| {"name":"MeasureTheory.ProbabilityMeasure.map_prod_map","declaration":"theorem MeasureTheory.ProbabilityMeasure.map_prod_map {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) {α' : Type u_3} [MeasurableSpace α'] {β' : Type u_4} [MeasurableSpace β'] {f : α → α'} {g : β → β'} (f_mble : Measurable f) (g_mble : Measurable g) : MeasureTheory.ProbabilityMeasure.prod (MeasureTheory.ProbabilityMeasure.map μ ⋯)\n (MeasureTheory.ProbabilityMeasure.map ν ⋯) =\n MeasureTheory.ProbabilityMeasure.map (MeasureTheory.ProbabilityMeasure.prod μ ν) ⋯"} | |
| {"name":"MeasureTheory.FiniteMeasure.prod","declaration":"/-- The binary product of finite measures. -/\ndef MeasureTheory.FiniteMeasure.prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) : MeasureTheory.FiniteMeasure (α × β)"} | |
| {"name":"MeasureTheory.ProbabilityMeasure.prod_prod","declaration":"theorem MeasureTheory.ProbabilityMeasure.prod_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) (s : Set α) (t : Set β) : (fun s => (↑↑↑(MeasureTheory.ProbabilityMeasure.prod μ ν) s).toNNReal) (s ×ˢ t) =\n (fun s => (↑↑↑μ s).toNNReal) s * (fun s => (↑↑↑ν s).toNNReal) t"} | |
| {"name":"MeasureTheory.ProbabilityMeasure.prod_swap","declaration":"theorem MeasureTheory.ProbabilityMeasure.prod_swap {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) : MeasureTheory.ProbabilityMeasure.map (MeasureTheory.ProbabilityMeasure.prod μ ν) ⋯ =\n MeasureTheory.ProbabilityMeasure.prod ν μ"} | |
| {"name":"MeasureTheory.FiniteMeasure.measure_ae_null_of_prod_null","declaration":"theorem MeasureTheory.FiniteMeasure.measure_ae_null_of_prod_null {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) {s : Set (α × β)} (h : (fun s => (↑↑↑(MeasureTheory.FiniteMeasure.prod μ ν) s).toNNReal) s = 0) : (fun x => (fun s => (↑↑↑ν s).toNNReal) (Prod.mk x ⁻¹' s)) =ᶠ[MeasureTheory.Measure.ae ↑μ] 0"} | |
| {"name":"MeasureTheory.ProbabilityMeasure.toMeasure_prod","declaration":"theorem MeasureTheory.ProbabilityMeasure.toMeasure_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) : ↑(MeasureTheory.ProbabilityMeasure.prod μ ν) = MeasureTheory.Measure.prod ↑μ ↑ν"} | |
| {"name":"MeasureTheory.FiniteMeasure.mass_prod","declaration":"theorem MeasureTheory.FiniteMeasure.mass_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) : MeasureTheory.FiniteMeasure.mass (MeasureTheory.FiniteMeasure.prod μ ν) =\n MeasureTheory.FiniteMeasure.mass μ * MeasureTheory.FiniteMeasure.mass ν"} | |
| {"name":"MeasureTheory.FiniteMeasure.prod_swap","declaration":"theorem MeasureTheory.FiniteMeasure.prod_swap {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) : MeasureTheory.FiniteMeasure.map (MeasureTheory.FiniteMeasure.prod μ ν) Prod.swap = MeasureTheory.FiniteMeasure.prod ν μ"} | |
| {"name":"MeasureTheory.FiniteMeasure.map_fst_prod","declaration":"theorem MeasureTheory.FiniteMeasure.map_fst_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) : MeasureTheory.FiniteMeasure.map (MeasureTheory.FiniteMeasure.prod μ ν) Prod.fst =\n (fun s => (↑↑↑ν s).toNNReal) Set.univ • μ"} | |
| {"name":"MeasureTheory.ProbabilityMeasure.map_fst_prod","declaration":"theorem MeasureTheory.ProbabilityMeasure.map_fst_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) : MeasureTheory.ProbabilityMeasure.map (MeasureTheory.ProbabilityMeasure.prod μ ν) ⋯ = μ"} | |
| {"name":"MeasureTheory.ProbabilityMeasure.prod_apply_null","declaration":"theorem MeasureTheory.ProbabilityMeasure.prod_apply_null {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) {s : Set (α × β)} (hs : MeasurableSet s) : (fun s => (↑↑↑(MeasureTheory.ProbabilityMeasure.prod μ ν) s).toNNReal) s = 0 ↔\n (fun x => (fun s => (↑↑↑ν s).toNNReal) (Prod.mk x ⁻¹' s)) =ᶠ[MeasureTheory.Measure.ae ↑μ] 0"} | |
| {"name":"MeasureTheory.FiniteMeasure.prod_apply","declaration":"theorem MeasureTheory.FiniteMeasure.prod_apply {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) (s : Set (α × β)) (s_mble : MeasurableSet s) : (fun s => (↑↑↑(MeasureTheory.FiniteMeasure.prod μ ν) s).toNNReal) s = (∫⁻ (x : α), ↑↑↑ν (Prod.mk x ⁻¹' s) ∂↑μ).toNNReal"} | |
| {"name":"MeasureTheory.FiniteMeasure.prod_apply_symm","declaration":"theorem MeasureTheory.FiniteMeasure.prod_apply_symm {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) (s : Set (α × β)) (s_mble : MeasurableSet s) : (fun s => (↑↑↑(MeasureTheory.FiniteMeasure.prod μ ν) s).toNNReal) s =\n (∫⁻ (y : β), ↑↑↑μ ((fun x => (x, y)) ⁻¹' s) ∂↑ν).toNNReal"} | |
| {"name":"MeasureTheory.FiniteMeasure.prod_prod","declaration":"theorem MeasureTheory.FiniteMeasure.prod_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) (s : Set α) (t : Set β) : (fun s => (↑↑↑(MeasureTheory.FiniteMeasure.prod μ ν) s).toNNReal) (s ×ˢ t) =\n (fun s => (↑↑↑μ s).toNNReal) s * (fun s => (↑↑↑ν s).toNNReal) t"} | |
| {"name":"MeasureTheory.FiniteMeasure.prod_apply_null","declaration":"theorem MeasureTheory.FiniteMeasure.prod_apply_null {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) {s : Set (α × β)} (hs : MeasurableSet s) : (fun s => (↑↑↑(MeasureTheory.FiniteMeasure.prod μ ν) s).toNNReal) s = 0 ↔\n (fun x => (fun s => (↑↑↑ν s).toNNReal) (Prod.mk x ⁻¹' s)) =ᶠ[MeasureTheory.Measure.ae ↑μ] 0"} | |
| {"name":"MeasureTheory.ProbabilityMeasure.map_snd_prod","declaration":"theorem MeasureTheory.ProbabilityMeasure.map_snd_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) : MeasureTheory.ProbabilityMeasure.map (MeasureTheory.ProbabilityMeasure.prod μ ν) ⋯ = ν"} | |
| {"name":"MeasureTheory.ProbabilityMeasure.prod","declaration":"/-- The binary product of probability measures. -/\ndef MeasureTheory.ProbabilityMeasure.prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) : MeasureTheory.ProbabilityMeasure (α × β)"} | |
| {"name":"MeasureTheory.ProbabilityMeasure.prod_apply_symm","declaration":"theorem MeasureTheory.ProbabilityMeasure.prod_apply_symm {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) (s : Set (α × β)) (s_mble : MeasurableSet s) : (fun s => (↑↑↑(MeasureTheory.ProbabilityMeasure.prod μ ν) s).toNNReal) s =\n (∫⁻ (y : β), ↑↑↑μ ((fun x => (x, y)) ⁻¹' s) ∂↑ν).toNNReal"} | |
| {"name":"MeasureTheory.FiniteMeasure.map_snd_prod","declaration":"theorem MeasureTheory.FiniteMeasure.map_snd_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) : MeasureTheory.FiniteMeasure.map (MeasureTheory.FiniteMeasure.prod μ ν) Prod.snd =\n (fun s => (↑↑↑μ s).toNNReal) Set.univ • ν"} | |
| {"name":"MeasureTheory.ProbabilityMeasure.measure_ae_null_of_prod_null","declaration":"theorem MeasureTheory.ProbabilityMeasure.measure_ae_null_of_prod_null {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) {s : Set (α × β)} (h : (fun s => (↑↑↑(MeasureTheory.ProbabilityMeasure.prod μ ν) s).toNNReal) s = 0) : (fun x => (fun s => (↑↑↑ν s).toNNReal) (Prod.mk x ⁻¹' s)) =ᶠ[MeasureTheory.Measure.ae ↑μ] 0"} | |
| {"name":"MeasureTheory.ProbabilityMeasure.prod_apply","declaration":"theorem MeasureTheory.ProbabilityMeasure.prod_apply {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) (s : Set (α × β)) (s_mble : MeasurableSet s) : (fun s => (↑↑↑(MeasureTheory.ProbabilityMeasure.prod μ ν) s).toNNReal) s =\n (∫⁻ (x : α), ↑↑↑ν (Prod.mk x ⁻¹' s) ∂↑μ).toNNReal"} | |
| {"name":"MeasureTheory.FiniteMeasure.map_prod_map","declaration":"theorem MeasureTheory.FiniteMeasure.map_prod_map {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) {α' : Type u_3} [MeasurableSpace α'] {β' : Type u_4} [MeasurableSpace β'] {f : α → α'} {g : β → β'} (f_mble : Measurable f) (g_mble : Measurable g) : MeasureTheory.FiniteMeasure.prod (MeasureTheory.FiniteMeasure.map μ f) (MeasureTheory.FiniteMeasure.map ν g) =\n MeasureTheory.FiniteMeasure.map (MeasureTheory.FiniteMeasure.prod μ ν) (Prod.map f g)"} | |
| {"name":"MeasureTheory.FiniteMeasure.zero_prod","declaration":"theorem MeasureTheory.FiniteMeasure.zero_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (ν : MeasureTheory.FiniteMeasure β) : MeasureTheory.FiniteMeasure.prod 0 ν = 0"} | |
| {"name":"MeasureTheory.FiniteMeasure.toMeasure_prod","declaration":"theorem MeasureTheory.FiniteMeasure.toMeasure_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) : ↑(MeasureTheory.FiniteMeasure.prod μ ν) = MeasureTheory.Measure.prod ↑μ ↑ν"} | |
| {"name":"MeasureTheory.FiniteMeasure.prod_zero","declaration":"theorem MeasureTheory.FiniteMeasure.prod_zero {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) : MeasureTheory.FiniteMeasure.prod μ 0 = 0"} | |