] [inst_6 : MeasurableSpace (G ⧸ Γ)] [inst_7 : BorelSpace (G ⧸ Γ)] [inst_8 : μ.IsMulRightInvariant] {g : G ⧸ Γ → ENNReal}, AEMeasurable g (MeasureTheory.Measure.map QuotientGroup.mk (μ.restrict 𝓕)) → essSup g (MeasureTheory.Measure.map QuotientGroup.mk (μ.restrict 𝓕)) = essSup (fun x => g ↑x) μ - MeasureTheory.AEStronglyMeasurable.comp_ae_measurable' Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable
∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : TopologicalSpace β] {mα : MeasurableSpace α} {x : MeasurableSpace γ} {f : α → β} {μ : MeasureTheory.Measure γ} {g : γ → α}, MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map g μ) → AEMeasurable g μ → MeasureTheory.AEStronglyMeasurable' (MeasurableSpace.comap g mα) (f ∘ g) μ - MeasureTheory.ProbabilityMeasure.toMeasure_map Mathlib.MeasureTheory.Measure.ProbabilityMeasure
∀ {Ω : Type u_1} {Ω' : Type u_2} [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace Ω'] (ν : MeasureTheory.ProbabilityMeasure Ω) {f : Ω → Ω'} (hf : AEMeasurable f ↑ν), ↑(ν.map hf) = MeasureTheory.Measure.map f ↑ν - ProbabilityTheory.indepFun_iff_map_prod_eq_prod_map_map Mathlib.Probability.Independence.Basic
∀ {Ω : Type u_1} {β : Type u_6} {β' : Type u_7} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ω → β} {g : Ω → β'} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} [inst : MeasureTheory.IsFiniteMeasure μ], AEMeasurable f μ → AEMeasurable g μ → (ProbabilityTheory.IndepFun f g μ ↔ MeasureTheory.Measure.map (fun ω => (f ω, g ω)) μ = (MeasureTheory.Measure.map f μ).prod (MeasureTheory.Measure.map g μ)) - Real.hasPDF_iff_of_aemeasurable Mathlib.Probability.Density
∀ {Ω : Type u_1} {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsFiniteMeasure ℙ] {X : Ω → ℝ}, AEMeasurable X ℙ → (MeasureTheory.HasPDF X ℙ MeasureTheory.volume ↔ (MeasureTheory.Measure.map X ℙ).AbsolutelyContinuous MeasureTheory.volume) - Real.hasPDF_iff Mathlib.Probability.Density
∀ {Ω : Type u_1} {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsFiniteMeasure ℙ] {X : Ω → ℝ}, MeasureTheory.HasPDF X ℙ MeasureTheory.volume ↔ AEMeasurable X ℙ ∧ (MeasureTheory.Measure.map X ℙ).AbsolutelyContinuous MeasureTheory.volume - MeasureTheory.hasPDF_iff_of_aemeasurable Mathlib.Probability.Density
∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {x : MeasurableSpace Ω} {X : Ω → E} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E}, AEMeasurable X ℙ → (MeasureTheory.HasPDF X ℙ μ ↔ (MeasureTheory.Measure.map X ℙ).HaveLebesgueDecomposition μ ∧ (MeasureTheory.Measure.map X ℙ).AbsolutelyContinuous μ) - MeasureTheory.hasPDF_iff Mathlib.Probability.Density
∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {x : MeasurableSpace Ω} {X : Ω → E} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E}, MeasureTheory.HasPDF X ℙ μ ↔ AEMeasurable X ℙ ∧ (MeasureTheory.Measure.map X ℙ).HaveLebesgueDecomposition μ ∧ (MeasureTheory.Measure.map X ℙ).AbsolutelyContinuous μ - MeasureTheory.hasPDF_of_map_eq_withDensity Mathlib.Probability.Density
∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {x : MeasurableSpace Ω} {X : Ω → E} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E}, AEMeasurable X ℙ → ∀ (f : E → ENNReal), AEMeasurable f μ → MeasureTheory.Measure.map X ℙ = μ.withDensity f → MeasureTheory.HasPDF X ℙ μ - MeasureTheory.HasPDF.mk Mathlib.Probability.Density
∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : MeasureTheory.Measure Ω} {μ : autoParam (MeasureTheory.Measure E) _auto✝}, AEMeasurable X ℙ ∧ (MeasureTheory.Measure.map X ℙ).HaveLebesgueDecomposition μ ∧ (MeasureTheory.Measure.map X ℙ).AbsolutelyContinuous μ → MeasureTheory.HasPDF X ℙ μ - MeasureTheory.HasPDF.pdf' Mathlib.Probability.Density
∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {m : MeasurableSpace Ω} {X : Ω → E} {ℙ : MeasureTheory.Measure Ω} {μ : autoParam (MeasureTheory.Measure E) _auto✝} [self : MeasureTheory.HasPDF X ℙ μ], AEMeasurable X ℙ ∧ (MeasureTheory.Measure.map X ℙ).HaveLebesgueDecomposition μ ∧ (MeasureTheory.Measure.map X ℙ).AbsolutelyContinuous μ - MeasureTheory.pdf.eq_of_map_eq_withDensity Mathlib.Probability.Density
∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E} [inst_1 : MeasureTheory.IsFiniteMeasure ℙ] {X : Ω → E} [inst_2 : MeasureTheory.HasPDF X ℙ μ] (f : E → ENNReal), AEMeasurable f μ → (MeasureTheory.Measure.map X ℙ = μ.withDensity f ↔ μ.ae.EventuallyEq (MeasureTheory.pdf X ℙ μ) f) - MeasureTheory.pdf.eq_of_map_eq_withDensity' Mathlib.Probability.Density
∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E} [inst_1 : MeasureTheory.SigmaFinite μ] {X : Ω → E} [inst_2 : MeasureTheory.HasPDF X ℙ μ] (f : E → ENNReal), AEMeasurable f μ → (MeasureTheory.Measure.map X ℙ = μ.withDensity f ↔ μ.ae.EventuallyEq (MeasureTheory.pdf X ℙ μ) f) - MeasureTheory.pdf.quasiMeasurePreserving_hasPDF Mathlib.Probability.Density
∀ {Ω : Type u_1} {E : Type u_2} [inst : MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E} {F : Type u_3} [inst_1 : MeasurableSpace F] {ν : MeasureTheory.Measure F} {X : Ω → E} [inst_2 : MeasureTheory.HasPDF X ℙ μ], AEMeasurable X ℙ → ∀ {g : E → F}, MeasureTheory.Measure.QuasiMeasurePreserving g μ ν → (MeasureTheory.Measure.map g (MeasureTheory.Measure.map X ℙ)).HaveLebesgueDecomposition ν → MeasureTheory.HasPDF (g ∘ X) ℙ ν - ProbabilityTheory.IdentDistrib.mk Mathlib.Probability.IdentDistrib
∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] {f : α → γ} {g : β → γ} {μ : autoParam (MeasureTheory.Measure α) _auto✝} {ν : autoParam (MeasureTheory.Measure β) _auto✝¹}, AEMeasurable f μ → AEMeasurable g ν → MeasureTheory.Measure.map f μ = MeasureTheory.Measure.map g ν → ProbabilityTheory.IdentDistrib f g μ ν - ProbabilityTheory.IdentDistrib.comp_of_aemeasurable Mathlib.Probability.IdentDistrib
∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [inst : MeasurableSpace α] [inst_1 : MeasurableSpace β] [inst_2 : MeasurableSpace γ] [inst_3 : MeasurableSpace δ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} {u : γ → δ}, ProbabilityTheory.IdentDistrib f g μ ν → AEMeasurable u (MeasureTheory.Measure.map f μ) → ProbabilityTheory.IdentDistrib (u ∘ f) (u ∘ g) μ ν - MeasureTheory.Integrable.condDistrib_ae_map Mathlib.Probability.Kernel.CondDistrib
∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] [inst_2 : Nonempty Ω] [inst_3 : NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F}, AEMeasurable Y μ → MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ) → ∀ᵐ (b : β) ∂MeasureTheory.Measure.map X μ, MeasureTheory.Integrable (fun ω => f (b, ω)) ((ProbabilityTheory.condDistrib Y X μ) b) - MeasureTheory.Integrable.condDistrib_ae Mathlib.Probability.Kernel.CondDistrib
∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] [inst_2 : Nonempty Ω] [inst_3 : NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F}, AEMeasurable X μ → AEMeasurable Y μ → MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ) → ∀ᵐ (a : α) ∂μ, MeasureTheory.Integrable (fun ω => f (X a, ω)) ((ProbabilityTheory.condDistrib Y X μ) (X a)) - MeasureTheory.Integrable.integral_condDistrib_map Mathlib.Probability.Kernel.CondDistrib
∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] [inst_2 : Nonempty Ω] [inst_3 : NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} [inst_5 : NormedSpace ℝ F], AEMeasurable Y μ → MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ) → MeasureTheory.Integrable (fun x => ∫ (y : Ω), f (x, y) ∂(ProbabilityTheory.condDistrib Y X μ) x) (MeasureTheory.Measure.map X μ) - MeasureTheory.Integrable.integral_condDistrib Mathlib.Probability.Kernel.CondDistrib
∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] [inst_2 : Nonempty Ω] [inst_3 : NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} [inst_5 : NormedSpace ℝ F], AEMeasurable X μ → AEMeasurable Y μ → MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ) → MeasureTheory.Integrable (fun a => ∫ (y : Ω), f (X a, y) ∂(ProbabilityTheory.condDistrib Y X μ) (X a)) μ - MeasureTheory.Integrable.integral_norm_condDistrib_map Mathlib.Probability.Kernel.CondDistrib
∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] [inst_2 : Nonempty Ω] [inst_3 : NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F}, AEMeasurable Y μ → MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ) → MeasureTheory.Integrable (fun x => ∫ (y : Ω), ‖f (x, y)‖ ∂(ProbabilityTheory.condDistrib Y X μ) x) (MeasureTheory.Measure.map X μ) - MeasureTheory.Integrable.norm_integral_condDistrib_map Mathlib.Probability.Kernel.CondDistrib
∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] [inst_2 : Nonempty Ω] [inst_3 : NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} [inst_5 : NormedSpace ℝ F], AEMeasurable Y μ → MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ) → MeasureTheory.Integrable (fun x => ‖∫ (y : Ω), f (x, y) ∂(ProbabilityTheory.condDistrib Y X μ) x‖) (MeasureTheory.Measure.map X μ) - MeasureTheory.Integrable.integral_norm_condDistrib Mathlib.Probability.Kernel.CondDistrib
∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] [inst_2 : Nonempty Ω] [inst_3 : NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F}, AEMeasurable X μ → AEMeasurable Y μ → MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ) → MeasureTheory.Integrable (fun a => ∫ (y : Ω), ‖f (X a, y)‖ ∂(ProbabilityTheory.condDistrib Y X μ) (X a)) μ - MeasureTheory.Integrable.norm_integral_condDistrib Mathlib.Probability.Kernel.CondDistrib
∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] [inst_2 : Nonempty Ω] [inst_3 : NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} [inst_5 : NormedSpace ℝ F], AEMeasurable X μ → AEMeasurable Y μ → MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ) → MeasureTheory.Integrable (fun a => ‖∫ (y : Ω), f (X a, y) ∂(ProbabilityTheory.condDistrib Y X μ) (X a)‖) μ - MeasureTheory.AEStronglyMeasurable.integral_condDistrib_map Mathlib.Probability.Kernel.CondDistrib
∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] [inst_2 : Nonempty Ω] [inst_3 : NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} [inst_5 : NormedSpace ℝ F], AEMeasurable Y μ → MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ) → MeasureTheory.AEStronglyMeasurable (fun x => ∫ (y : Ω), f (x, y) ∂(ProbabilityTheory.condDistrib Y X μ) x) (MeasureTheory.Measure.map X μ) - MeasureTheory.AEStronglyMeasurable.integral_condDistrib Mathlib.Probability.Kernel.CondDistrib
∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] [inst_2 : Nonempty Ω] [inst_3 : NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} [inst_5 : NormedSpace ℝ F], AEMeasurable X μ → AEMeasurable Y μ → MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ) → MeasureTheory.AEStronglyMeasurable (fun a => ∫ (y : Ω), f (X a, y) ∂(ProbabilityTheory.condDistrib Y X μ) (X a)) μ - ProbabilityTheory.aestronglyMeasurable'_integral_condDistrib Mathlib.Probability.Kernel.CondDistrib
∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] [inst_2 : Nonempty Ω] [inst_3 : NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} [inst_5 : NormedSpace ℝ F], AEMeasurable X μ → AEMeasurable Y μ → MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ) → MeasureTheory.AEStronglyMeasurable' (MeasurableSpace.comap X mβ) (fun a => ∫ (y : Ω), f (X a, y) ∂(ProbabilityTheory.condDistrib Y X μ) (X a)) μ - ProbabilityTheory.condexp_prod_ae_eq_integral_condDistrib' Mathlib.Probability.Kernel.CondDistrib
∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] [inst_2 : Nonempty Ω] [inst_3 : NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} [inst_5 : NormedSpace ℝ F] [inst_6 : CompleteSpace F], Measurable X → AEMeasurable Y μ → MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ) → μ.ae.EventuallyEq (MeasureTheory.condexp (MeasurableSpace.comap X mβ) μ fun a => f (X a, Y a)) fun a => ∫ (y : Ω), f (X a, y) ∂(ProbabilityTheory.condDistrib Y X μ) (X a) - ProbabilityTheory.condexp_prod_ae_eq_integral_condDistrib₀ Mathlib.Probability.Kernel.CondDistrib
∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] [inst_2 : Nonempty Ω] [inst_3 : NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F} [inst_5 : NormedSpace ℝ F] [inst_6 : CompleteSpace F], Measurable X → AEMeasurable Y μ → MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ) → MeasureTheory.Integrable (fun a => f (X a, Y a)) μ → μ.ae.EventuallyEq (MeasureTheory.condexp (MeasurableSpace.comap X mβ) μ fun a => f (X a, Y a)) fun a => ∫ (y : Ω), f (X a, y) ∂(ProbabilityTheory.condDistrib Y X μ) (X a) - MeasureTheory.AEStronglyMeasurable.ae_integrable_condDistrib_map_iff Mathlib.Probability.Kernel.CondDistrib
∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] [inst_2 : Nonempty Ω] [inst_3 : NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst_4 : MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F}, AEMeasurable Y μ → MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ) → ((∀ᵐ (a : β) ∂MeasureTheory.Measure.map X μ, MeasureTheory.Integrable (fun ω => f (a, ω)) ((ProbabilityTheory.condDistrib Y X μ) a)) ∧ MeasureTheory.Integrable (fun a => ∫ (ω : Ω), ‖f (a, ω)‖ ∂(ProbabilityTheory.condDistrib Y X μ) a) (MeasureTheory.Measure.map X μ) ↔ MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ)) - AEMeasurable.comp_aemeasurable' Mathlib.Tactic.FunProp.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
Did you maybe mean
About
Loogle searches of Lean and Mathlib definitions and theorems.
You may also want to try the CLI version, the VS
Code extension, the lean.nvim
integration or the Zulip bot.
Usage
Loogle finds definitions and lemmas in various ways:
By constant:
🔍Real.sin
finds all lemmas whose statement somehow mentions the sine function.By lemma name substring:
🔍"differ"
finds all lemmas that have"differ"
somewhere in their lemma name.By subexpression:
🔍_ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.The pattern can also be non-linear, as in
🔍Real.sqrt ?a * Real.sqrt ?a
If the pattern has parameters, they are matched in any order. Both of these will find
List.map
:
🔍(?a -> ?b) -> List ?a -> List ?b
🔍List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍|- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all→
and∀
) has the given shape.As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍|- _ < _ → tsum _ < tsum _
will findtsum_lt_tsum
even though the hypothesisf i < g i
is not the last.
If you pass more than one such search filter, separated by commas
Loogle will return lemmas which match all of them. The
search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
woould find all lemmas which mention the constants Real.sin
and tsum
, have "two"
as a substring of the
lemma name, include a product and a power somewhere in the type,
and have a hypothesis of the form _ < _
(if
there were any such lemmas). Metavariables (?a
) are
assigned independently in each filter.
The #lucky
button will directly send you to the
documentation of the first hit.
Source code
You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is currently provided by Joachim Breitner <mail@joachim-breitner.de>.
This is Loogle revision fa2ddf5
serving mathlib revision 38aa2fc