{"name":"MeasureTheory.Measure.map_prod_comap_swap","declaration":"/-- The law of $(X, Z)$ is the image of the law of $(Z,X)$.-/\ntheorem MeasureTheory.Measure.map_prod_comap_swap {Ω : Type u_1} {α : Type u_2} {γ : Type u_4} [MeasurableSpace Ω] [MeasurableSpace α] [MeasurableSpace γ] {X : Ω → α} {Z : Ω → γ} (hX : Measurable X) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) : MeasureTheory.Measure.comap Prod.swap (MeasureTheory.Measure.map (fun ω => (X ω, Z ω)) μ) =\n  MeasureTheory.Measure.map (fun ω => (Z ω, X ω)) μ"}
{"name":"MeasureTheory.Measure.prod_apply_singleton","declaration":"theorem MeasureTheory.Measure.prod_apply_singleton {α : Type u_5} {β : Type u_6} : ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace β} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure β)\n  [inst : MeasureTheory.SigmaFinite ν] (x_2 : α × β),\n  ↑↑(MeasureTheory.Measure.prod μ ν) {x_2} = ↑↑μ {x_2.1} * ↑↑ν {x_2.2}"}
{"name":"MeasureTheory.Measure.prod_of_full_measure_finset","declaration":"theorem MeasureTheory.Measure.prod_of_full_measure_finset {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SigmaFinite ν] {A : Finset α} {B : Finset β} (hA : ↑↑μ (↑A)ᶜ = 0) (hB : ↑↑ν (↑B)ᶜ = 0) : ↑↑(MeasureTheory.Measure.prod μ ν) (↑(A ×ˢ B))ᶜ = 0"}