File: Mathlib/Probability/Kernel/MeasurableIntegral.lean

full_name: ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left_of_finite
code: /-- This is an auxiliary lemma for `measurable_kernel_prod_mk_left`. -/
theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : MeasurableSet t)
    (hκs : ∀ a, IsFiniteMeasure (κ a)) : Measurable fun a => κ a (Prod.mk a ⁻¹' t) 
goal: Measurable fun a => (κ a) (Prod.mk a ⁻¹' t)

full_name: ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left
code: theorem measurable_kernel_prod_mk_left [IsSFiniteKernel κ] {t : Set (α × β)}
    (ht : MeasurableSet t) : Measurable fun a => κ a (Prod.mk a ⁻¹' t) 
goal: Measurable fun a => (κ a) (Prod.mk a ⁻¹' t)

full_name: ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left'
code: theorem measurable_kernel_prod_mk_left' [IsSFiniteKernel η] {s : Set (β × γ)} (hs : MeasurableSet s)
    (a : α) : Measurable fun b => η (a, b) (Prod.mk b ⁻¹' s) 
goal: Measurable fun b => (η (a, b)) (Prod.mk b ⁻¹' s)

full_name: ProbabilityTheory.Kernel.measurable_kernel_prod_mk_right
code: theorem measurable_kernel_prod_mk_right [IsSFiniteKernel κ] {s : Set (β × α)}
    (hs : MeasurableSet s) : Measurable fun y => κ y ((fun x => (x, y)) ⁻¹' s) 
goal: Measurable fun y => (κ y) ((fun x => (x, y)) ⁻¹' s)

full_name: ProbabilityTheory.Kernel.measurable_lintegral_indicator_const
code: /-- Auxiliary lemma for `Measurable.lintegral_kernel_prod_right`. -/
theorem Kernel.measurable_lintegral_indicator_const {t : Set (α × β)} (ht : MeasurableSet t)
    (c : ℝ≥0∞) : Measurable fun a => ∫⁻ b, t.indicator (Function.const (α × β) c) (a, b) ∂κ a 
goal: Measurable fun a => ∫⁻ (b : β), t.indicator (Function.const (α × β) c) (a, b) ∂κ a

full_name: Measurable.lintegral_kernel_prod_right
code: /-- For an s-finite kernel `κ` and a function `f : α → β → ℝ≥0∞` which is measurable when seen as a
map from `α × β` (hypothesis `Measurable (uncurry f)`), the integral `a ↦ ∫⁻ b, f a b ∂(κ a)` is
measurable. -/
theorem _root_.Measurable.lintegral_kernel_prod_right {f : α → β → ℝ≥0∞}
    (hf : Measurable (uncurry f)) : Measurable fun a => ∫⁻ b, f a b ∂κ a 
goal: Measurable fun a => ∫⁻ (b : β), f a b ∂κ a

full_name: Measurable.lintegral_kernel_prod_right'
code: theorem _root_.Measurable.lintegral_kernel_prod_right' {f : α × β → ℝ≥0∞} (hf : Measurable f) :
    Measurable fun a => ∫⁻ b, f (a, b) ∂κ a 
goal: Measurable fun a => ∫⁻ (b : β), f (a, b) ∂κ a

full_name: Measurable.lintegral_kernel_prod_right''
code: theorem _root_.Measurable.lintegral_kernel_prod_right'' {f : β × γ → ℝ≥0∞} (hf : Measurable f) :
    Measurable fun x => ∫⁻ y, f (x, y) ∂η (a, x) 
goal: Measurable fun x => ∫⁻ (y : γ), f (x, y) ∂η (a, x)

full_name: Measurable.setLIntegral_kernel_prod_right
code: theorem _root_.Measurable.setLIntegral_kernel_prod_right {f : α → β → ℝ≥0∞}
    (hf : Measurable (uncurry f)) {s : Set β} (hs : MeasurableSet s) :
    Measurable fun a => ∫⁻ b in s, f a b ∂κ a 
goal: Measurable fun a => ∫⁻ (b : β) in s, f a b ∂κ a

full_name: Measurable.lintegral_kernel_prod_left'
code: theorem _root_.Measurable.lintegral_kernel_prod_left' {f : β × α → ℝ≥0∞} (hf : Measurable f) :
    Measurable fun y => ∫⁻ x, f (x, y) ∂κ y 
goal: Measurable fun y => ∫⁻ (x : β), f (x, y) ∂κ y

full_name: Measurable.lintegral_kernel_prod_left
code: theorem _root_.Measurable.lintegral_kernel_prod_left {f : β → α → ℝ≥0∞}
    (hf : Measurable (uncurry f)) : Measurable fun y => ∫⁻ x, f x y ∂κ y 
goal: Measurable fun y => ∫⁻ (x : β), f x y ∂κ y

full_name: Measurable.setLIntegral_kernel_prod_left
code: theorem _root_.Measurable.setLIntegral_kernel_prod_left {f : β → α → ℝ≥0∞}
    (hf : Measurable (uncurry f)) {s : Set β} (hs : MeasurableSet s) :
    Measurable fun b => ∫⁻ a in s, f a b ∂κ b 
goal: Measurable fun b => ∫⁻ (a : β) in s, f a b ∂κ b

full_name: Measurable.lintegral_kernel
code: theorem _root_.Measurable.lintegral_kernel {f : β → ℝ≥0∞} (hf : Measurable f) :
    Measurable fun a => ∫⁻ b, f b ∂κ a 
goal: Measurable fun a => ∫⁻ (b : β), f b ∂κ a

full_name: Measurable.setLIntegral_kernel
code: theorem _root_.Measurable.setLIntegral_kernel {f : β → ℝ≥0∞} (hf : Measurable f) {s : Set β}
    (hs : MeasurableSet s) : Measurable fun a => ∫⁻ b in s, f b ∂κ a 
goal: Measurable fun a => ∫⁻ (b : β) in s, f b ∂κ a

full_name: ProbabilityTheory.measurableSet_kernel_integrable
code: theorem measurableSet_kernel_integrable ⦃f : α → β → E⦄ (hf : StronglyMeasurable (uncurry f)) :
    MeasurableSet {x | Integrable (f x) (κ x)} 
goal: MeasurableSet {x | MeasureTheory.Integrable (f x) (κ x)}

full_name: MeasureTheory.StronglyMeasurable.integral_kernel_prod_right
code: theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
    (hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun x => ∫ y, f x y ∂κ x 
goal: MeasureTheory.StronglyMeasurable fun x => ∫ (y : β), f x y ∂κ x

full_name: MeasureTheory.StronglyMeasurable.integral_kernel_prod_right'
code: theorem StronglyMeasurable.integral_kernel_prod_right' ⦃f : α × β → E⦄ (hf : StronglyMeasurable f) :
    StronglyMeasurable fun x => ∫ y, f (x, y) ∂κ x 
goal: MeasureTheory.StronglyMeasurable fun x => ∫ (y : β), f (x, y) ∂κ x

full_name: MeasureTheory.StronglyMeasurable.integral_kernel_prod_right''
code: theorem StronglyMeasurable.integral_kernel_prod_right'' {f : β × γ → E}
    (hf : StronglyMeasurable f) : StronglyMeasurable fun x => ∫ y, f (x, y) ∂η (a, x) 
goal: MeasureTheory.StronglyMeasurable fun x => ∫ (y : γ), f (x, y) ∂η (a, x)

full_name: MeasureTheory.StronglyMeasurable.integral_kernel_prod_left
code: theorem StronglyMeasurable.integral_kernel_prod_left ⦃f : β → α → E⦄
    (hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun y => ∫ x, f x y ∂κ y 
goal: MeasureTheory.StronglyMeasurable fun y => ∫ (x : β), f x y ∂κ y

full_name: MeasureTheory.StronglyMeasurable.integral_kernel_prod_left'
code: theorem StronglyMeasurable.integral_kernel_prod_left' ⦃f : β × α → E⦄ (hf : StronglyMeasurable f) :
    StronglyMeasurable fun y => ∫ x, f (x, y) ∂κ y 
goal: MeasureTheory.StronglyMeasurable fun y => ∫ (x : β), f (x, y) ∂κ y

full_name: MeasureTheory.StronglyMeasurable.integral_kernel_prod_left''
code: theorem StronglyMeasurable.integral_kernel_prod_left'' {f : γ × β → E} (hf : StronglyMeasurable f) :
    StronglyMeasurable fun y => ∫ x, f (x, y) ∂η (a, y) 
goal: MeasureTheory.StronglyMeasurable fun y => ∫ (x : γ), f (x, y) ∂η (a, y)