Documentation

Mathlib.Probability.Kernel.Defs

Markov Kernels #

A kernel from a measurable space α to another measurable space β is a measurable map α → MeasureTheory.Measure β, where the measurable space instance on measure β is the one defined in MeasureTheory.Measure.instMeasurableSpace. That is, a kernel κ verifies that for all measurable sets s of β, a ↦ κ a s is measurable.

Main definitions #

Classes of kernels:

ProbabilityTheory.Kernel α β: kernels from α to β.
ProbabilityTheory.IsMarkovKernel κ: a kernel from α to β is said to be a Markov kernel if for all a : α, k a is a probability measure.
ProbabilityTheory.IsZeroOrMarkovKernel κ: a kernel from α to β which is zero or a Markov kernel.
ProbabilityTheory.IsFiniteKernel κ: a kernel from α to β is said to be finite if there exists C : ℝ≥0∞ such that C < ∞ and for all a : α, κ a univ ≤ C. This implies in particular that all measures in the image of κ are finite, but is stronger since it requires a uniform bound. This stronger condition is necessary to ensure that the composition of two finite kernels is finite.
ProbabilityTheory.IsSFiniteKernel κ: a kernel is called s-finite if it is a countable sum of finite kernels.

Main statements #

ProbabilityTheory.Kernel.ext_fun: if ∫⁻ b, f b ∂(κ a) = ∫⁻ b, f b ∂(η a) for all measurable functions f and all a, then the two kernels κ and η are equal.

structure ProbabilityTheory.Kernel (α : Type u_1) (β : Type u_2) [MeasurableSpace α] [MeasurableSpace β] :

Type (max u_1 u_2)

A kernel from a measurable space α to another measurable space β is a measurable function κ : α → Measure β. The measurable space structure on MeasureTheory.Measure β is given by MeasureTheory.Measure.instMeasurableSpace. A map κ : α → MeasureTheory.Measure β is measurable iff ∀ s : Set β, MeasurableSet s → Measurable (fun a ↦ κ a s).

toFun : α → MeasureTheory.Measure β
The underlying function of a kernel.
Do not use this function directly. Instead use the coercion coming from the DFunLike instance.
measurable' : Measurable self.toFun
A kernel is a measurable map.
Do not use this lemma directly. Use Kernel.measurable instead.

Instances For

def ProbabilityTheory.«termKernel[_]__» :

Lean.ParserDescr

Notation for Kernel with respect to a non-standard σ-algebra in the domain.

Equations

One or more equations did not get rendered due to their size.

Instances For

def ProbabilityTheory.«termKernel[_,_]__» :

Lean.ParserDescr

Notation for Kernel with respect to a non-standard σ-algebra in the domain and codomain.

Equations

One or more equations did not get rendered due to their size.

Instances For

instance ProbabilityTheory.Kernel.instFunLike {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} :

FunLike (Kernel α β) α (MeasureTheory.Measure β)

Equations

ProbabilityTheory.Kernel.instFunLike = { coe := ProbabilityTheory.Kernel.toFun, coe_injective' := ⋯ }

theorem ProbabilityTheory.Kernel.measurable {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) :

Measurable ⇑κ

theorem ProbabilityTheory.Kernel.aemeasurable {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) {μ : MeasureTheory.Measure α} :

AEMeasurable (⇑κ) μ

@[simp]

theorem ProbabilityTheory.Kernel.coe_mk {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (f : α → MeasureTheory.Measure β) (hf : Measurable f) :

⇑{ toFun := f, measurable' := hf } = f

instance ProbabilityTheory.Kernel.instZero {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} :

Zero (Kernel α β)

Equations

ProbabilityTheory.Kernel.instZero = { zero := { toFun := 0, measurable' := ⋯ } }

noncomputable instance ProbabilityTheory.Kernel.instAdd {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} :

Add (Kernel α β)

Equations

ProbabilityTheory.Kernel.instAdd = { add := fun (κ η : ProbabilityTheory.Kernel α β) => { toFun := ⇑κ + ⇑η, measurable' := ⋯ } }

noncomputable instance ProbabilityTheory.Kernel.instSMulNat {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} :

SMul ℕ (Kernel α β)

Equations

ProbabilityTheory.Kernel.instSMulNat = { smul := fun (n : ℕ) (κ : ProbabilityTheory.Kernel α β) => { toFun := n • ⇑κ, measurable' := ⋯ } }

@[simp]

theorem ProbabilityTheory.Kernel.coe_zero {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} :

⇑0 = 0

@[simp]

theorem ProbabilityTheory.Kernel.coe_add {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ η : Kernel α β) :

⇑(κ + η) = ⇑κ + ⇑η

@[simp]

theorem ProbabilityTheory.Kernel.coe_nsmul {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (n : ℕ) (κ : Kernel α β) :

⇑(n • κ) = n • ⇑κ

@[simp]

theorem ProbabilityTheory.Kernel.zero_apply {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (a : α) :

0 a = 0

@[simp]

theorem ProbabilityTheory.Kernel.add_apply {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ η : Kernel α β) (a : α) :

(κ + η) a = κ a + η a

@[simp]

theorem ProbabilityTheory.Kernel.nsmul_apply {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (n : ℕ) (κ : Kernel α β) (a : α) :

(n • κ) a = n • κ a

noncomputable instance ProbabilityTheory.Kernel.instAddCommMonoid {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} :

AddCommMonoid (Kernel α β)

Equations

ProbabilityTheory.Kernel.instAddCommMonoid = Function.Injective.addCommMonoid (fun (f : ProbabilityTheory.Kernel α β) => ⇑f) ⋯ ⋯ ⋯ ⋯

instance ProbabilityTheory.Kernel.instPartialOrder {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} :

PartialOrder (Kernel α β)

Equations

ProbabilityTheory.Kernel.instPartialOrder = PartialOrder.lift (fun (f : ProbabilityTheory.Kernel α β) => ⇑f) ⋯

instance ProbabilityTheory.Kernel.instAddLeftMono {α : Type u_4} {β : Type u_5} [MeasurableSpace α] [MeasurableSpace β] :

AddLeftMono (Kernel α β)

noncomputable instance ProbabilityTheory.Kernel.instOrderBot {α : Type u_4} {β : Type u_5} [MeasurableSpace α] [MeasurableSpace β] :

OrderBot (Kernel α β)

Equations

ProbabilityTheory.Kernel.instOrderBot = { bot := 0, bot_le := ⋯ }

noncomputable def ProbabilityTheory.Kernel.coeAddHom (α : Type u_4) (β : Type u_5) [MeasurableSpace α] [MeasurableSpace β] :

Kernel α β →+ α → MeasureTheory.Measure β

Coercion to a function as an additive monoid homomorphism.

Equations

ProbabilityTheory.Kernel.coeAddHom α β = { toFun := DFunLike.coe, map_zero' := ⋯, map_add' := ⋯ }

Instances For

@[simp]

theorem ProbabilityTheory.Kernel.coeAddHom_apply (α : Type u_4) (β : Type u_5) [MeasurableSpace α] [MeasurableSpace β] (κ : Kernel α β) :

(coeAddHom α β) κ = ⇑κ

@[simp]

theorem ProbabilityTheory.Kernel.coe_finset_sum {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (I : Finset ι) (κ : ι → Kernel α β) :

⇑(∑ i ∈ I, κ i) = ∑ i ∈ I, ⇑(κ i)

theorem ProbabilityTheory.Kernel.finset_sum_apply {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (I : Finset ι) (κ : ι → Kernel α β) (a : α) :

(∑ i ∈ I, κ i) a = ∑ i ∈ I, (κ i) a

theorem ProbabilityTheory.Kernel.finset_sum_apply' {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (I : Finset ι) (κ : ι → Kernel α β) (a : α) (s : Set β) :

((∑ i ∈ I, κ i) a) s = ∑ i ∈ I, ((κ i) a) s

class ProbabilityTheory.IsMarkovKernel {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) :

A kernel is a Markov kernel if every measure in its image is a probability measure.

isProbabilityMeasure (a : α) : MeasureTheory.IsProbabilityMeasure (κ a)

Instances

class ProbabilityTheory.IsZeroOrMarkovKernel {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) :

A class for kernels which are zero or a Markov kernel.

eq_zero_or_isMarkovKernel' : κ = 0 ∨ IsMarkovKernel κ

Instances

class ProbabilityTheory.IsFiniteKernel {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) :

A kernel is finite if every measure in its image is finite, with a uniform bound.

exists_univ_le : ∃ C < ⊤, ∀ (a : α), (κ a) Set.univ ≤ C

Instances

theorem ProbabilityTheory.eq_zero_or_isMarkovKernel {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) [h : IsZeroOrMarkovKernel κ] :

κ = 0 ∨ IsMarkovKernel κ

noncomputable def ProbabilityTheory.Kernel.bound {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) :

A constant C : ℝ≥0∞ such that C < ∞ for a finite kernel (ProbabilityTheory.IsFiniteKernel.bound_lt_top κ) and for all a : α and s : Set β, κ a s ≤ C (ProbabilityTheory.Kernel.measure_le_bound κ a s).

Equations

κ.bound = ⨆ (a : α), (κ a) Set.univ

Instances For

@[deprecated ProbabilityTheory.Kernel.bound (since := "2025-09-13")]

def ProbabilityTheory.IsFiniteKernel.bound {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) :

Alias of ProbabilityTheory.Kernel.bound.

A constant C : ℝ≥0∞ such that C < ∞ for a finite kernel (ProbabilityTheory.IsFiniteKernel.bound_lt_top κ) and for all a : α and s : Set β, κ a s ≤ C (ProbabilityTheory.Kernel.measure_le_bound κ a s).

Equations

@ProbabilityTheory.IsFiniteKernel.bound = @ProbabilityTheory.Kernel.bound

Instances For

theorem ProbabilityTheory.Kernel.bound_lt_top {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) [h : IsFiniteKernel κ] :

@[deprecated ProbabilityTheory.Kernel.bound_lt_top (since := "2025-09-13")]

theorem ProbabilityTheory.IsFiniteKernel.bound_lt_top {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) [h : IsFiniteKernel κ] :

Alias of ProbabilityTheory.Kernel.bound_lt_top.

theorem ProbabilityTheory.Kernel.bound_ne_top {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) [IsFiniteKernel κ] :

κ.bound ≠ ⊤

@[deprecated ProbabilityTheory.Kernel.bound_ne_top (since := "2025-09-13")]

theorem ProbabilityTheory.IsFiniteKernel.bound_ne_top {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) [IsFiniteKernel κ] :

κ.bound ≠ ⊤

Alias of ProbabilityTheory.Kernel.bound_ne_top.

theorem ProbabilityTheory.Kernel.measure_le_bound {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) (a : α) (s : Set β) :

(κ a) s ≤ κ.bound

@[simp]

theorem ProbabilityTheory.Kernel.bound_eq_zero_of_isEmpty {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [IsEmpty α] (κ : Kernel α β) :

κ.bound = 0

@[deprecated ProbabilityTheory.Kernel.bound_eq_zero_of_isEmpty (since := "2025-09-13")]

theorem ProbabilityTheory.IsFiniteKernel.bound_eq_zero_of_isEmpty {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [IsEmpty α] (κ : Kernel α β) :

κ.bound = 0

Alias of ProbabilityTheory.Kernel.bound_eq_zero_of_isEmpty.

@[simp]

theorem ProbabilityTheory.Kernel.bound_eq_zero_of_isEmpty' {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [IsEmpty β] (κ : Kernel α β) :

κ.bound = 0

@[deprecated ProbabilityTheory.Kernel.bound_eq_zero_of_isEmpty' (since := "2025-09-13")]

theorem ProbabilityTheory.IsFiniteKernel.bound_eq_zero_of_isEmpty' {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [IsEmpty β] (κ : Kernel α β) :

κ.bound = 0

Alias of ProbabilityTheory.Kernel.bound_eq_zero_of_isEmpty'.

@[simp]

theorem ProbabilityTheory.Kernel.bound_zero {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} :

@[deprecated ProbabilityTheory.Kernel.bound_zero (since := "2025-09-13")]

theorem ProbabilityTheory.IsFiniteKernel.bound_zero {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} :

Kernel.bound 0 = 0

Alias of ProbabilityTheory.Kernel.bound_zero.

instance ProbabilityTheory.isFiniteKernel_zero (α : Type u_4) (β : Type u_5) {x✝ : MeasurableSpace α} {x✝¹ : MeasurableSpace β} :

IsFiniteKernel 0

instance ProbabilityTheory.IsFiniteKernel.add {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ η : Kernel α β) [IsFiniteKernel κ] [IsFiniteKernel η] :

IsFiniteKernel (κ + η)

theorem ProbabilityTheory.isFiniteKernel_of_le {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ ν : Kernel α β} [hν : IsFiniteKernel ν] (hκν : κ ≤ ν) :

IsFiniteKernel κ

instance ProbabilityTheory.IsMarkovKernel.is_probability_measure' {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : Kernel α β} [IsMarkovKernel κ] (a : α) :

MeasureTheory.IsProbabilityMeasure (κ a)

instance ProbabilityTheory.instIsZeroOrMarkovKernelOfNatKernel {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} :

IsZeroOrMarkovKernel 0

@[instance 100]

instance ProbabilityTheory.IsMarkovKernel.IsZeroOrMarkovKernel {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : Kernel α β} [h : IsMarkovKernel κ] :

ProbabilityTheory.IsZeroOrMarkovKernel κ

@[instance 100]

instance ProbabilityTheory.IsZeroOrMarkovKernel.isZeroOrProbabilityMeasure {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : Kernel α β} [IsZeroOrMarkovKernel κ] (a : α) :

MeasureTheory.IsZeroOrProbabilityMeasure (κ a)

instance ProbabilityTheory.IsFiniteKernel.isFiniteMeasure {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : Kernel α β} [IsFiniteKernel κ] (a : α) :

MeasureTheory.IsFiniteMeasure (κ a)

@[instance 100]

instance ProbabilityTheory.IsZeroOrMarkovKernel.isFiniteKernel {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : Kernel α β} [h : IsZeroOrMarkovKernel κ] :

IsFiniteKernel κ

@[simp]

theorem ProbabilityTheory.Kernel.bound_eq_one {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Nonempty α] (κ : Kernel α β) [IsMarkovKernel κ] :

κ.bound = 1

@[deprecated ProbabilityTheory.Kernel.bound_eq_one (since := "2025-09-13")]

theorem ProbabilityTheory.IsMarkovKernel.bound_eq_one {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Nonempty α] (κ : Kernel α β) [IsMarkovKernel κ] :

κ.bound = 1

Alias of ProbabilityTheory.Kernel.bound_eq_one.

@[simp]

theorem ProbabilityTheory.Kernel.bound_le_one {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) [IsZeroOrMarkovKernel κ] :

@[deprecated ProbabilityTheory.Kernel.bound_le_one (since := "2025-09-13")]

theorem ProbabilityTheory.IsZeroOrMarkovKernel.bound_le_one {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) [IsZeroOrMarkovKernel κ] :

Alias of ProbabilityTheory.Kernel.bound_le_one.

theorem ProbabilityTheory.Kernel.ext {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ η : Kernel α β} (h : ∀ (a : α), κ a = η a) :

κ = η

theorem ProbabilityTheory.Kernel.ext_iff {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ η : Kernel α β} :

κ = η ↔ ∀ (a : α), κ a = η a

theorem ProbabilityTheory.Kernel.ext_iff' {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ η : Kernel α β} :

κ = η ↔ ∀ (a : α) (s : Set β), MeasurableSet s → (κ a) s = (η a) s

theorem ProbabilityTheory.Kernel.ext_fun {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ η : Kernel α β} (h : ∀ (a : α) (f : β → ENNReal), Measurable f → ∫⁻ (b : β), f b ∂κ a = ∫⁻ (b : β), f b ∂η a) :

κ = η

theorem ProbabilityTheory.Kernel.ext_fun_iff {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ η : Kernel α β} :

κ = η ↔ ∀ (a : α) (f : β → ENNReal), Measurable f → ∫⁻ (b : β), f b ∂κ a = ∫⁻ (b : β), f b ∂η a

instance ProbabilityTheory.Kernel.instSubsingletonOfIsEmpty {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [IsEmpty β] :

Subsingleton (Kernel α β)

instance ProbabilityTheory.Kernel.instIsMarkovKernelOfIsEmpty {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [IsEmpty α] (κ : Kernel α β) :

IsMarkovKernel κ

instance ProbabilityTheory.Kernel.instIsZeroOrMarkovKernelOfIsEmpty {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [IsEmpty β] (κ : Kernel α β) :

IsZeroOrMarkovKernel κ

theorem ProbabilityTheory.Kernel.not_isMarkovKernel_zero {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Nonempty α] :

¬IsMarkovKernel 0

theorem ProbabilityTheory.Kernel.measurable_coe {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) {s : Set β} (hs : MeasurableSet s) :

Measurable fun (a : α) => (κ a) s

theorem ProbabilityTheory.Kernel.apply_congr_of_mem_measurableAtom {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) {y' y : α} (hy' : y' ∈ measurableAtom y) :

κ y' = κ y

theorem ProbabilityTheory.Kernel.eq_zero_of_isEmpty_left {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) [h : IsEmpty α] :

κ = 0

theorem ProbabilityTheory.Kernel.eq_zero_of_isEmpty_right {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) [IsEmpty β] :

κ = 0

noncomputable def ProbabilityTheory.Kernel.sum {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Countable ι] (κ : ι → Kernel α β) :

Kernel α β

Sum of an indexed family of kernels.

Equations

ProbabilityTheory.Kernel.sum κ = { toFun := fun (a : α) => MeasureTheory.Measure.sum fun (n : ι) => (κ n) a, measurable' := ⋯ }

Instances For

theorem ProbabilityTheory.Kernel.sum_apply {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Countable ι] (κ : ι → Kernel α β) (a : α) :

(Kernel.sum κ) a = MeasureTheory.Measure.sum fun (n : ι) => (κ n) a

theorem ProbabilityTheory.Kernel.sum_apply' {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Countable ι] (κ : ι → Kernel α β) (a : α) {s : Set β} (hs : MeasurableSet s) :

((Kernel.sum κ) a) s = ∑' (n : ι), ((κ n) a) s

@[simp]

theorem ProbabilityTheory.Kernel.sum_zero {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Countable ι] :

(Kernel.sum fun (x : ι) => 0) = 0

theorem ProbabilityTheory.Kernel.sum_comm {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Countable ι] (κ : ι → ι → Kernel α β) :

(Kernel.sum fun (n : ι) => Kernel.sum (κ n)) = Kernel.sum fun (m : ι) => Kernel.sum fun (n : ι) => κ n m

@[simp]

theorem ProbabilityTheory.Kernel.sum_fintype {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Fintype ι] (κ : ι → Kernel α β) :

Kernel.sum κ = ∑ i : ι, κ i

theorem ProbabilityTheory.Kernel.sum_add {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Countable ι] (κ η : ι → Kernel α β) :

(Kernel.sum fun (n : ι) => κ n + η n) = Kernel.sum κ + Kernel.sum η

class ProbabilityTheory.IsSFiniteKernel {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) :

A kernel is s-finite if it can be written as the sum of countably many finite kernels.

tsum_finite : ∃ (κs : ℕ → Kernel α β), (∀ (n : ℕ), IsFiniteKernel (κs n)) ∧ κ = Kernel.sum κs

Instances

@[instance 100]

instance ProbabilityTheory.Kernel.IsFiniteKernel.isSFiniteKernel {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : Kernel α β} [h : IsFiniteKernel κ] :

IsSFiniteKernel κ

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

ℕ → Kernel α β

A sequence of finite kernels such that κ = ProbabilityTheory.Kernel.sum (seq κ). See ProbabilityTheory.Kernel.isFiniteKernel_seq and ProbabilityTheory.Kernel.kernel_sum_seq.

Equations

κ.seq = ⋯.choose

Instances For

theorem ProbabilityTheory.Kernel.kernel_sum_seq {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) [h : IsSFiniteKernel κ] :

Kernel.sum κ.seq = κ

theorem ProbabilityTheory.Kernel.measure_sum_seq {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) [h : IsSFiniteKernel κ] (a : α) :

(MeasureTheory.Measure.sum fun (n : ℕ) => (κ.seq n) a) = κ a

instance ProbabilityTheory.Kernel.isFiniteKernel_seq {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ : Kernel α β) [h : IsSFiniteKernel κ] (n : ℕ) :

IsFiniteKernel (κ.seq n)

instance ProbabilityTheory.IsSFiniteKernel.sFinite {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : Kernel α β} [IsSFiniteKernel κ] (a : α) :

MeasureTheory.SFinite (κ a)

instance ProbabilityTheory.Kernel.IsSFiniteKernel.add {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (κ η : Kernel α β) [IsSFiniteKernel κ] [IsSFiniteKernel η] :

IsSFiniteKernel (κ + η)

theorem ProbabilityTheory.Kernel.IsSFiniteKernel.finset_sum {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κs : ι → Kernel α β} (I : Finset ι) (h : ∀ i ∈ I, IsSFiniteKernel (κs i)) :

IsSFiniteKernel (∑ i ∈ I, κs i)

theorem ProbabilityTheory.Kernel.isSFiniteKernel_sum_of_denumerable {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Denumerable ι] {κs : ι → Kernel α β} (hκs : ∀ (n : ι), IsSFiniteKernel (κs n)) :

IsSFiniteKernel (Kernel.sum κs)

instance ProbabilityTheory.Kernel.isSFiniteKernel_sum {α : Type u_1} {β : Type u_2} {ι : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [Countable ι] {κs : ι → Kernel α β} [hκs : ∀ (n : ι), IsSFiniteKernel (κs n)] :

IsSFiniteKernel (Kernel.sum κs)