fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
compProd_eq_tsum_compProd (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ)
[IsSFiniteKernel η] (a : α) (hs : MeasurableSet s) :
(κ ⊗ₖ η) a s = ∑' (n : ℕ) (m : ℕ), (seq κ n ⊗ₖ seq η m) a s := by
rw [compProd_eq_sum_compProd]
simp_rw [sum_apply' _ _ hs] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.Comp",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/CompProd.lean | compProd_eq_tsum_compProd | null |
IsMarkovKernel.compProd (κ : Kernel α β) [IsMarkovKernel κ] (η : Kernel (α × β) γ)
[IsMarkovKernel η] : IsMarkovKernel (κ ⊗ₖ η) where
isProbabilityMeasure a := ⟨by simp [compProd_apply]⟩ | instance | Probability | [
"Mathlib.Probability.Kernel.Composition.Comp",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/CompProd.lean | IsMarkovKernel.compProd | null |
IsZeroOrMarkovKernel.compProd (κ : Kernel α β) [IsZeroOrMarkovKernel κ]
(η : Kernel (α × β) γ) [IsZeroOrMarkovKernel η] : IsZeroOrMarkovKernel (κ ⊗ₖ η) := by
rw [compProd_def]
infer_instance | instance | Probability | [
"Mathlib.Probability.Kernel.Composition.Comp",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/CompProd.lean | IsZeroOrMarkovKernel.compProd | null |
compProd_apply_univ_le (κ : Kernel α β) (η : Kernel (α × β) γ) [IsFiniteKernel η] (a : α) :
(κ ⊗ₖ η) a Set.univ ≤ κ a Set.univ * η.bound := by
by_cases hκ : IsSFiniteKernel κ
swap
· rw [compProd_of_not_isSFiniteKernel_left _ _ hκ]
simp
rw [compProd_apply .univ]
let Cη := η.bound
calc
∫⁻ b, η (a, b) Set.univ ∂κ a ≤ ∫⁻ _, Cη ∂κ a :=
lintegral_mono fun b => measure_le_bound η (a, b) Set.univ
_ = Cη * κ a Set.univ := MeasureTheory.lintegral_const Cη
_ = κ a Set.univ * Cη := mul_comm _ _ | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.Comp",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/CompProd.lean | compProd_apply_univ_le | null |
IsFiniteKernel.compProd (κ : Kernel α β) [IsFiniteKernel κ] (η : Kernel (α × β) γ)
[IsFiniteKernel η] : IsFiniteKernel (κ ⊗ₖ η) := by
rw [compProd_def]
infer_instance | instance | Probability | [
"Mathlib.Probability.Kernel.Composition.Comp",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/CompProd.lean | IsFiniteKernel.compProd | null |
IsSFiniteKernel.compProd (κ : Kernel α β) (η : Kernel (α × β) γ) :
IsSFiniteKernel (κ ⊗ₖ η) := by
rw [compProd_def]
infer_instance | instance | Probability | [
"Mathlib.Probability.Kernel.Composition.Comp",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/CompProd.lean | IsSFiniteKernel.compProd | null |
compProd_assoc {δ : Type*} {mδ : MeasurableSpace δ}
{κ : Kernel α β} {η : Kernel (α × β) γ} {ξ : Kernel (α × β × γ) δ} :
(κ ⊗ₖ (η ⊗ₖ (ξ.comap MeasurableEquiv.prodAssoc (MeasurableEquiv.measurable _)))).map
MeasurableEquiv.prodAssoc.symm
= κ ⊗ₖ η ⊗ₖ ξ := by
by_cases hκ : IsSFiniteKernel κ
swap; · simp [hκ]
by_cases hη : IsSFiniteKernel η
swap; · simp [hη]
by_cases hξ : IsSFiniteKernel ξ
swap
· have : ¬ IsSFiniteKernel
(ξ.comap MeasurableEquiv.prodAssoc (MeasurableEquiv.measurable _)) := by
refine fun h_sfin ↦ hξ ?_
have : ξ = (ξ.comap MeasurableEquiv.prodAssoc (MeasurableEquiv.measurable _)).comap
MeasurableEquiv.prodAssoc.symm (MeasurableEquiv.measurable _) := by
simp [← comap_comp_right]
rw [this]
infer_instance
simp [hξ, this]
ext a s hs
rw [compProd_apply hs, map_apply' _ (by fun_prop) _ hs,
compProd_apply (hs.preimage (by fun_prop)), lintegral_compProd]
swap; · exact measurable_kernel_prodMk_left' hs a
congr with b
rw [compProd_apply]
· congr
· exact hs.preimage (by fun_prop) | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.Comp",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/CompProd.lean | compProd_assoc | `Kernel.compProd` is associative. We have to insert `MeasurableEquiv.prodAssoc` in two places
because the products of types `α × β × γ` and `(α × β) × γ` are different. |
compProd_add_left (μ κ : Kernel α β) (η : Kernel (α × β) γ)
[IsSFiniteKernel μ] [IsSFiniteKernel κ] :
(μ + κ) ⊗ₖ η = μ ⊗ₖ η + κ ⊗ₖ η := by
by_cases hη : IsSFiniteKernel η
· ext _ _ hs
simp [compProd_apply hs]
· simp [hη] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.Comp",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/CompProd.lean | compProd_add_left | null |
compProd_add_right (μ : Kernel α β) (κ η : Kernel (α × β) γ)
[IsSFiniteKernel κ] [IsSFiniteKernel η] :
μ ⊗ₖ (κ + η) = μ ⊗ₖ κ + μ ⊗ₖ η := by
by_cases hμ : IsSFiniteKernel μ
swap; · simp [hμ]
ext a s hs
simp only [compProd_apply hs, coe_add, Pi.add_apply, Measure.coe_add]
rw [lintegral_add_left]
exact measurable_kernel_prodMk_left' hs a | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.Comp",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/CompProd.lean | compProd_add_right | null |
compProd_sum_left {ι : Type*} [Countable ι]
{κ : ι → Kernel α β} {η : Kernel (α × β) γ} [∀ i, IsSFiniteKernel (κ i)] :
Kernel.sum κ ⊗ₖ η = Kernel.sum (fun i ↦ (κ i) ⊗ₖ η) := by
by_cases hη : IsSFiniteKernel η
· ext a s hs
simp_rw [sum_apply, compProd_apply hs, sum_apply, lintegral_sum_measure, Measure.sum_apply _ hs,
compProd_apply hs]
· simp [hη] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.Comp",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/CompProd.lean | compProd_sum_left | null |
compProd_sum_right {ι : Type*} [Countable ι]
{κ : Kernel α β} {η : ι → Kernel (α × β) γ} [∀ i, IsSFiniteKernel (η i)] :
κ ⊗ₖ Kernel.sum η = Kernel.sum (fun i ↦ κ ⊗ₖ (η i)) := by
by_cases hκ : IsSFiniteKernel κ
swap; · simp [hκ]
ext a s hs
simp_rw [sum_apply, compProd_apply hs, Measure.sum_apply _ hs, sum_apply, compProd_apply hs]
rw [← lintegral_tsum]
· congr with i
rw [Measure.sum_apply]
exact measurable_prodMk_left hs
· exact fun _ ↦ (measurable_kernel_prodMk_left' hs a).aemeasurable | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.Comp",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/CompProd.lean | compProd_sum_right | null |
comapRight_compProd_id_prod {δ : Type*} {mδ : MeasurableSpace δ}
(κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ) [IsSFiniteKernel η]
{f : δ → γ} (hf : MeasurableEmbedding f) :
comapRight (κ ⊗ₖ η) (MeasurableEmbedding.id.prodMap hf) = κ ⊗ₖ (comapRight η hf) := by
ext a t ht
rw [comapRight_apply' _ _ _ ht, compProd_apply, compProd_apply ht]
· refine lintegral_congr fun b ↦ ?_
rw [comapRight_apply']
· congr with x
aesop
· exact measurable_prodMk_left ht
· exact (MeasurableEmbedding.id.prodMap hf).measurableSet_image.mpr ht | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.Comp",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/CompProd.lean | comapRight_compProd_id_prod | null |
fst_compProd_apply (κ : Kernel α β) (η : Kernel (α × β) γ)
[IsSFiniteKernel κ] [IsSFiniteKernel η] (x : α) {s : Set β} (hs : MeasurableSet s) :
(κ ⊗ₖ η).fst x s = ∫⁻ b, s.indicator (fun b ↦ η (x, b) Set.univ) b ∂(κ x) := by
rw [Kernel.fst_apply' _ _ hs, Kernel.compProd_apply]
swap; · exact measurable_fst hs
have h_eq b : η (x, b) {c | b ∈ s} = s.indicator (fun b ↦ η (x, b) Set.univ) b := by
by_cases hb : b ∈ s <;> simp [hb]
simp_rw [Set.preimage, Set.mem_setOf_eq, h_eq]
@[simp] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.Comp",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/CompProd.lean | fst_compProd_apply | If `η` is a Markov kernel, use instead `fst_compProd` to get `(κ ⊗ₖ η).fst = κ`. |
fst_compProd (κ : Kernel α β) (η : Kernel (α × β) γ) [IsSFiniteKernel κ] [IsMarkovKernel η] :
fst (κ ⊗ₖ η) = κ := by
ext x s hs; simp [fst_compProd_apply, hs] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.Comp",
"Mathlib.Probability.Kernel.Composition.ParallelComp"
] | Mathlib/Probability/Kernel/Composition/CompProd.lean | fst_compProd | null |
hasFiniteIntegral_prodMk_left (a : α) {s : Set (β × γ)} (h2s : (κ ⊗ₖ η) a s ≠ ∞) :
HasFiniteIntegral (fun b => (η (a, b)).real (Prod.mk b ⁻¹' s)) (κ a) := by
let t := toMeasurable ((κ ⊗ₖ η) a) s
simp_rw [hasFiniteIntegral_iff_enorm, measureReal_def, enorm_eq_ofReal toReal_nonneg]
calc
∫⁻ b, ENNReal.ofReal (η (a, b) (Prod.mk b ⁻¹' s)).toReal ∂κ a
_ ≤ ∫⁻ b, η (a, b) (Prod.mk b ⁻¹' t) ∂κ a := by
refine lintegral_mono_ae ?_
filter_upwards [ae_kernel_lt_top a h2s] with b hb
rw [ofReal_toReal hb.ne]
exact measure_mono (preimage_mono (subset_toMeasurable _ _))
_ ≤ (κ ⊗ₖ η) a t := le_compProd_apply _ _ _ _
_ = (κ ⊗ₖ η) a s := measure_toMeasurable s
_ < ⊤ := h2s.lt_top
@[deprecated (since := "2025-03-05")]
alias hasFiniteIntegral_prod_mk_left := hasFiniteIntegral_prodMk_left | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | hasFiniteIntegral_prodMk_left | null |
integrable_kernel_prodMk_left (a : α) {s : Set (β × γ)} (hs : MeasurableSet s)
(h2s : (κ ⊗ₖ η) a s ≠ ∞) : Integrable (fun b => (η (a, b)).real (Prod.mk b ⁻¹' s)) (κ a) := by
constructor
· exact (measurable_kernel_prodMk_left' hs a).ennreal_toReal.aestronglyMeasurable
· exact hasFiniteIntegral_prodMk_left a h2s
@[deprecated (since := "2025-03-05")]
alias integrable_kernel_prod_mk_left := integrable_kernel_prodMk_left | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | integrable_kernel_prodMk_left | null |
_root_.MeasureTheory.AEStronglyMeasurable.integral_kernel_compProd [NormedSpace ℝ E]
⦃f : β × γ → E⦄ (hf : AEStronglyMeasurable f ((κ ⊗ₖ η) a)) :
AEStronglyMeasurable (fun x => ∫ y, f (x, y) ∂η (a, x)) (κ a) :=
⟨fun x => ∫ y, hf.mk f (x, y) ∂η (a, x), hf.stronglyMeasurable_mk.integral_kernel_prod_right'', by
filter_upwards [ae_ae_of_ae_compProd hf.ae_eq_mk] with _ hx using integral_congr_ae hx⟩ | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | _root_.MeasureTheory.AEStronglyMeasurable.integral_kernel_compProd | null |
_root_.MeasureTheory.AEStronglyMeasurable.compProd_mk_left {δ : Type*} [TopologicalSpace δ]
{f : β × γ → δ} (hf : AEStronglyMeasurable f ((κ ⊗ₖ η) a)) :
∀ᵐ x ∂κ a, AEStronglyMeasurable (fun y => f (x, y)) (η (a, x)) := by
filter_upwards [ae_ae_of_ae_compProd hf.ae_eq_mk] with x hx using
⟨fun y => hf.mk f (x, y), hf.stronglyMeasurable_mk.comp_measurable measurable_prodMk_left, hx⟩
/-! ### Integrability -/ | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | _root_.MeasureTheory.AEStronglyMeasurable.compProd_mk_left | null |
hasFiniteIntegral_compProd_iff ⦃f : β × γ → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f ((κ ⊗ₖ η) a) ↔
(∀ᵐ x ∂κ a, HasFiniteIntegral (fun y => f (x, y)) (η (a, x))) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂η (a, x)) (κ a) := by
simp only [hasFiniteIntegral_iff_enorm]
rw [lintegral_compProd _ _ _ h1f.enorm]
have : ∀ x, ∀ᵐ y ∂η (a, x), 0 ≤ ‖f (x, y)‖ := fun x => Eventually.of_forall fun y => norm_nonneg _
simp_rw [integral_eq_lintegral_of_nonneg_ae (this _)
(h1f.norm.comp_measurable measurable_prodMk_left).aestronglyMeasurable,
enorm_eq_ofReal toReal_nonneg, ofReal_norm_eq_enorm]
have : ∀ {p q r : Prop} (_ : r → p), (r ↔ p ∧ q) ↔ p → (r ↔ q) := fun {p q r} h1 => by
rw [← and_congr_right_iff, and_iff_right_of_imp h1]
rw [this]
· intro h2f; rw [lintegral_congr_ae]
filter_upwards [h2f] with x hx
rw [ofReal_toReal]; finiteness
· intro h2f; refine ae_lt_top ?_ h2f.ne; exact h1f.enorm.lintegral_kernel_prod_right'' | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | hasFiniteIntegral_compProd_iff | null |
hasFiniteIntegral_compProd_iff' ⦃f : β × γ → E⦄
(h1f : AEStronglyMeasurable f ((κ ⊗ₖ η) a)) :
HasFiniteIntegral f ((κ ⊗ₖ η) a) ↔
(∀ᵐ x ∂κ a, HasFiniteIntegral (fun y => f (x, y)) (η (a, x))) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂η (a, x)) (κ a) := by
rw [hasFiniteIntegral_congr h1f.ae_eq_mk,
hasFiniteIntegral_compProd_iff h1f.stronglyMeasurable_mk]
apply and_congr
· apply eventually_congr
filter_upwards [ae_ae_of_ae_compProd h1f.ae_eq_mk.symm] with x hx using
hasFiniteIntegral_congr hx
· apply hasFiniteIntegral_congr
filter_upwards [ae_ae_of_ae_compProd h1f.ae_eq_mk.symm] with _ hx using
integral_congr_ae (EventuallyEq.fun_comp hx _) | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | hasFiniteIntegral_compProd_iff' | null |
integrable_compProd_iff ⦃f : β × γ → E⦄ (hf : AEStronglyMeasurable f ((κ ⊗ₖ η) a)) :
Integrable f ((κ ⊗ₖ η) a) ↔
(∀ᵐ x ∂κ a, Integrable (fun y => f (x, y)) (η (a, x))) ∧
Integrable (fun x => ∫ y, ‖f (x, y)‖ ∂η (a, x)) (κ a) := by
simp only [Integrable, hasFiniteIntegral_compProd_iff' hf, hf.norm.integral_kernel_compProd,
hf, hf.compProd_mk_left, eventually_and, true_and] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | integrable_compProd_iff | null |
_root_.MeasureTheory.Integrable.ae_of_compProd ⦃f : β × γ → E⦄
(hf : Integrable f ((κ ⊗ₖ η) a)) : ∀ᵐ x ∂κ a, Integrable (fun y => f (x, y)) (η (a, x)) :=
((integrable_compProd_iff hf.aestronglyMeasurable).mp hf).1 | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | _root_.MeasureTheory.Integrable.ae_of_compProd | null |
_root_.MeasureTheory.Integrable.integral_norm_compProd ⦃f : β × γ → E⦄
(hf : Integrable f ((κ ⊗ₖ η) a)) : Integrable (fun x => ∫ y, ‖f (x, y)‖ ∂η (a, x)) (κ a) :=
((integrable_compProd_iff hf.aestronglyMeasurable).mp hf).2 | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | _root_.MeasureTheory.Integrable.integral_norm_compProd | null |
_root_.MeasureTheory.Integrable.integral_compProd [NormedSpace ℝ E]
⦃f : β × γ → E⦄ (hf : Integrable f ((κ ⊗ₖ η) a)) :
Integrable (fun x => ∫ y, f (x, y) ∂η (a, x)) (κ a) :=
Integrable.mono hf.integral_norm_compProd hf.aestronglyMeasurable.integral_kernel_compProd <|
Eventually.of_forall fun x =>
(norm_integral_le_integral_norm _).trans_eq <|
(norm_of_nonneg <|
integral_nonneg_of_ae <|
Eventually.of_forall fun y => (norm_nonneg (f (x, y)) :)).symm
/-! ### Bochner integral -/
variable [NormedSpace ℝ E] {E' : Type*} [NormedAddCommGroup E'] [NormedSpace ℝ E'] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | _root_.MeasureTheory.Integrable.integral_compProd | null |
Kernel.integral_fn_integral_add ⦃f g : β × γ → E⦄ (F : E → E')
(hf : Integrable f ((κ ⊗ₖ η) a)) (hg : Integrable g ((κ ⊗ₖ η) a)) :
∫ x, F (∫ y, f (x, y) + g (x, y) ∂η (a, x)) ∂κ a =
∫ x, F (∫ y, f (x, y) ∂η (a, x) + ∫ y, g (x, y) ∂η (a, x)) ∂κ a := by
refine integral_congr_ae ?_
filter_upwards [hf.ae_of_compProd, hg.ae_of_compProd] with _ h2f h2g
simp [integral_add h2f h2g] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | Kernel.integral_fn_integral_add | null |
Kernel.integral_fn_integral_sub ⦃f g : β × γ → E⦄ (F : E → E')
(hf : Integrable f ((κ ⊗ₖ η) a)) (hg : Integrable g ((κ ⊗ₖ η) a)) :
∫ x, F (∫ y, f (x, y) - g (x, y) ∂η (a, x)) ∂κ a =
∫ x, F (∫ y, f (x, y) ∂η (a, x) - ∫ y, g (x, y) ∂η (a, x)) ∂κ a := by
refine integral_congr_ae ?_
filter_upwards [hf.ae_of_compProd, hg.ae_of_compProd] with _ h2f h2g
simp [integral_sub h2f h2g] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | Kernel.integral_fn_integral_sub | null |
Kernel.lintegral_fn_integral_sub ⦃f g : β × γ → E⦄ (F : E → ℝ≥0∞)
(hf : Integrable f ((κ ⊗ₖ η) a)) (hg : Integrable g ((κ ⊗ₖ η) a)) :
∫⁻ x, F (∫ y, f (x, y) - g (x, y) ∂η (a, x)) ∂κ a =
∫⁻ x, F (∫ y, f (x, y) ∂η (a, x) - ∫ y, g (x, y) ∂η (a, x)) ∂κ a := by
refine lintegral_congr_ae ?_
filter_upwards [hf.ae_of_compProd, hg.ae_of_compProd] with _ h2f h2g
simp [integral_sub h2f h2g] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | Kernel.lintegral_fn_integral_sub | null |
Kernel.integral_integral_add ⦃f g : β × γ → E⦄ (hf : Integrable f ((κ ⊗ₖ η) a))
(hg : Integrable g ((κ ⊗ₖ η) a)) :
∫ x, ∫ y, f (x, y) + g (x, y) ∂η (a, x) ∂κ a =
∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a + ∫ x, ∫ y, g (x, y) ∂η (a, x) ∂κ a :=
(Kernel.integral_fn_integral_add id hf hg).trans <|
integral_add hf.integral_compProd hg.integral_compProd | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | Kernel.integral_integral_add | null |
Kernel.integral_integral_add' ⦃f g : β × γ → E⦄ (hf : Integrable f ((κ ⊗ₖ η) a))
(hg : Integrable g ((κ ⊗ₖ η) a)) :
∫ x, ∫ y, (f + g) (x, y) ∂η (a, x) ∂κ a =
∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a + ∫ x, ∫ y, g (x, y) ∂η (a, x) ∂κ a :=
Kernel.integral_integral_add hf hg | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | Kernel.integral_integral_add' | null |
Kernel.integral_integral_sub ⦃f g : β × γ → E⦄ (hf : Integrable f ((κ ⊗ₖ η) a))
(hg : Integrable g ((κ ⊗ₖ η) a)) :
∫ x, ∫ y, f (x, y) - g (x, y) ∂η (a, x) ∂κ a =
∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a - ∫ x, ∫ y, g (x, y) ∂η (a, x) ∂κ a :=
(Kernel.integral_fn_integral_sub id hf hg).trans <|
integral_sub hf.integral_compProd hg.integral_compProd | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | Kernel.integral_integral_sub | null |
Kernel.integral_integral_sub' ⦃f g : β × γ → E⦄ (hf : Integrable f ((κ ⊗ₖ η) a))
(hg : Integrable g ((κ ⊗ₖ η) a)) :
∫ x, ∫ y, (f - g) (x, y) ∂η (a, x) ∂κ a =
∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a - ∫ x, ∫ y, g (x, y) ∂η (a, x) ∂κ a :=
Kernel.integral_integral_sub hf hg | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | Kernel.integral_integral_sub' | null |
Kernel.continuous_integral_integral :
Continuous fun f : β × γ →₁[(κ ⊗ₖ η) a] E => ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
rw [continuous_iff_continuousAt]; intro g
refine
tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_compProd
(Eventually.of_forall fun h => (L1.integrable_coeFn h).integral_compProd) ?_
simp_rw [← lintegral_fn_integral_sub (‖·‖ₑ) (L1.integrable_coeFn _) (L1.integrable_coeFn g)]
apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds _ (fun i => zero_le _) _
· exact fun i => ∫⁻ x, ∫⁻ y, ‖i (x, y) - g (x, y)‖ₑ ∂η (a, x) ∂κ a
swap; · exact fun i => lintegral_mono fun x => enorm_integral_le_lintegral_enorm _
change
Tendsto
(fun i : β × γ →₁[(κ ⊗ₖ η) a] E => ∫⁻ x, ∫⁻ y : γ, ‖i (x, y) - g (x, y)‖ₑ ∂η (a, x) ∂κ a)
(𝓝 g) (𝓝 0)
have this (i : Lp (α := β × γ) E 1 (((κ ⊗ₖ η) a) : Measure (β × γ))) :
Measurable fun z => ‖i z - g z‖ₑ :=
((Lp.stronglyMeasurable i).sub (Lp.stronglyMeasurable g)).enorm
simp_rw [← lintegral_compProd _ _ _ (this _), ← L1.ofReal_norm_sub_eq_lintegral, ← ofReal_zero]
refine (continuous_ofReal.tendsto 0).comp ?_
rw [← tendsto_iff_norm_sub_tendsto_zero]
exact tendsto_id | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | Kernel.continuous_integral_integral | null |
integral_compProd :
∀ {f : β × γ → E} (_ : Integrable f ((κ ⊗ₖ η) a)),
∫ z, f z ∂(κ ⊗ₖ η) a = ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
by_cases hE : CompleteSpace E; swap
· simp [integral, hE]
apply Integrable.induction
· intro c s hs h2s
simp_rw [integral_indicator hs, ← indicator_comp_right, Function.comp_def,
integral_indicator (measurable_prodMk_left hs), MeasureTheory.setIntegral_const,
integral_smul_const, measureReal_def]
congr 1
rw [integral_toReal]
rotate_left
· exact (Kernel.measurable_kernel_prodMk_left' hs _).aemeasurable
· exact ae_kernel_lt_top a h2s.ne
rw [Kernel.compProd_apply hs]
· intro f g _ i_f i_g hf hg
simp_rw [integral_add' i_f i_g, Kernel.integral_integral_add' i_f i_g, hf, hg]
· exact isClosed_eq continuous_integral Kernel.continuous_integral_integral
· intro f g hfg _ hf
convert hf using 1
· exact integral_congr_ae hfg.symm
· apply integral_congr_ae
filter_upwards [ae_ae_of_ae_compProd hfg] with x hfgx using
integral_congr_ae (ae_eq_symm hfgx) | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | integral_compProd | null |
setIntegral_compProd {f : β × γ → E} {s : Set β} {t : Set γ} (hs : MeasurableSet s)
(ht : MeasurableSet t) (hf : IntegrableOn f (s ×ˢ t) ((κ ⊗ₖ η) a)) :
∫ z in s ×ˢ t, f z ∂(κ ⊗ₖ η) a = ∫ x in s, ∫ y in t, f (x, y) ∂η (a, x) ∂κ a := by
rw [← Kernel.restrict_apply (κ ⊗ₖ η) (hs.prod ht), ← compProd_restrict hs ht, integral_compProd]
· simp_rw [Kernel.restrict_apply]
· rw [compProd_restrict, Kernel.restrict_apply]; exact hf | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | setIntegral_compProd | null |
setIntegral_compProd_univ_right (f : β × γ → E) {s : Set β} (hs : MeasurableSet s)
(hf : IntegrableOn f (s ×ˢ univ) ((κ ⊗ₖ η) a)) :
∫ z in s ×ˢ univ, f z ∂(κ ⊗ₖ η) a = ∫ x in s, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
simp_rw [setIntegral_compProd hs MeasurableSet.univ hf, Measure.restrict_univ] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | setIntegral_compProd_univ_right | null |
setIntegral_compProd_univ_left (f : β × γ → E) {t : Set γ} (ht : MeasurableSet t)
(hf : IntegrableOn f (univ ×ˢ t) ((κ ⊗ₖ η) a)) :
∫ z in univ ×ˢ t, f z ∂(κ ⊗ₖ η) a = ∫ x, ∫ y in t, f (x, y) ∂η (a, x) ∂κ a := by
simp_rw [setIntegral_compProd MeasurableSet.univ ht hf, Measure.restrict_univ] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | setIntegral_compProd_univ_left | null |
_root_.MeasureTheory.AEStronglyMeasurable.integral_kernel_comp [NormedSpace ℝ E]
⦃f : γ → E⦄ (hf : AEStronglyMeasurable f ((η ∘ₖ κ) a)) :
AEStronglyMeasurable (fun x ↦ ∫ y, f y ∂η x) (κ a) :=
⟨fun x ↦ ∫ y, hf.mk f y ∂η x, hf.stronglyMeasurable_mk.integral_kernel, by
filter_upwards [ae_ae_of_ae_comp hf.ae_eq_mk] with _ hx using integral_congr_ae hx⟩ | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | _root_.MeasureTheory.AEStronglyMeasurable.integral_kernel_comp | null |
_root_.MeasureTheory.AEStronglyMeasurable.comp {δ : Type*} [TopologicalSpace δ]
{f : γ → δ} (hf : AEStronglyMeasurable f ((η ∘ₖ κ) a)) :
∀ᵐ x ∂κ a, AEStronglyMeasurable f (η x) := by
filter_upwards [ae_ae_of_ae_comp hf.ae_eq_mk] with x hx using
⟨hf.mk f, hf.stronglyMeasurable_mk, hx⟩
/-! ### Integrability with respect to composition -/ | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | _root_.MeasureTheory.AEStronglyMeasurable.comp | null |
hasFiniteIntegral_comp_iff ⦃f : γ → E⦄ (hf : StronglyMeasurable f) :
HasFiniteIntegral f ((η ∘ₖ κ) a) ↔
(∀ᵐ x ∂κ a, HasFiniteIntegral f (η x)) ∧ HasFiniteIntegral (fun x ↦ ∫ y, ‖f y‖ ∂η x) (κ a) := by
simp_rw [hasFiniteIntegral_iff_enorm, lintegral_comp _ _ _ hf.enorm]
simp_rw [integral_eq_lintegral_of_nonneg_ae (ae_of_all _ fun y ↦ norm_nonneg _)
hf.norm.aestronglyMeasurable, enorm_eq_ofReal toReal_nonneg, ofReal_norm_eq_enorm]
have : ∀ {p q r : Prop} (_ : r → p), (r ↔ p ∧ q) ↔ p → (r ↔ q) := fun h ↦ by
rw [← and_congr_right_iff, and_iff_right_of_imp h]
rw [this]
· intro h
rw [lintegral_congr_ae]
filter_upwards [h] with x hx
rw [ofReal_toReal]
finiteness
· exact fun h ↦ ae_lt_top hf.enorm.lintegral_kernel h.ne | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | hasFiniteIntegral_comp_iff | null |
hasFiniteIntegral_comp_iff' ⦃f : γ → E⦄ (hf : AEStronglyMeasurable f ((η ∘ₖ κ) a)) :
HasFiniteIntegral f ((η ∘ₖ κ) a) ↔
(∀ᵐ x ∂κ a, HasFiniteIntegral f (η x)) ∧ HasFiniteIntegral (fun x ↦ ∫ y, ‖f y‖ ∂η x) (κ a) := by
rw [hasFiniteIntegral_congr hf.ae_eq_mk, hasFiniteIntegral_comp_iff hf.stronglyMeasurable_mk]
refine and_congr (eventually_congr ?_) (hasFiniteIntegral_congr ?_)
· filter_upwards [ae_ae_of_ae_comp hf.ae_eq_mk.symm] with _ hx using
hasFiniteIntegral_congr hx
· filter_upwards [ae_ae_of_ae_comp hf.ae_eq_mk.symm] with _ hx using
integral_congr_ae (EventuallyEq.fun_comp hx _) | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | hasFiniteIntegral_comp_iff' | null |
integrable_comp_iff ⦃f : γ → E⦄ (hf : AEStronglyMeasurable f ((η ∘ₖ κ) a)) :
Integrable f ((η ∘ₖ κ) a) ↔
(∀ᵐ y ∂κ a, Integrable f (η y)) ∧ Integrable (fun y ↦ ∫ z, ‖f z‖ ∂η y) (κ a) := by
simp only [Integrable, hf, hasFiniteIntegral_comp_iff' hf, true_and, eventually_and, hf.comp,
hf.norm.integral_kernel_comp] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | integrable_comp_iff | null |
_root_.MeasureTheory.Integrable.ae_of_comp ⦃f : γ → E⦄ (hf : Integrable f ((η ∘ₖ κ) a)) :
∀ᵐ x ∂κ a, Integrable f (η x) := ((integrable_comp_iff hf.1).1 hf).1 | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | _root_.MeasureTheory.Integrable.ae_of_comp | null |
_root_.MeasureTheory.Integrable.integral_norm_comp ⦃f : γ → E⦄
(hf : Integrable f ((η ∘ₖ κ) a)) : Integrable (fun x ↦ ∫ y, ‖f y‖ ∂η x) (κ a) :=
((integrable_comp_iff hf.1).1 hf).2 | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | _root_.MeasureTheory.Integrable.integral_norm_comp | null |
_root_.MeasureTheory.Integrable.integral_comp [NormedSpace ℝ E] ⦃f : γ → E⦄
(hf : Integrable f ((η ∘ₖ κ) a)) : Integrable (fun x ↦ ∫ y, f y ∂η x) (κ a) :=
Integrable.mono hf.integral_norm_comp hf.1.integral_kernel_comp <|
ae_of_all _ fun _ ↦ (norm_integral_le_integral_norm _).trans_eq
(norm_of_nonneg <| integral_nonneg_of_ae <| ae_of_all _ fun _ ↦ norm_nonneg _).symm
/-! ### Bochner integral with respect to the composition -/
variable [NormedSpace ℝ E] {E' : Type*} [NormedAddCommGroup E'] [NormedSpace ℝ E'] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | _root_.MeasureTheory.Integrable.integral_comp | null |
integral_fn_integral_add_comp ⦃f g : γ → E⦄ (F : E → E')
(hf : Integrable f ((η ∘ₖ κ) a)) (hg : Integrable g ((η ∘ₖ κ) a)) :
∫ x, F (∫ y, f y + g y ∂η x) ∂κ a = ∫ x, F (∫ y, f y ∂η x + ∫ y, g y ∂η x) ∂κ a := by
refine integral_congr_ae ?_
filter_upwards [hf.ae_of_comp, hg.ae_of_comp] with _ h2f h2g
simp [integral_add h2f h2g] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | integral_fn_integral_add_comp | null |
integral_fn_integral_sub_comp ⦃f g : γ → E⦄ (F : E → E')
(hf : Integrable f ((η ∘ₖ κ) a)) (hg : Integrable g ((η ∘ₖ κ) a)) :
∫ x, F (∫ y, f y - g y ∂η x) ∂κ a = ∫ x, F (∫ y, f y ∂η x - ∫ y, g y ∂η x) ∂κ a := by
refine integral_congr_ae ?_
filter_upwards [hf.ae_of_comp, hg.ae_of_comp] with _ h2f h2g
simp [integral_sub h2f h2g] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | integral_fn_integral_sub_comp | null |
lintegral_fn_integral_sub_comp ⦃f g : γ → E⦄ (F : E → ℝ≥0∞)
(hf : Integrable f ((η ∘ₖ κ) a)) (hg : Integrable g ((η ∘ₖ κ) a)) :
∫⁻ x, F (∫ y, f y - g y ∂η x) ∂κ a = ∫⁻ x, F (∫ y, f y ∂η x - ∫ y, g y ∂η x) ∂κ a := by
refine lintegral_congr_ae ?_
filter_upwards [hf.ae_of_comp, hg.ae_of_comp] with _ h2f h2g
simp [integral_sub h2f h2g] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | lintegral_fn_integral_sub_comp | null |
integral_integral_add_comp ⦃f g : γ → E⦄ (hf : Integrable f ((η ∘ₖ κ) a))
(hg : Integrable g ((η ∘ₖ κ) a)) :
∫ x, ∫ y, f y + g y ∂η x ∂κ a = ∫ x, ∫ y, f y ∂η x ∂κ a + ∫ x, ∫ y, g y ∂η x ∂κ a :=
(integral_fn_integral_add_comp id hf hg).trans <| integral_add hf.integral_comp hg.integral_comp | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | integral_integral_add_comp | null |
integral_integral_add'_comp ⦃f g : γ → E⦄ (hf : Integrable f ((η ∘ₖ κ) a))
(hg : Integrable g ((η ∘ₖ κ) a)) :
∫ x, ∫ y, (f + g) y ∂η x ∂κ a = ∫ x, ∫ y, f y ∂η x ∂κ a + ∫ x, ∫ y, g y ∂η x ∂κ a :=
integral_integral_add_comp hf hg | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | integral_integral_add'_comp | null |
integral_integral_sub_comp ⦃f g : γ → E⦄ (hf : Integrable f ((η ∘ₖ κ) a))
(hg : Integrable g ((η ∘ₖ κ) a)) :
∫ x, ∫ y, f y - g y ∂η x ∂κ a = ∫ x, ∫ y, f y ∂η x ∂κ a - ∫ x, ∫ y, g y ∂η x ∂κ a :=
(integral_fn_integral_sub_comp id hf hg).trans <| integral_sub hf.integral_comp hg.integral_comp | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | integral_integral_sub_comp | null |
integral_integral_sub'_comp ⦃f g : γ → E⦄ (hf : Integrable f ((η ∘ₖ κ) a))
(hg : Integrable g ((η ∘ₖ κ) a)) :
∫ x, ∫ y, (f - g) y ∂η x ∂κ a = ∫ x, ∫ y, f y ∂η x ∂κ a - ∫ x, ∫ y, g y ∂η x ∂κ a :=
integral_integral_sub_comp hf hg | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | integral_integral_sub'_comp | null |
continuous_integral_integral_comp :
Continuous fun f : γ →₁[(η ∘ₖ κ) a] E ↦ ∫ x, ∫ y, f y ∂η x ∂κ a := by
refine continuous_iff_continuousAt.2 fun g ↦ ?_
refine tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_comp
(Eventually.of_forall fun h ↦ (L1.integrable_coeFn h).integral_comp) ?_
simp_rw [← lintegral_fn_integral_sub_comp (‖·‖ₑ) (L1.integrable_coeFn _) (L1.integrable_coeFn g)]
refine tendsto_of_tendsto_of_tendsto_of_le_of_le
(h := fun i ↦ ∫⁻ x, ∫⁻ y, ‖i y - g y‖ₑ ∂η x ∂κ a)
tendsto_const_nhds ?_ (fun _ ↦ zero_le _) ?_
swap; · exact fun _ ↦ lintegral_mono fun _ ↦ enorm_integral_le_lintegral_enorm _
have (i : γ →₁[(η ∘ₖ κ) a] E) : Measurable fun z ↦ ‖i z - g z‖ₑ :=
((Lp.stronglyMeasurable i).sub (Lp.stronglyMeasurable g)).enorm
simp_rw [← lintegral_comp _ _ _ (this _), ← L1.ofReal_norm_sub_eq_lintegral, ← ofReal_zero]
exact (continuous_ofReal.tendsto 0).comp (tendsto_iff_norm_sub_tendsto_zero.1 tendsto_id) | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | continuous_integral_integral_comp | null |
integral_comp : ∀ {f : γ → E} (_ : Integrable f ((η ∘ₖ κ) a)),
∫ z, f z ∂(η ∘ₖ κ) a = ∫ x, ∫ y, f y ∂η x ∂κ a := by
by_cases hE : CompleteSpace E; swap
· simp [integral, hE]
apply Integrable.induction
· intro c s hs ms
simp_rw [integral_indicator hs, MeasureTheory.setIntegral_const, integral_smul_const,
measureReal_def]
congr
rw [integral_toReal, Kernel.comp_apply' _ _ _ hs]
· exact (Kernel.measurable_coe _ hs).aemeasurable
· exact ae_lt_top_of_comp_ne_top a ms.ne
· rintro f g - i_f i_g hf hg
simp_rw [integral_add' i_f i_g, integral_integral_add'_comp i_f i_g, hf, hg]
· exact isClosed_eq continuous_integral Kernel.continuous_integral_integral_comp
· rintro f g hfg - hf
convert hf using 1
· exact integral_congr_ae hfg.symm
· apply integral_congr_ae
filter_upwards [ae_ae_of_ae_comp hfg] with x hfgx using integral_congr_ae (ae_eq_symm hfgx) | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | integral_comp | null |
setIntegral_comp {f : γ → E} {s : Set γ} (hs : MeasurableSet s)
(hf : IntegrableOn f s ((η ∘ₖ κ) a)) :
∫ z in s, f z ∂(η ∘ₖ κ) a = ∫ x, ∫ y in s, f y ∂η x ∂κ a := by
rw [← restrict_apply (η ∘ₖ κ) hs, ← comp_restrict hs, integral_comp]
· simp_rw [restrict_apply]
· rwa [comp_restrict, restrict_apply] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | setIntegral_comp | null |
_root_.MeasureTheory.AEStronglyMeasurable.ae_of_compProd [SFinite μ] [IsSFiniteKernel κ]
{E : Type*} [NormedAddCommGroup E] {f : α → β → E}
(hf : AEStronglyMeasurable f.uncurry (μ ⊗ₘ κ)) :
∀ᵐ x ∂μ, AEStronglyMeasurable (f x) (κ x) := by
simpa using hf.compProd_mk_left | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | _root_.MeasureTheory.AEStronglyMeasurable.ae_of_compProd | null |
integrable_compProd_iff [SFinite μ] [IsSFiniteKernel κ] {E : Type*} [NormedAddCommGroup E]
{f : α × β → E} (hf : AEStronglyMeasurable f (μ ⊗ₘ κ)) :
Integrable f (μ ⊗ₘ κ) ↔
(∀ᵐ x ∂μ, Integrable (fun y => f (x, y)) (κ x)) ∧
Integrable (fun x => ∫ y, ‖f (x, y)‖ ∂(κ x)) μ := by
simp_rw [Measure.compProd, ProbabilityTheory.integrable_compProd_iff hf, Kernel.prodMkLeft_apply,
Kernel.const_apply] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | integrable_compProd_iff | null |
integral_compProd [SFinite μ] [IsSFiniteKernel κ] {E : Type*}
[NormedAddCommGroup E] [NormedSpace ℝ E]
{f : α × β → E} (hf : Integrable f (μ ⊗ₘ κ)) :
∫ x, f x ∂(μ ⊗ₘ κ) = ∫ a, ∫ b, f (a, b) ∂(κ a) ∂μ := by
rw [Measure.compProd, ProbabilityTheory.integral_compProd hf]
simp | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | integral_compProd | null |
setIntegral_compProd [SFinite μ] [IsSFiniteKernel κ] {E : Type*}
[NormedAddCommGroup E] [NormedSpace ℝ E]
{s : Set α} (hs : MeasurableSet s) {t : Set β} (ht : MeasurableSet t)
{f : α × β → E} (hf : IntegrableOn f (s ×ˢ t) (μ ⊗ₘ κ)) :
∫ x in s ×ˢ t, f x ∂(μ ⊗ₘ κ) = ∫ a in s, ∫ b in t, f (a, b) ∂(κ a) ∂μ := by
rw [Measure.compProd, ProbabilityTheory.setIntegral_compProd hs ht hf]
simp | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | setIntegral_compProd | null |
integrable_compProd_snd_iff [SFinite μ] [IsSFiniteKernel κ]
(hf : AEStronglyMeasurable f (κ ∘ₘ μ)) :
Integrable (fun p ↦ f p.2) (μ ⊗ₘ κ) ↔ Integrable f (κ ∘ₘ μ) := by
rw [← Measure.snd_compProd, Measure.snd, integrable_map_measure _ measurable_snd.aemeasurable,
Function.comp_def]
rwa [← Measure.snd, Measure.snd_compProd] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | integrable_compProd_snd_iff | null |
ae_integrable_of_integrable_comp (h_int : Integrable f (κ ∘ₘ μ)) :
∀ᵐ x ∂μ, Integrable f (κ x) := by
rw [Measure.comp_eq_comp_const_apply, integrable_comp_iff h_int.1] at h_int
exact h_int.1 | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | ae_integrable_of_integrable_comp | null |
integrable_integral_norm_of_integrable_comp (h_int : Integrable f (κ ∘ₘ μ)) :
Integrable (fun x ↦ ∫ y, ‖f y‖ ∂κ x) μ := by
rw [Measure.comp_eq_comp_const_apply, integrable_comp_iff h_int.1] at h_int
exact h_int.2 | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.MeasurableIntegral"
] | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | integrable_integral_norm_of_integrable_comp | null |
comp_eq_snd_compProd (η : Kernel Y Z) [IsSFiniteKernel η] (κ : Kernel X Y)
[IsSFiniteKernel κ] : η ∘ₖ κ = snd (κ ⊗ₖ prodMkLeft X η) := by
ext a s hs
rw [comp_apply' _ _ _ hs, snd_apply' _ _ hs, compProd_apply (measurable_snd hs)]
simp [← Set.preimage_comp]
@[simp] lemma snd_compProd_prodMkLeft
(κ : Kernel X Y) (η : Kernel Y Z) [IsSFiniteKernel κ] [IsSFiniteKernel η] :
snd (κ ⊗ₖ prodMkLeft X η) = η ∘ₖ κ := (comp_eq_snd_compProd η κ).symm | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Composition.Prod"
] | Mathlib/Probability/Kernel/Composition/KernelLemmas.lean | comp_eq_snd_compProd | null |
compProd_prodMkLeft_eq_comp
(κ : Kernel X Y) [IsSFiniteKernel κ] (η : Kernel Y Z) [IsSFiniteKernel η] :
κ ⊗ₖ (prodMkLeft X η) = (Kernel.id ×ₖ η) ∘ₖ κ := by
ext a s hs
rw [comp_eq_snd_compProd, compProd_apply hs, snd_apply' _ _ hs, compProd_apply]
swap; · exact measurable_snd hs
simp only [prodMkLeft_apply, ← Set.preimage_comp, Prod.snd_comp_mk, Set.preimage_id_eq, id_eq,
prod_apply' _ _ _ hs, id_apply]
congr with b
rw [lintegral_dirac']
exact measurable_measure_prodMk_left hs | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Composition.Prod"
] | Mathlib/Probability/Kernel/Composition/KernelLemmas.lean | compProd_prodMkLeft_eq_comp | null |
swap_parallelComp : swap Y T ∘ₖ (κ ∥ₖ η) = η ∥ₖ κ ∘ₖ swap X Z := by
by_cases hκ : IsSFiniteKernel κ
swap; · simp [hκ]
by_cases hη : IsSFiniteKernel η
swap; · simp [hη]
ext ac s hs
simp_rw [comp_apply, parallelComp_apply, Measure.bind_apply hs (Kernel.aemeasurable _),
swap_apply, lintegral_dirac' _ (Kernel.measurable_coe _ hs), parallelComp_apply' hs,
Prod.fst_swap, Prod.snd_swap]
rw [MeasureTheory.lintegral_prod_symm]
swap; · exact ((Kernel.id.measurable_coe hs).comp measurable_swap).aemeasurable
congr with d
simp_rw [Prod.swap_prod_mk, Measure.dirac_apply' _ hs, ← Set.indicator_comp_right,
lintegral_indicator (measurable_prodMk_left hs)]
simp | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Composition.Prod"
] | Mathlib/Probability/Kernel/Composition/KernelLemmas.lean | swap_parallelComp | null |
deterministic_comp_copy {f : X → Y} (hf : Measurable f) :
(deterministic f hf ∥ₖ deterministic f hf) ∘ₖ copy X = copy Y ∘ₖ deterministic f hf := by
simp_rw [parallelComp_comp_copy, deterministic_prod_deterministic, copy,
deterministic_comp_deterministic, Function.comp_def] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Composition.Prod"
] | Mathlib/Probability/Kernel/Composition/KernelLemmas.lean | deterministic_comp_copy | For a deterministic kernel, copying then applying the kernel to the two copies is the same
as first applying the kernel then copying. |
parallelComp_id_left_comp_parallelComp
{η : Kernel X' Z} [IsSFiniteKernel η] {ξ : Kernel Z T} [IsSFiniteKernel ξ] :
(Kernel.id ∥ₖ ξ) ∘ₖ (κ ∥ₖ η) = κ ∥ₖ (ξ ∘ₖ η) := by
by_cases hκ : IsSFiniteKernel κ
swap; · simp [hκ]
ext a s hs
rw [comp_apply' _ _ _ hs, parallelComp_apply,
MeasureTheory.lintegral_prod _ (Kernel.measurable_coe _ hs).aemeasurable]
rw [parallelComp_apply, Measure.prod_apply hs]
congr with x
rw [comp_apply' _ _ _ (measurable_prodMk_left hs)]
congr with y
rw [parallelComp_apply' hs, Kernel.id_apply,
lintegral_dirac' _ (measurable_measure_prodMk_left hs)] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Composition.Prod"
] | Mathlib/Probability/Kernel/Composition/KernelLemmas.lean | parallelComp_id_left_comp_parallelComp | null |
parallelComp_id_right_comp_parallelComp {η : Kernel X' Z} [IsSFiniteKernel η]
{ξ : Kernel Z T} [IsSFiniteKernel ξ] :
(ξ ∥ₖ Kernel.id) ∘ₖ (η ∥ₖ κ) = (ξ ∘ₖ η) ∥ₖ κ := by
suffices swap T Y ∘ₖ (ξ ∥ₖ Kernel.id) ∘ₖ (η ∥ₖ κ) = swap T Y ∘ₖ ((ξ ∘ₖ η) ∥ₖ κ) by
calc ξ ∥ₖ Kernel.id ∘ₖ (η ∥ₖ κ)
_ = swap Y T ∘ₖ (swap T Y ∘ₖ (ξ ∥ₖ Kernel.id) ∘ₖ (η ∥ₖ κ)) := by
simp_rw [← comp_assoc, swap_swap, id_comp]
_ = swap Y T ∘ₖ (swap T Y ∘ₖ ((ξ ∘ₖ η) ∥ₖ κ)) := by rw [this]
_ = ξ ∘ₖ η ∥ₖ κ := by simp_rw [← comp_assoc, swap_swap, id_comp]
simp_rw [swap_parallelComp, comp_assoc, swap_parallelComp, ← comp_assoc,
parallelComp_id_left_comp_parallelComp] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Composition.Prod"
] | Mathlib/Probability/Kernel/Composition/KernelLemmas.lean | parallelComp_id_right_comp_parallelComp | null |
parallelComp_comp_parallelComp [IsSFiniteKernel κ] {η : Kernel Y Z} [IsSFiniteKernel η]
{κ' : Kernel X' Y'} [IsSFiniteKernel κ'] {η' : Kernel Y' Z'} [IsSFiniteKernel η'] :
(η ∥ₖ η') ∘ₖ (κ ∥ₖ κ') = (η ∘ₖ κ) ∥ₖ (η' ∘ₖ κ') := by
rw [← parallelComp_id_left_comp_parallelComp, ← parallelComp_id_right_comp_parallelComp,
← comp_assoc, parallelComp_id_left_comp_parallelComp, comp_id] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Composition.Prod"
] | Mathlib/Probability/Kernel/Composition/KernelLemmas.lean | parallelComp_comp_parallelComp | null |
parallelComp_comp_prod [IsSFiniteKernel κ] {η : Kernel Y Z} [IsSFiniteKernel η]
{κ' : Kernel X Y'} [IsSFiniteKernel κ'] {η' : Kernel Y' Z'} [IsSFiniteKernel η'] :
(η ∥ₖ η') ∘ₖ (κ ×ₖ κ') = (η ∘ₖ κ) ×ₖ (η' ∘ₖ κ') := by
rw [← parallelComp_comp_copy, ← comp_assoc, parallelComp_comp_parallelComp,
← parallelComp_comp_copy] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Composition.Prod"
] | Mathlib/Probability/Kernel/Composition/KernelLemmas.lean | parallelComp_comp_prod | null |
parallelComp_comm {η : Kernel Z T} :
(Kernel.id ∥ₖ κ) ∘ₖ (η ∥ₖ Kernel.id) = (η ∥ₖ Kernel.id) ∘ₖ (Kernel.id ∥ₖ κ) := by
by_cases hκ : IsSFiniteKernel κ
swap; · simp [hκ]
by_cases hη : IsSFiniteKernel η
swap; · simp [hη]
rw [parallelComp_id_left_comp_parallelComp, parallelComp_id_right_comp_parallelComp,
comp_id, comp_id] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.CompProd",
"Mathlib.Probability.Kernel.Composition.Prod"
] | Mathlib/Probability/Kernel/Composition/KernelLemmas.lean | parallelComp_comm | null |
prod_prodMkLeft_comp_prod_deterministic {β' ε : Type*}
{mβ' : MeasurableSpace β'} {mε : MeasurableSpace ε}
(κ : Kernel γ β) [IsSFiniteKernel κ] (η : Kernel ε β') [IsSFiniteKernel η]
(ξ : Kernel (β × ε) δ) [IsSFiniteKernel ξ] {f : γ → ε} (hf : Measurable f) :
(ξ ×ₖ η.prodMkLeft β) ∘ₖ (κ ×ₖ deterministic f hf)
= (ξ ∘ₖ (κ ×ₖ deterministic f hf)) ×ₖ (η ∘ₖ deterministic f hf) := by
ext ω s hs
rw [prod_apply' _ _ _ hs, comp_apply' _ _ _ hs, lintegral_prod_deterministic,
lintegral_comp, lintegral_prod_deterministic]
· congr with b
rw [prod_apply' _ _ _ hs, prodMkLeft_apply, comp_deterministic_eq_comap, comap_apply]
· exact (measurable_measure_prodMk_left hs).lintegral_kernel
· exact measurable_measure_prodMk_left hs
· exact Kernel.measurable_coe _ hs | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp"
] | Mathlib/Probability/Kernel/Composition/Lemmas.lean | prod_prodMkLeft_comp_prod_deterministic | The composition of two product kernels `(ξ ×ₖ η') ∘ₖ (κ ×ₖ ζ)` is the product of the
compositions `(ξ ∘ₖ (κ ×ₖ ζ)) ×ₖ (η' ∘ₖ (κ ×ₖ ζ))`, if `ζ` is deterministic (of the form
`.deterministic f hf`) and `η'` does not depend on the output of `κ`.
That is, `η'` has the form `η.prodMkLeft β` for a kernel `η`.
If `κ` was deterministic, this would be true even if `η.prodMkLeft β` was a more general
kernel since `κ ×ₖ Kernel.deterministic f hf` would be deterministic and commute with copying.
Here `κ` is not deterministic, but it is discarded in one branch of the copy. |
prod_prodMkRight_comp_deterministic_prod {β' ε : Type*}
{mβ' : MeasurableSpace β'} {mε : MeasurableSpace ε}
(κ : Kernel γ β) [IsSFiniteKernel κ] (η : Kernel ε β') [IsSFiniteKernel η]
(ξ : Kernel (ε × β) δ) [IsSFiniteKernel ξ] {f : γ → ε} (hf : Measurable f) :
(ξ ×ₖ η.prodMkRight β) ∘ₖ (deterministic f hf ×ₖ κ)
= (ξ ∘ₖ (deterministic f hf ×ₖ κ)) ×ₖ (η ∘ₖ deterministic f hf) := by
ext ω s hs
rw [prod_apply' _ _ _ hs, comp_apply' _ _ _ hs, lintegral_deterministic_prod,
lintegral_comp, lintegral_deterministic_prod]
· congr with b
rw [prod_apply' _ _ _ hs, prodMkRight_apply, comp_deterministic_eq_comap, comap_apply]
· exact (measurable_measure_prodMk_left hs).lintegral_kernel
· exact measurable_measure_prodMk_left hs
· exact Kernel.measurable_coe _ hs | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp"
] | Mathlib/Probability/Kernel/Composition/Lemmas.lean | prod_prodMkRight_comp_deterministic_prod | The composition of two product kernels `(ξ ×ₖ η') ∘ₖ (ζ ×ₖ κ)` is the product of the
compositions, `(ξ ∘ₖ (ζ ×ₖ κ)) ×ₖ (η' ∘ₖ (ζ ×ₖ κ))`, if `ζ` is deterministic (of the form
`.deterministic f hf`) and `η'` does not depend on the output of `κ`.
That is, `η'` has the form `η.prodMkRight β` for a kernel `η`.
If `κ` was deterministic, this would be true even if `η.prodMkRight β` was a more general
kernel since `Kernel.deterministic f hf ×ₖ κ` would be deterministic and commute with copying.
Here `κ` is not deterministic, but it is discarded in one branch of the copy. |
compProd_eq_parallelComp_comp_copy_comp [SFinite μ] :
μ ⊗ₘ κ = (Kernel.id ∥ₖ κ) ∘ₘ Kernel.copy α ∘ₘ μ := by
by_cases hκ : IsSFiniteKernel κ
swap; · simp [hκ]
rw [compProd_eq_comp_prod, ← Kernel.parallelComp_comp_copy, Measure.comp_assoc] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp"
] | Mathlib/Probability/Kernel/Composition/Lemmas.lean | compProd_eq_parallelComp_comp_copy_comp | null |
prod_comp_right [SFinite ν] {κ : Kernel β γ} [IsSFiniteKernel κ] :
μ.prod (κ ∘ₘ ν) = (Kernel.id ∥ₖ κ) ∘ₘ (μ.prod ν) := by
ext s hs
rw [Measure.prod_apply hs, Measure.bind_apply hs (Kernel.aemeasurable _)]
simp_rw [Measure.bind_apply (measurable_prodMk_left hs) (Kernel.aemeasurable _)]
rw [MeasureTheory.lintegral_prod]
swap; · exact (Kernel.measurable_coe _ hs).aemeasurable
congr with a
congr with b
rw [Kernel.parallelComp_apply, Kernel.id_apply, Measure.prod_apply hs, lintegral_dirac']
exact measurable_measure_prodMk_left hs | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp"
] | Mathlib/Probability/Kernel/Composition/Lemmas.lean | prod_comp_right | null |
prod_comp_left [SFinite μ] [SFinite ν] {κ : Kernel α γ} [IsSFiniteKernel κ] :
(κ ∘ₘ μ).prod ν = (κ ∥ₖ Kernel.id) ∘ₘ (μ.prod ν) := by
have h1 : (κ ∘ₘ μ).prod ν = (ν.prod (κ ∘ₘ μ)).map Prod.swap := by rw [Measure.prod_swap]
have h2 : (κ ∥ₖ Kernel.id) ∘ₘ (μ.prod ν) = ((Kernel.id ∥ₖ κ) ∘ₘ (ν.prod μ)).map Prod.swap := by
calc (κ ∥ₖ Kernel.id) ∘ₘ (μ.prod ν)
_ = (κ ∥ₖ Kernel.id) ∘ₘ ((ν.prod μ).map Prod.swap) := by rw [Measure.prod_swap]
_ = (κ ∥ₖ Kernel.id) ∘ₘ ((Kernel.swap _ _) ∘ₘ (ν.prod μ)) := by
rw [Kernel.swap, Measure.deterministic_comp_eq_map]
_ = (Kernel.swap _ _) ∘ₘ ((Kernel.id ∥ₖ κ) ∘ₘ (ν.prod μ)) := by
rw [Measure.comp_assoc, Measure.comp_assoc, Kernel.swap_parallelComp]
_ = ((Kernel.id ∥ₖ κ) ∘ₘ (ν.prod μ)).map Prod.swap := by
rw [Kernel.swap, Measure.deterministic_comp_eq_map]
rw [← Measure.prod_comp_right, ← h1] at h2
exact h2.symm | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp"
] | Mathlib/Probability/Kernel/Composition/Lemmas.lean | prod_comp_left | null |
parallelComp_comp_compProd [IsSFiniteKernel κ] {η : Kernel β γ} [IsSFiniteKernel η] :
(Kernel.id ∥ₖ η) ∘ₘ (μ ⊗ₘ κ) = μ ⊗ₘ (η ∘ₖ κ) := by
by_cases hμ : SFinite μ
swap; · simp [hμ]
rw [Measure.compProd_eq_comp_prod, Measure.compProd_eq_comp_prod, Measure.comp_assoc,
Kernel.parallelComp_comp_prod, Kernel.id_comp] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp"
] | Mathlib/Probability/Kernel/Composition/Lemmas.lean | parallelComp_comp_compProd | null |
compProd_map [SFinite μ] [IsSFiniteKernel κ] {f : β → γ} (hf : Measurable f) :
μ ⊗ₘ (κ.map f) = (μ ⊗ₘ κ).map (Prod.map id f) := by
calc μ ⊗ₘ (κ.map f)
_ = (Kernel.id ∥ₖ Kernel.deterministic f hf) ∘ₘ (Kernel.id ×ₖ κ) ∘ₘ μ := by
rw [comp_assoc, Kernel.parallelComp_comp_prod, compProd_eq_comp_prod,
Kernel.id_comp, Kernel.deterministic_comp_eq_map]
_ = (Kernel.id ∥ₖ Kernel.deterministic f hf) ∘ₘ (μ ⊗ₘ κ) := by rw [compProd_eq_comp_prod]
_ = (μ ⊗ₘ κ).map (Prod.map id f) := by
rw [Kernel.id, Kernel.deterministic_parallelComp_deterministic, deterministic_comp_eq_map] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp"
] | Mathlib/Probability/Kernel/Composition/Lemmas.lean | compProd_map | null |
noncomputable mapOfMeasurable (κ : Kernel α β) (f : β → γ) (hf : Measurable f) :
Kernel α γ where
toFun a := (κ a).map f
measurable' := by fun_prop
open Classical in | def | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | mapOfMeasurable | The pushforward of a kernel along a measurable function. This is an implementation detail,
use `map κ f` instead. |
noncomputable map [MeasurableSpace γ] (κ : Kernel α β) (f : β → γ) : Kernel α γ :=
if hf : Measurable f then mapOfMeasurable κ f hf else 0 | def | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | map | The pushforward of a kernel along a function.
If the function is not measurable, we use zero instead. This choice of junk
value ensures that typeclass inference can infer that the `map` of a kernel
satisfying `IsZeroOrMarkovKernel` again satisfies this property. |
map_of_not_measurable (κ : Kernel α β) {f : β → γ} (hf : ¬(Measurable f)) :
map κ f = 0 := by
simp [map, hf]
@[simp] theorem mapOfMeasurable_eq_map (κ : Kernel α β) {f : β → γ} (hf : Measurable f) :
mapOfMeasurable κ f hf = map κ f := by
simp [map, hf] | theorem | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | map_of_not_measurable | null |
map_apply (κ : Kernel α β) (hf : Measurable f) (a : α) : map κ f a = (κ a).map f := by
simp only [map, hf, ↓reduceDIte, mapOfMeasurable, coe_mk] | theorem | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | map_apply | null |
map_apply' (κ : Kernel α β) (hf : Measurable f) (a : α) {s : Set γ} (hs : MeasurableSet s) :
map κ f a s = κ a (f ⁻¹' s) := by rw [map_apply _ hf, Measure.map_apply hf hs] | theorem | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | map_apply' | null |
map_comp_right (κ : Kernel α β) {f : β → γ} (hf : Measurable f) {g : γ → δ}
(hg : Measurable g) : κ.map (g ∘ f) = (κ.map f).map g := by
ext1 x
rw [map_apply _ hg, map_apply _ hf, Measure.map_map hg hf, ← map_apply _ (hg.comp hf)]
@[simp] | lemma | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | map_comp_right | null |
map_zero : Kernel.map (0 : Kernel α β) f = 0 := by
ext
by_cases hf : Measurable f
· simp [map_apply, hf]
· simp [map_of_not_measurable _ hf]
@[simp] | lemma | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | map_zero | null |
map_id (κ : Kernel α β) : map κ id = κ := by
ext a
simp [map_apply, measurable_id]
@[simp] | lemma | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | map_id | null |
map_id' (κ : Kernel α β) : map κ (fun a ↦ a) = κ := map_id κ
nonrec theorem lintegral_map (κ : Kernel α β) (hf : Measurable f) (a : α) {g' : γ → ℝ≥0∞}
(hg : Measurable g') : ∫⁻ b, g' b ∂map κ f a = ∫⁻ a, g' (f a) ∂κ a := by
rw [map_apply _ hf, lintegral_map hg hf] | lemma | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | map_id' | null |
map_apply_eq_iff_map_symm_apply_eq (κ : Kernel α β) {f : β ≃ᵐ γ} (η : Kernel α γ) :
κ.map f = η ↔ κ = η.map f.symm := by
simp_rw [Kernel.ext_iff, map_apply _ f.measurable, map_apply _ f.symm.measurable,
f.map_apply_eq_iff_map_symm_apply_eq] | lemma | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | map_apply_eq_iff_map_symm_apply_eq | null |
sum_map_seq (κ : Kernel α β) [IsSFiniteKernel κ] (f : β → γ) :
(Kernel.sum fun n => map (seq κ n) f) = map κ f := by
by_cases hf : Measurable f
· ext a s hs
rw [Kernel.sum_apply, map_apply' κ hf a hs, Measure.sum_apply _ hs, ← measure_sum_seq κ,
Measure.sum_apply _ (hf hs)]
simp_rw [map_apply' _ hf _ hs]
· simp [map_of_not_measurable _ hf] | theorem | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | sum_map_seq | null |
IsMarkovKernel.map (κ : Kernel α β) [IsMarkovKernel κ] (hf : Measurable f) :
IsMarkovKernel (map κ f) :=
⟨fun a => ⟨by rw [map_apply' κ hf a MeasurableSet.univ, Set.preimage_univ, measure_univ]⟩⟩ | lemma | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | IsMarkovKernel.map | null |
IsZeroOrMarkovKernel.map (κ : Kernel α β) [IsZeroOrMarkovKernel κ] (f : β → γ) :
IsZeroOrMarkovKernel (map κ f) := by
by_cases hf : Measurable f
· rcases eq_zero_or_isMarkovKernel κ with rfl | h
· simp only [map_zero]; infer_instance
· have := IsMarkovKernel.map κ hf; infer_instance
· simp only [map_of_not_measurable _ hf]; infer_instance | instance | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | IsZeroOrMarkovKernel.map | null |
IsFiniteKernel.map (κ : Kernel α β) [IsFiniteKernel κ] (f : β → γ) :
IsFiniteKernel (map κ f) := by
refine ⟨⟨κ.bound, κ.bound_lt_top, fun a => ?_⟩⟩
by_cases hf : Measurable f
· rw [map_apply' κ hf a MeasurableSet.univ]
exact measure_le_bound κ a _
· simp [map_of_not_measurable _ hf] | instance | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | IsFiniteKernel.map | null |
IsSFiniteKernel.map (κ : Kernel α β) [IsSFiniteKernel κ] (f : β → γ) :
IsSFiniteKernel (map κ f) :=
⟨⟨fun n => Kernel.map (seq κ n) f, inferInstance, (sum_map_seq κ f).symm⟩⟩
@[simp] | instance | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | IsSFiniteKernel.map | null |
map_const (μ : Measure α) {f : α → β} (hf : Measurable f) :
map (const γ μ) f = const γ (μ.map f) := by
ext x s hs
rw [map_apply' _ hf _ hs, const_apply, const_apply, Measure.map_apply hf hs] | lemma | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | map_const | null |
comap (κ : Kernel α β) (g : γ → α) (hg : Measurable g) : Kernel γ β where
toFun a := κ (g a)
measurable' := κ.measurable.comp hg
@[simp, norm_cast] | def | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | comap | Pullback of a kernel, such that for each set s `comap κ g hg c s = κ (g c) s`.
We include measurability in the assumptions instead of using junk values
to make sure that typeclass inference can infer that the `comap` of a Markov kernel
is again a Markov kernel. |
coe_comap (κ : Kernel α β) (g : γ → α) (hg : Measurable g) : κ.comap g hg = κ ∘ g := rfl | lemma | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | coe_comap | null |
comap_apply (κ : Kernel α β) (hg : Measurable g) (c : γ) : comap κ g hg c = κ (g c) :=
rfl | theorem | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | comap_apply | null |
comap_apply' (κ : Kernel α β) (hg : Measurable g) (c : γ) (s : Set β) :
comap κ g hg c s = κ (g c) s :=
rfl
@[simp] | theorem | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | comap_apply' | null |
comap_zero (hg : Measurable g) : Kernel.comap (0 : Kernel α β) g hg = 0 := by
ext; rw [Kernel.comap_apply]; simp
@[simp] | lemma | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | comap_zero | null |
comap_id (κ : Kernel α β) : comap κ id measurable_id = κ := by ext a; rw [comap_apply]; simp
@[simp] | lemma | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | comap_id | null |
comap_id' (κ : Kernel α β) : comap κ (fun a ↦ a) measurable_id = κ := comap_id κ | lemma | Probability | [
"Mathlib.Probability.Kernel.Basic"
] | Mathlib/Probability/Kernel/Composition/MapComap.lean | comap_id' | null |
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