statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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comp_prod_fun_tsum_left (κ : kernel α β) (η : kernel (α × β) γ) [is_s_finite_kernel κ]
(a : α) (s : set (β × γ)) :
comp_prod_fun κ η a s = ∑' n, comp_prod_fun (seq κ n) η a s | by simp_rw [comp_prod_fun, (measure_sum_seq κ _).symm, lintegral_sum_measure] | lemma | probability_theory.kernel.comp_prod_fun_tsum_left | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_prod_fun_eq_tsum (κ : kernel α β) [is_s_finite_kernel κ]
(η : kernel (α × β) γ) [is_s_finite_kernel η] (a : α) (hs : measurable_set s) :
comp_prod_fun κ η a s = ∑' n m, comp_prod_fun (seq κ n) (seq η m) a s | by simp_rw [comp_prod_fun_tsum_left κ η a s, comp_prod_fun_tsum_right _ η a hs] | lemma | probability_theory.kernel.comp_prod_fun_eq_tsum | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_comp_prod_fun_of_finite (κ : kernel α β) [is_finite_kernel κ]
(η : kernel (α × β) γ) [is_finite_kernel η] (hs : measurable_set s) :
measurable (λ a, comp_prod_fun κ η a s) | begin
simp only [comp_prod_fun],
have h_meas : measurable (function.uncurry (λ a b, η (a, b) {c : γ | (b, c) ∈ s})),
{ have : function.uncurry (λ a b, η (a, b) {c : γ | (b, c) ∈ s})
= λ p, η p {c : γ | (p.2, c) ∈ s},
{ ext1 p,
have hp_eq_mk : p = (p.fst, p.snd) := prod.mk.eta.symm,
rw [hp_eq... | lemma | probability_theory.kernel.measurable_comp_prod_fun_of_finite | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable",
"measurable_set",
"measurable_snd"
] | Auxiliary lemma for `measurable_comp_prod_fun`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
measurable_comp_prod_fun (κ : kernel α β) [is_s_finite_kernel κ]
(η : kernel (α × β) γ) [is_s_finite_kernel η] (hs : measurable_set s) :
measurable (λ a, comp_prod_fun κ η a s) | begin
simp_rw comp_prod_fun_tsum_right κ η _ hs,
refine measurable.ennreal_tsum (λ n, _),
simp only [comp_prod_fun],
have h_meas : measurable (function.uncurry (λ a b, seq η n (a, b) {c : γ | (b, c) ∈ s})),
{ have : function.uncurry (λ a b, seq η n (a, b) {c : γ | (b, c) ∈ s})
= λ p, seq η n p {c : γ | ... | lemma | probability_theory.kernel.measurable_comp_prod_fun | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable",
"measurable.ennreal_tsum",
"measurable_set",
"measurable_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_prod (κ : kernel α β) [is_s_finite_kernel κ]
(η : kernel (α × β) γ) [is_s_finite_kernel η] :
kernel α (β × γ) | { val := λ a, measure.of_measurable (λ s hs, comp_prod_fun κ η a s) (comp_prod_fun_empty κ η a)
(comp_prod_fun_Union κ η a),
property :=
begin
refine measure.measurable_of_measurable_coe _ (λ s hs, _),
have : (λ a, measure.of_measurable (λ s hs, comp_prod_fun κ η a s) (comp_prod_fun_empty κ η a)
... | def | probability_theory.kernel.comp_prod | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | Composition-Product of kernels. It verifies
`∫⁻ bc, f bc ∂(comp_prod κ η a) = ∫⁻ b, ∫⁻ c, f (b, c) ∂(η (a, b)) ∂(κ a)`
(see `lintegral_comp_prod`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_prod_apply_eq_comp_prod_fun (κ : kernel α β) [is_s_finite_kernel κ]
(η : kernel (α × β) γ) [is_s_finite_kernel η] (a : α) (hs : measurable_set s) :
(κ ⊗ₖ η) a s = comp_prod_fun κ η a s | begin
rw [comp_prod],
change measure.of_measurable (λ s hs, comp_prod_fun κ η a s) (comp_prod_fun_empty κ η a)
(comp_prod_fun_Union κ η a) s = ∫⁻ b, η (a, b) {c | (b, c) ∈ s} ∂(κ a),
rw measure.of_measurable_apply _ hs,
refl,
end | lemma | probability_theory.kernel.comp_prod_apply_eq_comp_prod_fun | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_prod_apply (κ : kernel α β) [is_s_finite_kernel κ] (η : kernel (α × β) γ)
[is_s_finite_kernel η] (a : α) (hs : measurable_set s) :
(κ ⊗ₖ η) a s = ∫⁻ b, η (a, b) {c | (b, c) ∈ s} ∂(κ a) | comp_prod_apply_eq_comp_prod_fun κ η a hs | lemma | probability_theory.kernel.comp_prod_apply | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_comp_prod_apply (κ : kernel α β) [is_s_finite_kernel κ] (η : kernel (α × β) γ)
[is_s_finite_kernel η] (a : α) (s : set (β × γ)) :
∫⁻ b, η (a, b) {c | (b, c) ∈ s} ∂(κ a) ≤ (κ ⊗ₖ η) a s | calc ∫⁻ b, η (a, b) {c | (b, c) ∈ s} ∂(κ a)
≤ ∫⁻ b, η (a, b) {c | (b, c) ∈ (to_measurable ((κ ⊗ₖ η) a) s)} ∂(κ a) :
lintegral_mono (λ b, measure_mono (λ _ h_mem, subset_to_measurable _ _ h_mem))
... = (κ ⊗ₖ η) a (to_measurable ((κ ⊗ₖ η) a) s) :
(kernel.comp_prod_apply_eq_comp_prod_fun κ η a (measurable_... | lemma | probability_theory.kernel.le_comp_prod_apply | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ae_kernel_lt_top (a : α) (h2s : (κ ⊗ₖ η) a s ≠ ∞) :
∀ᵐ b ∂(κ a), η (a, b) (prod.mk b ⁻¹' s) < ∞ | begin
let t := to_measurable ((κ ⊗ₖ η) a) s,
have : ∀ (b : β), η (a, b) (prod.mk b ⁻¹' s) ≤ η (a, b) (prod.mk b ⁻¹' t),
{ exact λ b, measure_mono (set.preimage_mono (subset_to_measurable _ _)), },
have ht : measurable_set t := measurable_set_to_measurable _ _,
have h2t : (κ ⊗ₖ η) a t ≠ ∞, by rwa measure_to_me... | lemma | probability_theory.kernel.ae_kernel_lt_top | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_set",
"set.preimage_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_prod_null (a : α) (hs : measurable_set s) :
(κ ⊗ₖ η) a s = 0 ↔ (λ b, η (a, b) (prod.mk b ⁻¹' s)) =ᵐ[κ a] 0 | begin
rw [kernel.comp_prod_apply _ _ _ hs, lintegral_eq_zero_iff],
{ refl, },
{ exact kernel.measurable_kernel_prod_mk_left' hs a, },
end | lemma | probability_theory.kernel.comp_prod_null | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ae_null_of_comp_prod_null (h : (κ ⊗ₖ η) a s = 0) :
(λ b, η (a, b) (prod.mk b ⁻¹' s)) =ᵐ[κ a] 0 | begin
obtain ⟨t, hst, mt, ht⟩ := exists_measurable_superset_of_null h,
simp_rw [comp_prod_null a mt] at ht,
rw [filter.eventually_le_antisymm_iff],
exact ⟨filter.eventually_le.trans_eq
(filter.eventually_of_forall $ λ x, (measure_mono (set.preimage_mono hst) : _)) ht,
filter.eventually_of_forall $ λ x, ... | lemma | probability_theory.kernel.ae_null_of_comp_prod_null | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"filter.eventually_le_antisymm_iff",
"filter.eventually_of_forall",
"set.preimage_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ae_ae_of_ae_comp_prod {p : β × γ → Prop} (h : ∀ᵐ bc ∂((κ ⊗ₖ η) a), p bc) :
∀ᵐ b ∂(κ a), ∀ᵐ c ∂(η (a, b)), p (b, c) | ae_null_of_comp_prod_null h | lemma | probability_theory.kernel.ae_ae_of_ae_comp_prod | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_prod_restrict {s : set β} {t : set γ} (hs : measurable_set s) (ht : measurable_set t) :
(kernel.restrict κ hs) ⊗ₖ (kernel.restrict η ht) = kernel.restrict (κ ⊗ₖ η) (hs.prod ht) | begin
ext a u hu : 2,
rw [comp_prod_apply _ _ _ hu, restrict_apply' _ _ _ hu,
comp_prod_apply _ _ _ (hu.inter (hs.prod ht))],
simp only [kernel.restrict_apply, measure.restrict_apply' ht, set.mem_inter_iff,
set.prod_mk_mem_set_prod_eq],
have : ∀ b, η (a, b) {c : γ | (b, c) ∈ u ∧ b ∈ s ∧ c ∈ t}
= s.i... | lemma | probability_theory.kernel.comp_prod_restrict | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_set",
"set.mem_inter_iff",
"set.prod_mk_mem_set_prod_eq",
"set.set_of_false"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_prod_restrict_left {s : set β} (hs : measurable_set s) :
(kernel.restrict κ hs) ⊗ₖ η = kernel.restrict (κ ⊗ₖ η) (hs.prod measurable_set.univ) | by { rw ← comp_prod_restrict, congr, exact kernel.restrict_univ.symm, } | lemma | probability_theory.kernel.comp_prod_restrict_left | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_set",
"measurable_set.univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_prod_restrict_right {t : set γ} (ht : measurable_set t) :
κ ⊗ₖ (kernel.restrict η ht) = kernel.restrict (κ ⊗ₖ η) (measurable_set.univ.prod ht) | by { rw ← comp_prod_restrict, congr, exact kernel.restrict_univ.symm, } | lemma | probability_theory.kernel.comp_prod_restrict_right | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_comp_prod' (κ : kernel α β) [is_s_finite_kernel κ] (η : kernel (α × β) γ)
[is_s_finite_kernel η] (a : α) {f : β → γ → ℝ≥0∞} (hf : measurable (function.uncurry f)) :
∫⁻ bc, f bc.1 bc.2 ∂((κ ⊗ₖ η) a) = ∫⁻ b, ∫⁻ c, f b c ∂(η (a, b)) ∂(κ a) | begin
let F : ℕ → simple_func (β × γ) ℝ≥0∞ := simple_func.eapprox (function.uncurry f),
have h : ∀ a, (⨆ n, F n a) = function.uncurry f a,
from simple_func.supr_eapprox_apply (function.uncurry f) hf,
simp only [prod.forall, function.uncurry_apply_pair] at h,
simp_rw [← h, prod.mk.eta],
have h_mono : monot... | theorem | probability_theory.kernel.lintegral_comp_prod' | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable",
"measurable.comp",
"measurable.lintegral_kernel_prod_right",
"measurable_prod_mk_left",
"measurable_snd",
"monotone"
] | Lebesgue integral against the composition-product of two kernels. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lintegral_comp_prod (κ : kernel α β) [is_s_finite_kernel κ] (η : kernel (α × β) γ)
[is_s_finite_kernel η] (a : α) {f : β × γ → ℝ≥0∞} (hf : measurable f) :
∫⁻ bc, f bc ∂((κ ⊗ₖ η) a) = ∫⁻ b, ∫⁻ c, f (b, c) ∂(η (a, b)) ∂(κ a) | begin
let g := function.curry f,
change ∫⁻ bc, f bc ∂((κ ⊗ₖ η) a) = ∫⁻ b, ∫⁻ c, g b c ∂(η (a, b)) ∂(κ a),
rw ← lintegral_comp_prod',
{ simp_rw [g, function.curry_apply, prod.mk.eta], },
{ simp_rw [g, function.uncurry_curry], exact hf, },
end | theorem | probability_theory.kernel.lintegral_comp_prod | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable"
] | Lebesgue integral against the composition-product of two kernels. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lintegral_comp_prod₀ (κ : kernel α β) [is_s_finite_kernel κ] (η : kernel (α × β) γ)
[is_s_finite_kernel η] (a : α) {f : β × γ → ℝ≥0∞} (hf : ae_measurable f ((κ ⊗ₖ η) a)) :
∫⁻ z, f z ∂((κ ⊗ₖ η) a) = ∫⁻ x, ∫⁻ y, f (x, y) ∂(η (a, x)) ∂(κ a) | begin
have A : ∫⁻ z, f z ∂((κ ⊗ₖ η) a) = ∫⁻ z, hf.mk f z ∂((κ ⊗ₖ η) a) :=
lintegral_congr_ae hf.ae_eq_mk,
have B : ∫⁻ x, ∫⁻ y, f (x, y) ∂(η (a, x)) ∂(κ a) = ∫⁻ x, ∫⁻ y, hf.mk f (x, y) ∂(η (a, x)) ∂(κ a),
{ apply lintegral_congr_ae,
filter_upwards [ae_ae_of_ae_comp_prod hf.ae_eq_mk] with _ ha using lintegr... | lemma | probability_theory.kernel.lintegral_comp_prod₀ | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"ae_measurable"
] | Lebesgue integral against the composition-product of two kernels. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_lintegral_comp_prod (κ : kernel α β) [is_s_finite_kernel κ] (η : kernel (α × β) γ)
[is_s_finite_kernel η] (a : α) {f : β × γ → ℝ≥0∞} (hf : measurable f)
{s : set β} {t : set γ} (hs : measurable_set s) (ht : measurable_set t) :
∫⁻ z in s ×ˢ t, f z ∂((κ ⊗ₖ η) a) = ∫⁻ x in s, ∫⁻ y in t, f (x, y) ∂(η (a, x)) ∂(κ ... | by simp_rw [← kernel.restrict_apply (κ ⊗ₖ η) (hs.prod ht), ← comp_prod_restrict,
lintegral_comp_prod _ _ _ hf, kernel.restrict_apply] | lemma | probability_theory.kernel.set_lintegral_comp_prod | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_lintegral_comp_prod_univ_right (κ : kernel α β) [is_s_finite_kernel κ]
(η : kernel (α × β) γ) [is_s_finite_kernel η] (a : α) {f : β × γ → ℝ≥0∞} (hf : measurable f)
{s : set β} (hs : measurable_set s) :
∫⁻ z in s ×ˢ set.univ, f z ∂((κ ⊗ₖ η) a) = ∫⁻ x in s, ∫⁻ y, f (x, y) ∂(η (a, x)) ∂(κ a) | by simp_rw [set_lintegral_comp_prod κ η a hf hs measurable_set.univ, measure.restrict_univ] | lemma | probability_theory.kernel.set_lintegral_comp_prod_univ_right | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable",
"measurable_set",
"measurable_set.univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_lintegral_comp_prod_univ_left (κ : kernel α β) [is_s_finite_kernel κ]
(η : kernel (α × β) γ) [is_s_finite_kernel η] (a : α) {f : β × γ → ℝ≥0∞} (hf : measurable f)
{t : set γ} (ht : measurable_set t) :
∫⁻ z in set.univ ×ˢ t, f z ∂((κ ⊗ₖ η) a) = ∫⁻ x, ∫⁻ y in t, f (x, y) ∂(η (a, x)) ∂(κ a) | by simp_rw [set_lintegral_comp_prod κ η a hf measurable_set.univ ht, measure.restrict_univ] | lemma | probability_theory.kernel.set_lintegral_comp_prod_univ_left | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable",
"measurable_set",
"measurable_set.univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_prod_eq_tsum_comp_prod (κ : kernel α β) [is_s_finite_kernel κ] (η : kernel (α × β) γ)
[is_s_finite_kernel η] (a : α) (hs : measurable_set s) :
(κ ⊗ₖ η) a s = ∑' (n m : ℕ), (seq κ n ⊗ₖ seq η m) a s | by { simp_rw comp_prod_apply_eq_comp_prod_fun _ _ _ hs, exact comp_prod_fun_eq_tsum κ η a hs, } | lemma | probability_theory.kernel.comp_prod_eq_tsum_comp_prod | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_prod_eq_sum_comp_prod (κ : kernel α β) [is_s_finite_kernel κ]
(η : kernel (α × β) γ) [is_s_finite_kernel η] :
κ ⊗ₖ η = kernel.sum (λ n, kernel.sum (λ m, seq κ n ⊗ₖ seq η m)) | by { ext a s hs : 2, simp_rw [kernel.sum_apply' _ a hs], rw comp_prod_eq_tsum_comp_prod κ η a hs, } | lemma | probability_theory.kernel.comp_prod_eq_sum_comp_prod | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_prod_eq_sum_comp_prod_left (κ : kernel α β) [is_s_finite_kernel κ]
(η : kernel (α × β) γ) [is_s_finite_kernel η] :
κ ⊗ₖ η = kernel.sum (λ n, seq κ n ⊗ₖ η) | begin
rw comp_prod_eq_sum_comp_prod,
congr' with n a s hs,
simp_rw [kernel.sum_apply' _ _ hs, comp_prod_apply_eq_comp_prod_fun _ _ _ hs,
comp_prod_fun_tsum_right _ η a hs],
end | lemma | probability_theory.kernel.comp_prod_eq_sum_comp_prod_left | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_prod_eq_sum_comp_prod_right (κ : kernel α β) [is_s_finite_kernel κ]
(η : kernel (α × β) γ) [is_s_finite_kernel η] :
κ ⊗ₖ η = kernel.sum (λ n, κ ⊗ₖ seq η n) | begin
rw comp_prod_eq_sum_comp_prod,
simp_rw comp_prod_eq_sum_comp_prod_left κ _,
rw kernel.sum_comm,
end | lemma | probability_theory.kernel.comp_prod_eq_sum_comp_prod_right | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_markov_kernel.comp_prod (κ : kernel α β) [is_markov_kernel κ]
(η : kernel (α × β) γ) [is_markov_kernel η] :
is_markov_kernel (κ ⊗ₖ η) | ⟨λ a, ⟨begin
rw comp_prod_apply κ η a measurable_set.univ,
simp only [set.mem_univ, set.set_of_true, measure_univ, lintegral_one],
end⟩⟩ | instance | probability_theory.kernel.is_markov_kernel.comp_prod | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_set.univ",
"set.mem_univ",
"set.set_of_true"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_prod_apply_univ_le (κ : kernel α β) [is_s_finite_kernel κ]
(η : kernel (α × β) γ) [is_finite_kernel η] (a : α) :
(κ ⊗ₖ η) a set.univ ≤ (κ a set.univ) * (is_finite_kernel.bound η) | begin
rw comp_prod_apply κ η a measurable_set.univ,
simp only [set.mem_univ, set.set_of_true],
let Cη := is_finite_kernel.bound η,
calc ∫⁻ b, η (a, b) set.univ ∂(κ a)
≤ ∫⁻ b, Cη ∂(κ a) : lintegral_mono (λ b, measure_le_bound η (a, b) set.univ)
... = Cη * κ a set.univ : measure_theory.lintegral_const Cη
... | lemma | probability_theory.kernel.comp_prod_apply_univ_le | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_set.univ",
"measure_theory.lintegral_const",
"mul_comm",
"set.mem_univ",
"set.set_of_true"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel.comp_prod (κ : kernel α β) [is_finite_kernel κ]
(η : kernel (α × β) γ) [is_finite_kernel η] :
is_finite_kernel (κ ⊗ₖ η) | ⟨⟨is_finite_kernel.bound κ * is_finite_kernel.bound η,
ennreal.mul_lt_top (is_finite_kernel.bound_ne_top κ) (is_finite_kernel.bound_ne_top η),
λ a, calc (κ ⊗ₖ η) a set.univ
≤ (κ a set.univ) * is_finite_kernel.bound η : comp_prod_apply_univ_le κ η a
... ≤ is_finite_kernel.bound κ * is_finite_kernel.bound η :
... | instance | probability_theory.kernel.is_finite_kernel.comp_prod | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"ennreal.mul_lt_top",
"le_rfl",
"mul_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_s_finite_kernel.comp_prod (κ : kernel α β) [is_s_finite_kernel κ]
(η : kernel (α × β) γ) [is_s_finite_kernel η] :
is_s_finite_kernel (κ ⊗ₖ η) | begin
rw comp_prod_eq_sum_comp_prod,
exact kernel.is_s_finite_kernel_sum (λ n, kernel.is_s_finite_kernel_sum infer_instance),
end | instance | probability_theory.kernel.is_s_finite_kernel.comp_prod | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (κ : kernel α β) (f : β → γ) (hf : measurable f) : kernel α γ | { val := λ a, (κ a).map f,
property := (measure.measurable_map _ hf).comp (kernel.measurable κ) } | def | probability_theory.kernel.map | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable"
] | The pushforward of a kernel along a measurable function.
We include measurability in the assumptions instead of using junk values
to make sure that typeclass inference can infer that the `map` of a Markov kernel
is again a Markov kernel. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_apply (κ : kernel α β) (hf : measurable f) (a : α) :
map κ f hf a = (κ a).map f | rfl | lemma | probability_theory.kernel.map_apply | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_apply' (κ : kernel α β) (hf : measurable f) (a : α) {s : set γ} (hs : measurable_set s) :
map κ f hf a s = κ a (f ⁻¹' s) | by rw [map_apply, measure.map_apply hf hs] | lemma | probability_theory.kernel.map_apply' | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_map (κ : kernel α β) (hf : measurable f) (a : α)
{g' : γ → ℝ≥0∞} (hg : measurable g') :
∫⁻ b, g' b ∂(map κ f hf a) = ∫⁻ a, g' (f a) ∂(κ a) | by rw [map_apply _ hf, lintegral_map hg hf] | lemma | probability_theory.kernel.lintegral_map | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_map_seq (κ : kernel α β) [is_s_finite_kernel κ] (hf : measurable f) :
kernel.sum (λ n, map (seq κ n) f hf) = map κ f hf | begin
ext a s hs : 2,
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,
end | lemma | probability_theory.kernel.sum_map_seq | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_markov_kernel.map (κ : kernel α β) [is_markov_kernel κ] (hf : measurable f) :
is_markov_kernel (map κ f hf) | ⟨λ a, ⟨by rw [map_apply' κ hf a measurable_set.univ, set.preimage_univ, measure_univ]⟩⟩ | instance | probability_theory.kernel.is_markov_kernel.map | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable",
"measurable_set.univ",
"set.preimage_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel.map (κ : kernel α β) [is_finite_kernel κ] (hf : measurable f) :
is_finite_kernel (map κ f hf) | begin
refine ⟨⟨is_finite_kernel.bound κ, is_finite_kernel.bound_lt_top κ, λ a, _⟩⟩,
rw map_apply' κ hf a measurable_set.univ,
exact measure_le_bound κ a _,
end | instance | probability_theory.kernel.is_finite_kernel.map | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable",
"measurable_set.univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_s_finite_kernel.map (κ : kernel α β) [is_s_finite_kernel κ] (hf : measurable f) :
is_s_finite_kernel (map κ f hf) | ⟨⟨λ n, map (seq κ n) f hf, infer_instance, (sum_map_seq κ hf).symm⟩⟩ | instance | probability_theory.kernel.is_s_finite_kernel.map | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap (κ : kernel α β) (g : γ → α) (hg : measurable g) : kernel γ β | { val := λ a, κ (g a),
property := (kernel.measurable κ).comp hg } | def | probability_theory.kernel.comap | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_apply (κ : kernel α β) (hg : measurable g) (c : γ) :
comap κ g hg c = κ (g c) | rfl | lemma | probability_theory.kernel.comap_apply | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 | lemma | probability_theory.kernel.comap_apply' | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_comap (κ : kernel α β) (hg : measurable g) (c : γ) (g' : β → ℝ≥0∞) :
∫⁻ b, g' b ∂(comap κ g hg c) = ∫⁻ b, g' b ∂(κ (g c)) | rfl | lemma | probability_theory.kernel.lintegral_comap | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_comap_seq (κ : kernel α β) [is_s_finite_kernel κ] (hg : measurable g) :
kernel.sum (λ n, comap (seq κ n) g hg) = comap κ g hg | begin
ext a s hs : 2,
rw [kernel.sum_apply, comap_apply' κ hg a s, measure.sum_apply _ hs, ← measure_sum_seq κ,
measure.sum_apply _ hs],
simp_rw comap_apply' _ hg _ s,
end | lemma | probability_theory.kernel.sum_comap_seq | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_markov_kernel.comap (κ : kernel α β) [is_markov_kernel κ] (hg : measurable g) :
is_markov_kernel (comap κ g hg) | ⟨λ a, ⟨by rw [comap_apply' κ hg a set.univ, measure_univ]⟩⟩ | instance | probability_theory.kernel.is_markov_kernel.comap | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel.comap (κ : kernel α β) [is_finite_kernel κ] (hg : measurable g) :
is_finite_kernel (comap κ g hg) | begin
refine ⟨⟨is_finite_kernel.bound κ, is_finite_kernel.bound_lt_top κ, λ a, _⟩⟩,
rw comap_apply' κ hg a set.univ,
exact measure_le_bound κ _ _,
end | instance | probability_theory.kernel.is_finite_kernel.comap | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_s_finite_kernel.comap (κ : kernel α β) [is_s_finite_kernel κ] (hg : measurable g) :
is_s_finite_kernel (comap κ g hg) | ⟨⟨λ n, comap (seq κ n) g hg, infer_instance, (sum_comap_seq κ hg).symm⟩⟩ | instance | probability_theory.kernel.is_s_finite_kernel.comap | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mk_left (γ : Type*) [measurable_space γ] (κ : kernel α β) : kernel (γ × α) β | comap κ prod.snd measurable_snd | def | probability_theory.kernel.prod_mk_left | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_snd",
"measurable_space"
] | Define a `kernel (γ × α) β` from a `kernel α β` by taking the comap of the projection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_mk_left_apply (κ : kernel α β) (ca : γ × α) :
prod_mk_left γ κ ca = κ ca.snd | rfl | lemma | probability_theory.kernel.prod_mk_left_apply | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mk_left_apply' (κ : kernel α β) (ca : γ × α) (s : set β) :
prod_mk_left γ κ ca s = κ ca.snd s | rfl | lemma | probability_theory.kernel.prod_mk_left_apply' | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_prod_mk_left (κ : kernel α β) (ca : γ × α) (g : β → ℝ≥0∞) :
∫⁻ b, g b ∂(prod_mk_left γ κ ca) = ∫⁻ b, g b ∂(κ ca.snd) | rfl | lemma | probability_theory.kernel.lintegral_prod_mk_left | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_markov_kernel.prod_mk_left (κ : kernel α β) [is_markov_kernel κ] :
is_markov_kernel (prod_mk_left γ κ) | by { rw prod_mk_left, apply_instance, } | instance | probability_theory.kernel.is_markov_kernel.prod_mk_left | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel.prod_mk_left (κ : kernel α β) [is_finite_kernel κ] :
is_finite_kernel (prod_mk_left γ κ) | by { rw prod_mk_left, apply_instance, } | instance | probability_theory.kernel.is_finite_kernel.prod_mk_left | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_s_finite_kernel.prod_mk_left (κ : kernel α β) [is_s_finite_kernel κ] :
is_s_finite_kernel (prod_mk_left γ κ) | by { rw prod_mk_left, apply_instance, } | instance | probability_theory.kernel.is_s_finite_kernel.prod_mk_left | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
swap_left (κ : kernel (α × β) γ) : kernel (β × α) γ | comap κ prod.swap measurable_swap | def | probability_theory.kernel.swap_left | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_swap",
"prod.swap"
] | Define a `kernel (β × α) γ` from a `kernel (α × β) γ` by taking the comap of `prod.swap`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
swap_left_apply (κ : kernel (α × β) γ) (a : β × α) :
swap_left κ a = (κ a.swap) | rfl | lemma | probability_theory.kernel.swap_left_apply | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
swap_left_apply' (κ : kernel (α × β) γ) (a : β × α) (s : set γ) :
swap_left κ a s = κ a.swap s | rfl | lemma | probability_theory.kernel.swap_left_apply' | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_swap_left (κ : kernel (α × β) γ) (a : β × α) (g : γ → ℝ≥0∞) :
∫⁻ c, g c ∂(swap_left κ a) = ∫⁻ c, g c ∂(κ a.swap) | by { rw [swap_left, lintegral_comap _ measurable_swap a], } | lemma | probability_theory.kernel.lintegral_swap_left | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_markov_kernel.swap_left (κ : kernel (α × β) γ) [is_markov_kernel κ] :
is_markov_kernel (swap_left κ) | by { rw swap_left, apply_instance, } | instance | probability_theory.kernel.is_markov_kernel.swap_left | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel.swap_left (κ : kernel (α × β) γ) [is_finite_kernel κ] :
is_finite_kernel (swap_left κ) | by { rw swap_left, apply_instance, } | instance | probability_theory.kernel.is_finite_kernel.swap_left | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_s_finite_kernel.swap_left (κ : kernel (α × β) γ) [is_s_finite_kernel κ] :
is_s_finite_kernel (swap_left κ) | by { rw swap_left, apply_instance, } | instance | probability_theory.kernel.is_s_finite_kernel.swap_left | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
swap_right (κ : kernel α (β × γ)) : kernel α (γ × β) | map κ prod.swap measurable_swap | def | probability_theory.kernel.swap_right | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_swap",
"prod.swap",
"swap_right"
] | Define a `kernel α (γ × β)` from a `kernel α (β × γ)` by taking the map of `prod.swap`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
swap_right_apply (κ : kernel α (β × γ)) (a : α) :
swap_right κ a = (κ a).map prod.swap | rfl | lemma | probability_theory.kernel.swap_right_apply | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"prod.swap",
"swap_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
swap_right_apply' (κ : kernel α (β × γ)) (a : α) {s : set (γ × β)} (hs : measurable_set s) :
swap_right κ a s = κ a {p | p.swap ∈ s} | by { rw [swap_right_apply, measure.map_apply measurable_swap hs], refl, } | lemma | probability_theory.kernel.swap_right_apply' | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_set",
"measurable_swap",
"swap_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_swap_right (κ : kernel α (β × γ)) (a : α) {g : γ × β → ℝ≥0∞} (hg : measurable g) :
∫⁻ c, g c ∂(swap_right κ a) = ∫⁻ (bc : β × γ), g bc.swap ∂(κ a) | by rw [swap_right, lintegral_map _ measurable_swap a hg] | lemma | probability_theory.kernel.lintegral_swap_right | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable",
"measurable_swap",
"swap_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_markov_kernel.swap_right (κ : kernel α (β × γ)) [is_markov_kernel κ] :
is_markov_kernel (swap_right κ) | by { rw swap_right, apply_instance, } | instance | probability_theory.kernel.is_markov_kernel.swap_right | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"swap_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel.swap_right (κ : kernel α (β × γ)) [is_finite_kernel κ] :
is_finite_kernel (swap_right κ) | by { rw swap_right, apply_instance, } | instance | probability_theory.kernel.is_finite_kernel.swap_right | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"swap_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_s_finite_kernel.swap_right (κ : kernel α (β × γ)) [is_s_finite_kernel κ] :
is_s_finite_kernel (swap_right κ) | by { rw swap_right, apply_instance, } | instance | probability_theory.kernel.is_s_finite_kernel.swap_right | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"swap_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst (κ : kernel α (β × γ)) : kernel α β | map κ prod.fst measurable_fst | def | probability_theory.kernel.fst | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_fst"
] | Define a `kernel α β` from a `kernel α (β × γ)` by taking the map of the first projection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst_apply (κ : kernel α (β × γ)) (a : α) :
fst κ a = (κ a).map prod.fst | rfl | lemma | probability_theory.kernel.fst_apply | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_apply' (κ : kernel α (β × γ)) (a : α) {s : set β} (hs : measurable_set s) :
fst κ a s = κ a {p | p.1 ∈ s} | by { rw [fst_apply, measure.map_apply measurable_fst hs], refl, } | lemma | probability_theory.kernel.fst_apply' | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_fst",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_fst (κ : kernel α (β × γ)) (a : α) {g : β → ℝ≥0∞} (hg : measurable g) :
∫⁻ c, g c ∂(fst κ a) = ∫⁻ (bc : β × γ), g bc.fst ∂(κ a) | by rw [fst, lintegral_map _ measurable_fst a hg] | lemma | probability_theory.kernel.lintegral_fst | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable",
"measurable_fst"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_markov_kernel.fst (κ : kernel α (β × γ)) [is_markov_kernel κ] :
is_markov_kernel (fst κ) | by { rw fst, apply_instance, } | instance | probability_theory.kernel.is_markov_kernel.fst | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel.fst (κ : kernel α (β × γ)) [is_finite_kernel κ] :
is_finite_kernel (fst κ) | by { rw fst, apply_instance, } | instance | probability_theory.kernel.is_finite_kernel.fst | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_s_finite_kernel.fst (κ : kernel α (β × γ)) [is_s_finite_kernel κ] :
is_s_finite_kernel (fst κ) | by { rw fst, apply_instance, } | instance | probability_theory.kernel.is_s_finite_kernel.fst | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd (κ : kernel α (β × γ)) : kernel α γ | map κ prod.snd measurable_snd | def | probability_theory.kernel.snd | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_snd"
] | Define a `kernel α γ` from a `kernel α (β × γ)` by taking the map of the second projection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd_apply (κ : kernel α (β × γ)) (a : α) :
snd κ a = (κ a).map prod.snd | rfl | lemma | probability_theory.kernel.snd_apply | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_apply' (κ : kernel α (β × γ)) (a : α) {s : set γ} (hs : measurable_set s) :
snd κ a s = κ a {p | p.2 ∈ s} | by { rw [snd_apply, measure.map_apply measurable_snd hs], refl, } | lemma | probability_theory.kernel.snd_apply' | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_set",
"measurable_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_snd (κ : kernel α (β × γ)) (a : α) {g : γ → ℝ≥0∞} (hg : measurable g) :
∫⁻ c, g c ∂(snd κ a) = ∫⁻ (bc : β × γ), g bc.snd ∂(κ a) | by rw [snd, lintegral_map _ measurable_snd a hg] | lemma | probability_theory.kernel.lintegral_snd | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable",
"measurable_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_markov_kernel.snd (κ : kernel α (β × γ)) [is_markov_kernel κ] :
is_markov_kernel (snd κ) | by { rw snd, apply_instance, } | instance | probability_theory.kernel.is_markov_kernel.snd | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel.snd (κ : kernel α (β × γ)) [is_finite_kernel κ] :
is_finite_kernel (snd κ) | by { rw snd, apply_instance, } | instance | probability_theory.kernel.is_finite_kernel.snd | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_s_finite_kernel.snd (κ : kernel α (β × γ)) [is_s_finite_kernel κ] :
is_s_finite_kernel (snd κ) | by { rw snd, apply_instance, } | instance | probability_theory.kernel.is_s_finite_kernel.snd | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (η : kernel β γ) (κ : kernel α β) : kernel α γ | { val := λ a, (κ a).bind η,
property := (measure.measurable_bind' (kernel.measurable _)).comp (kernel.measurable _) } | def | probability_theory.kernel.comp | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | Composition of two s-finite kernels. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_apply (η : kernel β γ) (κ : kernel α β) (a : α) :
(η ∘ₖ κ) a = (κ a).bind η | rfl | lemma | probability_theory.kernel.comp_apply | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply' (η : kernel β γ) (κ : kernel α β) (a : α) {s : set γ} (hs : measurable_set s) :
(η ∘ₖ κ) a s = ∫⁻ b, η b s ∂(κ a) | by rw [comp_apply, measure.bind_apply hs (kernel.measurable _)] | lemma | probability_theory.kernel.comp_apply' | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_eq_snd_comp_prod (η : kernel β γ) [is_s_finite_kernel η]
(κ : kernel α β) [is_s_finite_kernel κ] :
η ∘ₖ κ = snd (κ ⊗ₖ prod_mk_left α η) | begin
ext a s hs : 2,
rw [comp_apply' _ _ _ hs, snd_apply' _ _ hs, comp_prod_apply],
swap, { exact measurable_snd hs, },
simp only [set.mem_set_of_eq, set.set_of_mem_eq, prod_mk_left_apply' _ _ s],
end | lemma | probability_theory.kernel.comp_eq_snd_comp_prod | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_snd",
"set.set_of_mem_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_comp (η : kernel β γ) (κ : kernel α β)
(a : α) {g : γ → ℝ≥0∞} (hg : measurable g) :
∫⁻ c, g c ∂((η ∘ₖ κ) a) = ∫⁻ b, ∫⁻ c, g c ∂(η b) ∂(κ a) | by rw [comp_apply, measure.lintegral_bind (kernel.measurable _) hg] | lemma | probability_theory.kernel.lintegral_comp | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_markov_kernel.comp (η : kernel β γ) [is_markov_kernel η]
(κ : kernel α β) [is_markov_kernel κ] :
is_markov_kernel (η ∘ₖ κ) | by { rw comp_eq_snd_comp_prod, apply_instance, } | instance | probability_theory.kernel.is_markov_kernel.comp | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel.comp (η : kernel β γ) [is_finite_kernel η]
(κ : kernel α β) [is_finite_kernel κ] :
is_finite_kernel (η ∘ₖ κ) | by { rw comp_eq_snd_comp_prod, apply_instance, } | instance | probability_theory.kernel.is_finite_kernel.comp | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_s_finite_kernel.comp (η : kernel β γ) [is_s_finite_kernel η]
(κ : kernel α β) [is_s_finite_kernel κ] :
is_s_finite_kernel (η ∘ₖ κ) | by { rw comp_eq_snd_comp_prod, apply_instance, } | instance | probability_theory.kernel.is_s_finite_kernel.comp | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc {δ : Type*} {mδ : measurable_space δ} (ξ : kernel γ δ) [is_s_finite_kernel ξ]
(η : kernel β γ) (κ : kernel α β) :
(ξ ∘ₖ η ∘ₖ κ) = ξ ∘ₖ (η ∘ₖ κ) | begin
refine ext_fun (λ a f hf, _),
simp_rw [lintegral_comp _ _ _ hf, lintegral_comp _ _ _ hf.lintegral_kernel],
end | lemma | probability_theory.kernel.comp_assoc | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_space"
] | Composition of kernels is associative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
deterministic_comp_eq_map (hf : measurable f) (κ : kernel α β) :
(deterministic f hf ∘ₖ κ) = map κ f hf | begin
ext a s hs : 2,
simp_rw [map_apply' _ _ _ hs, comp_apply' _ _ _ hs, deterministic_apply' hf _ hs,
lintegral_indicator_const_comp hf hs, one_mul],
end | lemma | probability_theory.kernel.deterministic_comp_eq_map | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_deterministic_eq_comap (κ : kernel α β) (hg : measurable g) :
(κ ∘ₖ deterministic g hg) = comap κ g hg | begin
ext a s hs : 2,
simp_rw [comap_apply' _ _ _ s, comp_apply' _ _ _ hs, deterministic_apply hg a,
lintegral_dirac' _ (kernel.measurable_coe κ hs)],
end | lemma | probability_theory.kernel.comp_deterministic_eq_comap | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod (κ : kernel α β) [is_s_finite_kernel κ] (η : kernel α γ) [is_s_finite_kernel η] :
kernel α (β × γ) | κ ⊗ₖ (swap_left (prod_mk_left β η)) | def | probability_theory.kernel.prod | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | Product of two s-finite kernels. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_apply (κ : kernel α β) [is_s_finite_kernel κ] (η : kernel α γ) [is_s_finite_kernel η]
(a : α) {s : set (β × γ)} (hs : measurable_set s) :
(κ ×ₖ η) a s = ∫⁻ (b : β), (η a) {c : γ | (b, c) ∈ s} ∂(κ a) | by simp_rw [prod, comp_prod_apply _ _ _ hs, swap_left_apply _ _, prod_mk_left_apply,
prod.swap_prod_mk] | lemma | probability_theory.kernel.prod_apply | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable_set",
"prod.swap_prod_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_prod (κ : kernel α β) [is_s_finite_kernel κ] (η : kernel α γ) [is_s_finite_kernel η]
(a : α) {g : (β × γ) → ℝ≥0∞} (hg : measurable g) :
∫⁻ c, g c ∂((κ ×ₖ η) a) = ∫⁻ b, ∫⁻ c, g (b, c) ∂(η a) ∂(κ a) | by simp_rw [prod, lintegral_comp_prod _ _ _ hg, swap_left_apply, prod_mk_left_apply,
prod.swap_prod_mk] | lemma | probability_theory.kernel.lintegral_prod | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measurable",
"prod.swap_prod_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_markov_kernel.prod (κ : kernel α β) [is_markov_kernel κ]
(η : kernel α γ) [is_markov_kernel η] :
is_markov_kernel (κ ×ₖ η) | by { rw prod, apply_instance, } | instance | probability_theory.kernel.is_markov_kernel.prod | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel.prod (κ : kernel α β) [is_finite_kernel κ]
(η : kernel α γ) [is_finite_kernel η] :
is_finite_kernel (κ ×ₖ η) | by { rw prod, apply_instance, } | instance | probability_theory.kernel.is_finite_kernel.prod | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_s_finite_kernel.prod (κ : kernel α β) [is_s_finite_kernel κ]
(η : kernel α γ) [is_s_finite_kernel η] :
is_s_finite_kernel (κ ×ₖ η) | by { rw prod, apply_instance, } | instance | probability_theory.kernel.is_s_finite_kernel.prod | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_id'' (hm : m ≤ mΩ) :
@measurable Ω Ω mΩ m id | measurable_id.mono le_rfl hm | lemma | probability_theory.measurable_id'' | probability.kernel | src/probability/kernel/condexp.lean | [
"probability.kernel.cond_distrib"
] | [
"le_rfl",
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ae_measurable_id'' (μ : measure Ω) (hm : m ≤ mΩ) :
@ae_measurable Ω Ω m mΩ id μ | @measurable.ae_measurable Ω Ω mΩ m id μ (measurable_id'' hm) | lemma | probability_theory.ae_measurable_id'' | probability.kernel | src/probability/kernel/condexp.lean | [
"probability.kernel.cond_distrib"
] | [
"ae_measurable",
"measurable.ae_measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.measure_theory.ae_strongly_measurable.comp_snd_map_prod_id [topological_space F]
(hm : m ≤ mΩ) (hf : ae_strongly_measurable f μ) :
ae_strongly_measurable (λ x : Ω × Ω, f x.2)
(@measure.map Ω (Ω × Ω) (m.prod mΩ) mΩ (λ ω, (id ω, id ω)) μ) | begin
rw ← ae_strongly_measurable_comp_snd_map_prod_mk_iff (measurable_id'' hm) at hf,
simp_rw [id.def] at hf ⊢,
exact hf,
end | lemma | measure_theory.ae_strongly_measurable.comp_snd_map_prod_id | probability.kernel | src/probability/kernel/condexp.lean | [
"probability.kernel.cond_distrib"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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