Documentation

Mathlib.Probability.Kernel.WithDensity

With Density #

For an s-finite kernel κ : Kernel α β and a function f : α → β → ℝ≥0∞ which is finite everywhere, we define withDensity κ f as the kernel a ↦ (κ a).withDensity (f a). This is an s-finite kernel.

Main definitions #

Main statements #

noncomputable def ProbabilityTheory.Kernel.withDensity {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (f : αβENNReal) :

Kernel with image (κ a).withDensity (f a) if Function.uncurry f is measurable, and with image 0 otherwise. If Function.uncurry f is measurable, it satisfies ∫⁻ b, g b ∂(withDensity κ f hf a) = ∫⁻ b, f a b * g b ∂(κ a).

Equations
  • κ.withDensity f = if hf : Measurable (Function.uncurry f) then { toFun := fun (a : α) => (κ a).withDensity (f a), measurable' := } else 0
Instances For
    theorem ProbabilityTheory.Kernel.withDensity_of_not_measurable {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : αβENNReal} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (hf : ¬Measurable (Function.uncurry f)) :
    κ.withDensity f = 0
    theorem ProbabilityTheory.Kernel.withDensity_apply {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : αβENNReal} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (hf : Measurable (Function.uncurry f)) (a : α) :
    (κ.withDensity f) a = (κ a).withDensity (f a)
    theorem ProbabilityTheory.Kernel.withDensity_apply' {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : αβENNReal} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (hf : Measurable (Function.uncurry f)) (a : α) (s : Set β) :
    ((κ.withDensity f) a) s = ∫⁻ (b : β) in s, f a bκ a
    theorem ProbabilityTheory.Kernel.withDensity_congr_ae {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] {f : αβENNReal} {g : αβENNReal} (hf : Measurable (Function.uncurry f)) (hg : Measurable (Function.uncurry g)) (hfg : ∀ (a : α), f a =ᵐ[κ a] g a) :
    κ.withDensity f = κ.withDensity g
    theorem ProbabilityTheory.Kernel.withDensity_absolutelyContinuous {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] (f : αβENNReal) (a : α) :
    ((κ.withDensity f) a).AbsolutelyContinuous (κ a)
    @[simp]
    theorem ProbabilityTheory.Kernel.withDensity_one {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] :
    κ.withDensity 1 = κ
    @[simp]
    theorem ProbabilityTheory.Kernel.withDensity_one' {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] :
    (κ.withDensity fun (x : α) (x : β) => 1) = κ
    @[simp]
    theorem ProbabilityTheory.Kernel.withDensity_zero {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] :
    κ.withDensity 0 = 0
    @[simp]
    theorem ProbabilityTheory.Kernel.withDensity_zero' {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] :
    (κ.withDensity fun (x : α) (x : β) => 0) = 0
    theorem ProbabilityTheory.Kernel.lintegral_withDensity {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : αβENNReal} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (hf : Measurable (Function.uncurry f)) (a : α) {g : βENNReal} (hg : Measurable g) :
    ∫⁻ (b : β), g b(κ.withDensity f) a = ∫⁻ (b : β), f a b * g bκ a
    theorem ProbabilityTheory.Kernel.integral_withDensity {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α β} {E : Type u_4} [NormedAddCommGroup E] [NormedSpace E] {f : βE} [ProbabilityTheory.IsSFiniteKernel κ] {a : α} {g : αβNNReal} (hg : Measurable (Function.uncurry g)) :
    ∫ (b : β), f b(κ.withDensity fun (a : α) (b : β) => (g a b)) a = ∫ (b : β), g a b f bκ a
    theorem ProbabilityTheory.Kernel.withDensity_add_left {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) (η : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] [ProbabilityTheory.IsSFiniteKernel η] (f : αβENNReal) :
    (κ + η).withDensity f = κ.withDensity f + η.withDensity f
    theorem ProbabilityTheory.Kernel.withDensity_kernel_sum {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Countable ι] (κ : ιProbabilityTheory.Kernel α β) (hκ : ∀ (i : ι), ProbabilityTheory.IsSFiniteKernel (κ i)) (f : αβENNReal) :
    (ProbabilityTheory.Kernel.sum κ).withDensity f = ProbabilityTheory.Kernel.sum fun (i : ι) => (κ i).withDensity f
    theorem ProbabilityTheory.Kernel.withDensity_add_right {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {f : αβENNReal} {g : αβENNReal} (hf : Measurable (Function.uncurry f)) (hg : Measurable (Function.uncurry g)) :
    κ.withDensity (f + g) = κ.withDensity f + κ.withDensity g
    theorem ProbabilityTheory.Kernel.withDensity_sub_add_cancel {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {f : αβENNReal} {g : αβENNReal} (hf : Measurable (Function.uncurry f)) (hg : Measurable (Function.uncurry g)) (hfg : ∀ (a : α), g a ≤ᵐ[κ a] f a) :
    (κ.withDensity fun (a : α) (x : β) => f a x - g a x) + κ.withDensity g = κ.withDensity f
    theorem ProbabilityTheory.Kernel.withDensity_tsum {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Countable ι] (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] {f : ιαβENNReal} (hf : ∀ (i : ι), Measurable (Function.uncurry (f i))) :
    κ.withDensity (∑' (n : ι), f n) = ProbabilityTheory.Kernel.sum fun (n : ι) => κ.withDensity (f n)
    theorem ProbabilityTheory.Kernel.isFiniteKernel_withDensity_of_bounded {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : αβENNReal} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsFiniteKernel κ] {B : ENNReal} (hB_top : B ) (hf_B : ∀ (a : α) (b : β), f a b B) :

    If a kernel κ is finite and a function f : α → β → ℝ≥0∞ is bounded, then withDensity κ f is finite.

    theorem ProbabilityTheory.Kernel.isSFiniteKernel_withDensity_of_isFiniteKernel {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : αβENNReal} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsFiniteKernel κ] (hf_ne_top : ∀ (a : α) (b : β), f a b ) :

    Auxiliary lemma for IsSFiniteKernel.withDensity. If a kernel κ is finite, then withDensity κ f is s-finite.

    theorem ProbabilityTheory.Kernel.IsSFiniteKernel.withDensity {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {f : αβENNReal} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (hf_ne_top : ∀ (a : α) (b : β), f a b ) :

    For an s-finite kernel κ and a function f : α → β → ℝ≥0∞ which is everywhere finite, withDensity κ f is s-finite.

    instance ProbabilityTheory.Kernel.instIsSFiniteKernelWithDensityOfNNReal {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] (f : αβNNReal) :
    ProbabilityTheory.IsSFiniteKernel (κ.withDensity fun (a : α) (b : β) => (f a b))

    For an s-finite kernel κ and a function f : α → β → ℝ≥0, withDensity κ f is s-finite.

    Equations
    • =
    theorem ProbabilityTheory.Kernel.withDensity_mul {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {f : αβNNReal} {g : αβENNReal} (hf : Measurable (Function.uncurry f)) (hg : Measurable (Function.uncurry g)) :
    (κ.withDensity fun (a : α) (x : β) => (f a x) * g a x) = (κ.withDensity fun (a : α) (x : β) => (f a x)).withDensity g