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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indep_supr_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
(hns : directed (≤) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
indep (⨆ n, s n) (limsup s f) μ | begin
suffices : (⨆ a, ⨆ n ∈ (ns a), s n) = ⨆ n, s n,
{ rw ← this,
exact indep_supr_directed_limsup h_le h_indep hf hns hnsp, },
rw supr_comm,
refine supr_congr (λ n, _),
have : (⨆ (i : α) (H : n ∈ ns i), s n) = (⨆ (h : ∃ i, n ∈ ns i), s n), by rw supr_exists,
haveI : nonempty (∃ (i : α), n ∈ ns i) := ⟨... | lemma | probability_theory.indep_supr_limsup | probability.independence | src/probability/independence/zero_one.lean | [
"probability.independence.basic"
] | [
"directed",
"supr_comm",
"supr_congr",
"supr_const",
"supr_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_limsup_self (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
(hns : directed (≤) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
indep (limsup s f) (limsup s f) μ | indep_of_indep_of_le_left (indep_supr_limsup h_le h_indep hf hns hnsp hns_univ) limsup_le_supr | lemma | probability_theory.indep_limsup_self | probability.independence | src/probability/independence/zero_one.lean | [
"probability.independence.basic"
] | [
"directed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_zero_or_one_of_measurable_set_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ)
(hf : ∀ t, p t → tᶜ ∈ f) (hns : directed (≤) ns) (hnsp : ∀ a, p (ns a))
(hns_univ : ∀ n, ∃ a, n ∈ ns a) {t : set Ω} (ht_tail : measurable_set[limsup s f] t) :
μ t = 0 ∨ μ t = 1 | measure_eq_zero_or_one_of_indep_set_self
((indep_limsup_self h_le h_indep hf hns hnsp hns_univ).indep_set_of_measurable_set
ht_tail ht_tail) | theorem | probability_theory.measure_zero_or_one_of_measurable_set_limsup | probability.independence | src/probability/independence/zero_one.lean | [
"probability.independence.basic"
] | [
"directed",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_limsup_at_top_self (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) :
indep (limsup s at_top) (limsup s at_top) μ | begin
let ns : ι → set ι := set.Iic,
have hnsp : ∀ i, bdd_above (ns i) := λ i, bdd_above_Iic,
refine indep_limsup_self h_le h_indep _ _ hnsp _,
{ simp only [mem_at_top_sets, ge_iff_le, set.mem_compl_iff, bdd_above, upper_bounds,
set.nonempty],
rintros t ⟨a, ha⟩,
obtain ⟨b, hb⟩ : ∃ b, a < b := exis... | lemma | probability_theory.indep_limsup_at_top_self | probability.independence | src/probability/independence/zero_one.lean | [
"probability.independence.basic"
] | [
"bdd_above",
"bdd_above_Iic",
"ge_iff_le",
"monotone.directed_le",
"set.Iic",
"set.mem_compl_iff",
"set.nonempty",
"upper_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_zero_or_one_of_measurable_set_limsup_at_top (h_le : ∀ n, s n ≤ m0)
(h_indep : Indep s μ) {t : set Ω} (ht_tail : measurable_set[limsup s at_top] t) :
μ t = 0 ∨ μ t = 1 | measure_eq_zero_or_one_of_indep_set_self
((indep_limsup_at_top_self h_le h_indep).indep_set_of_measurable_set ht_tail ht_tail) | theorem | probability_theory.measure_zero_or_one_of_measurable_set_limsup_at_top | probability.independence | src/probability/independence/zero_one.lean | [
"probability.independence.basic"
] | [
"measurable_set"
] | **Kolmogorov's 0-1 law** : any event in the tail σ-algebra of an independent sequence of
sub-σ-algebras has probability 0 or 1.
The tail σ-algebra `limsup s at_top` is the same as `⋂ n, ⋃ i ≥ n, s i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indep_limsup_at_bot_self (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) :
indep (limsup s at_bot) (limsup s at_bot) μ | begin
let ns : ι → set ι := set.Ici,
have hnsp : ∀ i, bdd_below (ns i) := λ i, bdd_below_Ici,
refine indep_limsup_self h_le h_indep _ _ hnsp _,
{ simp only [mem_at_bot_sets, ge_iff_le, set.mem_compl_iff, bdd_below, lower_bounds,
set.nonempty],
rintros t ⟨a, ha⟩,
obtain ⟨b, hb⟩ : ∃ b, b < a := exis... | lemma | probability_theory.indep_limsup_at_bot_self | probability.independence | src/probability/independence/zero_one.lean | [
"probability.independence.basic"
] | [
"bdd_below",
"bdd_below_Ici",
"directed_of_inf",
"ge_iff_le",
"lower_bounds",
"set.Ici",
"set.mem_compl_iff",
"set.nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_zero_or_one_of_measurable_set_limsup_at_bot (h_le : ∀ n, s n ≤ m0)
(h_indep : Indep s μ) {t : set Ω} (ht_tail : measurable_set[limsup s at_bot] t) :
μ t = 0 ∨ μ t = 1 | measure_eq_zero_or_one_of_indep_set_self
((indep_limsup_at_bot_self h_le h_indep).indep_set_of_measurable_set ht_tail ht_tail) | theorem | probability_theory.measure_zero_or_one_of_measurable_set_limsup_at_bot | probability.independence | src/probability/independence/zero_one.lean | [
"probability.independence.basic"
] | [
"measurable_set"
] | **Kolmogorov's 0-1 law** : any event in the tail σ-algebra of an independent sequence of
sub-σ-algebras has probability 0 or 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel (α β : Type*) [measurable_space α] [measurable_space β] :
add_submonoid (α → measure β) | { carrier := measurable,
zero_mem' := measurable_zero,
add_mem' := λ f g hf hg, measurable.add hf hg, } | def | probability_theory.kernel | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"add_submonoid",
"measurable",
"measurable_space"
] | A kernel from a measurable space `α` to another measurable space `β` is a measurable function
`κ : α → measure β`. The measurable space structure on `measure β` is given by
`measure_theory.measure.measurable_space`. A map `κ : α → measure β` is measurable iff
`∀ s : set β, measurable_set s → measurable (λ a, κ a s)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fn_zero : ⇑(0 : kernel α β) = 0 | rfl | lemma | probability_theory.kernel.coe_fn_zero | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_add (κ η : kernel α β) : ⇑(κ + η) = κ + η | rfl | lemma | probability_theory.kernel.coe_fn_add | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add_hom (α β : Type*) [measurable_space α] [measurable_space β] :
kernel α β →+ (α → measure β) | ⟨coe_fn, coe_fn_zero, coe_fn_add⟩ | def | probability_theory.kernel.coe_add_hom | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable_space"
] | Coercion to a function as an additive monoid homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_apply (a : α) : (0 : kernel α β) a = 0 | rfl | lemma | probability_theory.kernel.zero_apply | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_finset_sum (I : finset ι) (κ : ι → kernel α β) :
⇑(∑ i in I, κ i) = ∑ i in I, κ i | (coe_add_hom α β).map_sum _ _ | lemma | probability_theory.kernel.coe_finset_sum | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset_sum_apply (I : finset ι) (κ : ι → kernel α β) (a : α) :
(∑ i in I, κ i) a = ∑ i in I, κ i a | by rw [coe_finset_sum, finset.sum_apply] | lemma | probability_theory.kernel.finset_sum_apply | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset_sum_apply' (I : finset ι) (κ : ι → kernel α β) (a : α) (s : set β) :
(∑ i in I, κ i) a s = ∑ i in I, κ i a s | by rw [finset_sum_apply, measure.finset_sum_apply] | lemma | probability_theory.kernel.finset_sum_apply' | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_markov_kernel (κ : kernel α β) : Prop | (is_probability_measure : ∀ a, is_probability_measure (κ a)) | class | probability_theory.is_markov_kernel | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | A kernel is a Markov kernel if every measure in its image is a probability measure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_finite_kernel (κ : kernel α β) : Prop | (exists_univ_le : ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ a, κ a set.univ ≤ C) | class | probability_theory.is_finite_kernel | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | A kernel is finite if every measure in its image is finite, with a uniform bound. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_finite_kernel.bound (κ : kernel α β) [h : is_finite_kernel κ] : ℝ≥0∞ | h.exists_univ_le.some | def | probability_theory.is_finite_kernel.bound | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | A constant `C : ℝ≥0∞` such that `C < ∞` (`is_finite_kernel.bound_lt_top κ`) and for all
`a : α` and `s : set β`, `κ a s ≤ C` (`measure_le_bound κ a s`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_finite_kernel.bound_lt_top (κ : kernel α β) [h : is_finite_kernel κ] :
is_finite_kernel.bound κ < ∞ | h.exists_univ_le.some_spec.1 | lemma | probability_theory.is_finite_kernel.bound_lt_top | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel.bound_ne_top (κ : kernel α β) [h : is_finite_kernel κ] :
is_finite_kernel.bound κ ≠ ∞ | (is_finite_kernel.bound_lt_top κ).ne | lemma | probability_theory.is_finite_kernel.bound_ne_top | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
kernel.measure_le_bound (κ : kernel α β) [h : is_finite_kernel κ] (a : α) (s : set β) :
κ a s ≤ is_finite_kernel.bound κ | (measure_mono (set.subset_univ s)).trans (h.exists_univ_le.some_spec.2 a) | lemma | probability_theory.kernel.measure_le_bound | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"set.subset_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel_zero (α β : Type*) {mα : measurable_space α} {mβ : measurable_space β} :
is_finite_kernel (0 : kernel α β) | ⟨⟨0, ennreal.coe_lt_top,
λ a, by simp only [kernel.zero_apply, measure.coe_zero, pi.zero_apply, le_zero_iff]⟩⟩ | instance | probability_theory.is_finite_kernel_zero | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"ennreal.coe_lt_top",
"le_zero_iff",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel.add (κ η : kernel α β) [is_finite_kernel κ] [is_finite_kernel η] :
is_finite_kernel (κ + η) | begin
refine ⟨⟨is_finite_kernel.bound κ + is_finite_kernel.bound η,
ennreal.add_lt_top.mpr ⟨is_finite_kernel.bound_lt_top κ, is_finite_kernel.bound_lt_top η⟩,
λ a, _⟩⟩,
simp_rw [kernel.coe_fn_add, pi.add_apply, measure.coe_add, pi.add_apply],
exact add_le_add (kernel.measure_le_bound _ _ _) (kernel.measur... | instance | probability_theory.is_finite_kernel.add | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_markov_kernel.is_probability_measure' [h : is_markov_kernel κ] (a : α) :
is_probability_measure (κ a) | is_markov_kernel.is_probability_measure a | instance | probability_theory.is_markov_kernel.is_probability_measure' | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel.is_finite_measure [h : is_finite_kernel κ] (a : α) :
is_finite_measure (κ a) | ⟨(kernel.measure_le_bound κ a set.univ).trans_lt (is_finite_kernel.bound_lt_top κ)⟩ | instance | probability_theory.is_finite_kernel.is_finite_measure | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_markov_kernel.is_finite_kernel [h : is_markov_kernel κ] : is_finite_kernel κ | ⟨⟨1, ennreal.one_lt_top, λ a, prob_le_one⟩⟩ | instance | probability_theory.is_markov_kernel.is_finite_kernel | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"ennreal.one_lt_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {η : kernel α β} (h : ∀ a, κ a = η a) : κ = η | by { ext1, ext1 a, exact h a, } | lemma | probability_theory.kernel.ext | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {η : kernel α β} : κ = η ↔ ∀ a, κ a = η a | ⟨λ h a, by rw h, ext⟩ | lemma | probability_theory.kernel.ext_iff | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff' {η : kernel α β} : κ = η ↔ ∀ a (s : set β) (hs : measurable_set s), κ a s = η a s | by simp_rw [ext_iff, measure.ext_iff] | lemma | probability_theory.kernel.ext_iff' | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_fun {η : kernel α β} (h : ∀ a f, measurable f → ∫⁻ b, f b ∂(κ a) = ∫⁻ b, f b ∂(η a)) :
κ = η | begin
ext a s hs,
specialize h a (s.indicator (λ _, 1)) (measurable.indicator measurable_const hs),
simp_rw [lintegral_indicator_const hs, one_mul] at h,
rw h,
end | lemma | probability_theory.kernel.ext_fun | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable",
"measurable.indicator",
"measurable_const",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_fun_iff {η : kernel α β} :
κ = η ↔ ∀ a f, measurable f → ∫⁻ b, f b ∂(κ a) = ∫⁻ b, f b ∂(η a) | ⟨λ h a f hf, by rw h, ext_fun⟩ | lemma | probability_theory.kernel.ext_fun_iff | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable (κ : kernel α β) : measurable κ | κ.prop | lemma | probability_theory.kernel.measurable | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_coe (κ : kernel α β) {s : set β} (hs : measurable_set s) :
measurable (λ a, κ a s) | (measure.measurable_coe hs).comp (kernel.measurable κ) | lemma | probability_theory.kernel.measurable_coe | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum [countable ι] (κ : ι → kernel α β) : kernel α β | { val := λ a, measure.sum (λ n, κ n a),
property :=
begin
refine measure.measurable_of_measurable_coe _ (λ s hs, _),
simp_rw measure.sum_apply _ hs,
exact measurable.ennreal_tsum (λ n, kernel.measurable_coe (κ n) hs),
end, } | def | probability_theory.kernel.sum | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"countable",
"measurable.ennreal_tsum"
] | Sum of an indexed family of kernels. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_apply [countable ι] (κ : ι → kernel α β) (a : α) :
kernel.sum κ a = measure.sum (λ n, κ n a) | rfl | lemma | probability_theory.kernel.sum_apply | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_apply' [countable ι] (κ : ι → kernel α β) (a : α) {s : set β} (hs : measurable_set s) :
kernel.sum κ a s = ∑' n, κ n a s | by rw [sum_apply κ a, measure.sum_apply _ hs] | lemma | probability_theory.kernel.sum_apply' | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"countable",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_zero [countable ι] : kernel.sum (λ (i : ι), (0 : kernel α β)) = 0 | begin
ext a s hs : 2,
rw [sum_apply' _ a hs],
simp only [zero_apply, measure.coe_zero, pi.zero_apply, tsum_zero],
end | lemma | probability_theory.kernel.sum_zero | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"countable",
"tsum_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_comm [countable ι] (κ : ι → ι → kernel α β) :
kernel.sum (λ n, kernel.sum (κ n)) = kernel.sum (λ m, kernel.sum (λ n, κ n m)) | by { ext a s hs, simp_rw [sum_apply], rw measure.sum_comm, } | lemma | probability_theory.kernel.sum_comm | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_fintype [fintype ι] (κ : ι → kernel α β) : kernel.sum κ = ∑ i, κ i | by { ext a s hs, simp only [sum_apply' κ a hs, finset_sum_apply' _ κ a s, tsum_fintype], } | lemma | probability_theory.kernel.sum_fintype | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"fintype",
"tsum_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_add [countable ι] (κ η : ι → kernel α β) :
kernel.sum (λ n, κ n + η n) = kernel.sum κ + kernel.sum η | begin
ext a s hs,
simp only [coe_fn_add, pi.add_apply, sum_apply, measure.sum_apply _ hs, pi.add_apply,
measure.coe_add, tsum_add ennreal.summable ennreal.summable],
end | lemma | probability_theory.kernel.sum_add | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"countable",
"ennreal.summable",
"tsum_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.probability_theory.is_s_finite_kernel (κ : kernel α β) : Prop | (tsum_finite : ∃ κs : ℕ → kernel α β, (∀ n, is_finite_kernel (κs n)) ∧ κ = kernel.sum κs) | class | probability_theory.is_s_finite_kernel | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | A kernel is s-finite if it can be written as the sum of countably many finite kernels. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_finite_kernel.is_s_finite_kernel [h : is_finite_kernel κ] : is_s_finite_kernel κ | ⟨⟨λ n, if n = 0 then κ else 0,
λ n, by { split_ifs, exact h, apply_instance, },
begin
ext a s hs,
rw kernel.sum_apply' _ _ hs,
have : (λ i, ((ite (i = 0) κ 0) a) s) = λ i, ite (i = 0) (κ a s) 0,
{ ext1 i, split_ifs; refl, },
rw [this, tsum_ite_eq],
end⟩⟩ | instance | probability_theory.kernel.is_finite_kernel.is_s_finite_kernel | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"tsum_ite_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seq (κ : kernel α β) [h : is_s_finite_kernel κ] :
ℕ → kernel α β | h.tsum_finite.some | def | probability_theory.kernel.seq | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | A sequence of finite kernels such that `κ = kernel.sum (seq κ)`. See `is_finite_kernel_seq`
and `kernel_sum_seq`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
kernel_sum_seq (κ : kernel α β) [h : is_s_finite_kernel κ] :
kernel.sum (seq κ) = κ | h.tsum_finite.some_spec.2.symm | lemma | probability_theory.kernel.kernel_sum_seq | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_sum_seq (κ : kernel α β) [h : is_s_finite_kernel κ] (a : α) :
measure.sum (λ n, seq κ n a) = κ a | by rw [← kernel.sum_apply, kernel_sum_seq κ] | lemma | probability_theory.kernel.measure_sum_seq | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel_seq (κ : kernel α β) [h : is_s_finite_kernel κ] (n : ℕ) :
is_finite_kernel (kernel.seq κ n) | h.tsum_finite.some_spec.1 n | instance | probability_theory.kernel.is_finite_kernel_seq | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_s_finite_kernel.add (κ η : kernel α β) [is_s_finite_kernel κ] [is_s_finite_kernel η] :
is_s_finite_kernel (κ + η) | begin
refine ⟨⟨λ n, seq κ n + seq η n, λ n, infer_instance, _⟩⟩,
rw [sum_add, kernel_sum_seq κ, kernel_sum_seq η],
end | instance | probability_theory.kernel.is_s_finite_kernel.add | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_s_finite_kernel.finset_sum {κs : ι → kernel α β} (I : finset ι)
(h : ∀ i ∈ I, is_s_finite_kernel (κs i)) :
is_s_finite_kernel (∑ i in I, κs i) | begin
classical,
unfreezingI
{ induction I using finset.induction with i I hi_nmem_I h_ind h,
{ rw [finset.sum_empty], apply_instance, },
{ rw finset.sum_insert hi_nmem_I,
haveI : is_s_finite_kernel (κs i) := h i (finset.mem_insert_self _ _),
haveI : is_s_finite_kernel (∑ (x : ι) in I, κs x),
... | lemma | probability_theory.kernel.is_s_finite_kernel.finset_sum | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"finset",
"finset.induction",
"finset.mem_insert_of_mem",
"finset.mem_insert_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_s_finite_kernel_sum_of_denumerable [denumerable ι] {κs : ι → kernel α β}
(hκs : ∀ n, is_s_finite_kernel (κs n)) :
is_s_finite_kernel (kernel.sum κs) | begin
let e : ℕ ≃ (ι × ℕ) := denumerable.equiv₂ ℕ (ι × ℕ),
refine ⟨⟨λ n, seq (κs (e n).1) (e n).2, infer_instance, _⟩⟩,
have hκ_eq : kernel.sum κs = kernel.sum (λ n, kernel.sum (seq (κs n))),
{ simp_rw kernel_sum_seq, },
ext a s hs : 2,
rw hκ_eq,
simp_rw kernel.sum_apply' _ _ hs,
change ∑' i m, seq (κs ... | lemma | probability_theory.kernel.is_s_finite_kernel_sum_of_denumerable | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"denumerable",
"denumerable.equiv₂",
"ennreal.summable",
"tsum_prod'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_s_finite_kernel_sum [countable ι] {κs : ι → kernel α β}
(hκs : ∀ n, is_s_finite_kernel (κs n)) :
is_s_finite_kernel (kernel.sum κs) | begin
casesI fintype_or_infinite ι,
{ rw sum_fintype,
exact is_s_finite_kernel.finset_sum finset.univ (λ i _, hκs i), },
haveI : encodable ι := encodable.of_countable ι,
haveI : denumerable ι := denumerable.of_encodable_of_infinite ι,
exact is_s_finite_kernel_sum_of_denumerable hκs,
end | lemma | probability_theory.kernel.is_s_finite_kernel_sum | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"countable",
"denumerable",
"denumerable.of_encodable_of_infinite",
"encodable",
"encodable.of_countable",
"finset.univ",
"fintype_or_infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deterministic (f : α → β) (hf : measurable f) :
kernel α β | { val := λ a, measure.dirac (f a),
property :=
begin
refine measure.measurable_of_measurable_coe _ (λ s hs, _),
simp_rw measure.dirac_apply' _ hs,
exact measurable_one.indicator (hf hs),
end, } | def | probability_theory.kernel.deterministic | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable"
] | Kernel which to `a` associates the dirac measure at `f a`. This is a Markov kernel. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
deterministic_apply {f : α → β} (hf : measurable f) (a : α) :
deterministic f hf a = measure.dirac (f a) | rfl | lemma | probability_theory.kernel.deterministic_apply | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deterministic_apply' {f : α → β} (hf : measurable f) (a : α) {s : set β}
(hs : measurable_set s) :
deterministic f hf a s = s.indicator (λ _, 1) (f a) | begin
rw [deterministic],
change measure.dirac (f a) s = s.indicator 1 (f a),
simp_rw measure.dirac_apply' _ hs,
end | lemma | probability_theory.kernel.deterministic_apply' | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_markov_kernel_deterministic {f : α → β} (hf : measurable f) :
is_markov_kernel (deterministic f hf) | ⟨λ a, by { rw deterministic_apply hf, apply_instance, }⟩ | instance | probability_theory.kernel.is_markov_kernel_deterministic | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_deterministic' {f : β → ℝ≥0∞} {g : α → β} {a : α}
(hg : measurable g) (hf : measurable f) :
∫⁻ x, f x ∂(kernel.deterministic g hg a) = f (g a) | by rw [kernel.deterministic_apply, lintegral_dirac' _ hf] | lemma | probability_theory.kernel.lintegral_deterministic' | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_deterministic {f : β → ℝ≥0∞} {g : α → β} {a : α}
(hg : measurable g) [measurable_singleton_class β] :
∫⁻ x, f x ∂(kernel.deterministic g hg a) = f (g a) | by rw [kernel.deterministic_apply, lintegral_dirac (g a) f] | lemma | probability_theory.kernel.lintegral_deterministic | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable",
"measurable_singleton_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_lintegral_deterministic' {f : β → ℝ≥0∞} {g : α → β} {a : α}
(hg : measurable g) (hf : measurable f) {s : set β} (hs : measurable_set s)
[decidable (g a ∈ s)] :
∫⁻ x in s, f x ∂(kernel.deterministic g hg a) = if g a ∈ s then f (g a) else 0 | by rw [kernel.deterministic_apply, set_lintegral_dirac' hf hs] | lemma | probability_theory.kernel.set_lintegral_deterministic' | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_lintegral_deterministic {f : β → ℝ≥0∞} {g : α → β} {a : α}
(hg : measurable g) [measurable_singleton_class β] (s : set β) [decidable (g a ∈ s)] :
∫⁻ x in s, f x ∂(kernel.deterministic g hg a) = if g a ∈ s then f (g a) else 0 | by rw [kernel.deterministic_apply, set_lintegral_dirac f s] | lemma | probability_theory.kernel.set_lintegral_deterministic | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable",
"measurable_singleton_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_deterministic' {E : Type*} [normed_add_comm_group E] [normed_space ℝ E]
[complete_space E] {f : β → E} {g : α → β} {a : α}
(hg : measurable g) (hf : strongly_measurable f) :
∫ x, f x ∂(kernel.deterministic g hg a) = f (g a) | by rw [kernel.deterministic_apply, integral_dirac' _ _ hf] | lemma | probability_theory.kernel.integral_deterministic' | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"complete_space",
"measurable",
"normed_add_comm_group",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_deterministic {E : Type*} [normed_add_comm_group E] [normed_space ℝ E]
[complete_space E] {f : β → E} {g : α → β} {a : α}
(hg : measurable g) [measurable_singleton_class β] :
∫ x, f x ∂(kernel.deterministic g hg a) = f (g a) | by rw [kernel.deterministic_apply, integral_dirac _ (g a)] | lemma | probability_theory.kernel.integral_deterministic | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"complete_space",
"measurable",
"measurable_singleton_class",
"normed_add_comm_group",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_integral_deterministic' {E : Type*} [normed_add_comm_group E] [normed_space ℝ E]
[complete_space E] {f : β → E} {g : α → β} {a : α}
(hg : measurable g) (hf : strongly_measurable f) {s : set β} (hs : measurable_set s)
[decidable (g a ∈ s)] :
∫ x in s, f x ∂(kernel.deterministic g hg a) = if g a ∈ s then f (g... | by rw [kernel.deterministic_apply, set_integral_dirac' hf _ hs] | lemma | probability_theory.kernel.set_integral_deterministic' | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"complete_space",
"measurable",
"measurable_set",
"normed_add_comm_group",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_integral_deterministic {E : Type*} [normed_add_comm_group E] [normed_space ℝ E]
[complete_space E] {f : β → E} {g : α → β} {a : α}
(hg : measurable g) [measurable_singleton_class β] (s : set β) [decidable (g a ∈ s)] :
∫ x in s, f x ∂(kernel.deterministic g hg a) = if g a ∈ s then f (g a) else 0 | by rw [kernel.deterministic_apply, set_integral_dirac f _ s] | lemma | probability_theory.kernel.set_integral_deterministic | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"complete_space",
"measurable",
"measurable_singleton_class",
"normed_add_comm_group",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const (α : Type*) {β : Type*} [measurable_space α] {mβ : measurable_space β} (μβ : measure β) :
kernel α β | { val := λ _, μβ,
property := measure.measurable_of_measurable_coe _ (λ s hs, measurable_const), } | def | probability_theory.kernel.const | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable_const",
"measurable_space"
] | Constant kernel, which always returns the same measure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_apply (μβ : measure β) (a : α) :
const α μβ a = μβ | rfl | lemma | probability_theory.kernel.const_apply | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel_const {μβ : measure β} [hμβ : is_finite_measure μβ] :
is_finite_kernel (const α μβ) | ⟨⟨μβ set.univ, measure_lt_top _ _, λ a, le_rfl⟩⟩ | instance | probability_theory.kernel.is_finite_kernel_const | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_markov_kernel_const {μβ : measure β} [hμβ : is_probability_measure μβ] :
is_markov_kernel (const α μβ) | ⟨λ a, hμβ⟩ | instance | probability_theory.kernel.is_markov_kernel_const | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_const {f : β → ℝ≥0∞} {μ : measure β} {a : α} :
∫⁻ x, f x ∂(kernel.const α μ a) = ∫⁻ x, f x ∂μ | by rw kernel.const_apply | lemma | probability_theory.kernel.lintegral_const | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_lintegral_const {f : β → ℝ≥0∞} {μ : measure β} {a : α} {s : set β} :
∫⁻ x in s, f x ∂(kernel.const α μ a) = ∫⁻ x in s, f x ∂μ | by rw kernel.const_apply | lemma | probability_theory.kernel.set_lintegral_const | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_const {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [complete_space E]
{f : β → E} {μ : measure β} {a : α} :
∫ x, f x ∂(kernel.const α μ a) = ∫ x, f x ∂μ | by rw kernel.const_apply | lemma | probability_theory.kernel.integral_const | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"complete_space",
"normed_add_comm_group",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_integral_const {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [complete_space E]
{f : β → E} {μ : measure β} {a : α} {s : set β} :
∫ x in s, f x ∂(kernel.const α μ a) = ∫ x in s, f x ∂μ | by rw kernel.const_apply | lemma | probability_theory.kernel.set_integral_const | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"complete_space",
"normed_add_comm_group",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_fun_of_countable [measurable_space α] {mβ : measurable_space β}
[countable α] [measurable_singleton_class α] (f : α → measure β) :
kernel α β | { val := f,
property := measurable_of_countable f } | def | probability_theory.kernel.of_fun_of_countable | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"countable",
"measurable_of_countable",
"measurable_singleton_class",
"measurable_space"
] | In a countable space with measurable singletons, every function `α → measure β` defines a
kernel. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict (κ : kernel α β) (hs : measurable_set s) : kernel α β | { val := λ a, (κ a).restrict s,
property :=
begin
refine measure.measurable_of_measurable_coe _ (λ t ht, _),
simp_rw measure.restrict_apply ht,
exact kernel.measurable_coe κ (ht.inter hs),
end, } | def | probability_theory.kernel.restrict | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable_set"
] | Kernel given by the restriction of the measures in the image of a kernel to a set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_apply (κ : kernel α β) (hs : measurable_set s) (a : α) :
kernel.restrict κ hs a = (κ a).restrict s | rfl | lemma | probability_theory.kernel.restrict_apply | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_apply' (κ : kernel α β) (hs : measurable_set s) (a : α) (ht : measurable_set t) :
kernel.restrict κ hs a t = (κ a) (t ∩ s) | by rw [restrict_apply κ hs a, measure.restrict_apply ht] | lemma | probability_theory.kernel.restrict_apply' | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_univ : kernel.restrict κ measurable_set.univ = κ | by { ext1 a, rw [kernel.restrict_apply, measure.restrict_univ], } | lemma | probability_theory.kernel.restrict_univ | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable_set.univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_restrict (κ : kernel α β) (hs : measurable_set s) (a : α) (f : β → ℝ≥0∞) :
∫⁻ b, f b ∂(kernel.restrict κ hs a) = ∫⁻ b in s, f b ∂(κ a) | by rw restrict_apply | lemma | probability_theory.kernel.lintegral_restrict | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_lintegral_restrict (κ : kernel α β) (hs : measurable_set s) (a : α) (f : β → ℝ≥0∞)
(t : set β) :
∫⁻ b in t, f b ∂(kernel.restrict κ hs a) = ∫⁻ b in (t ∩ s), f b ∂(κ a) | by rw [restrict_apply, measure.restrict_restrict' hs] | lemma | probability_theory.kernel.set_lintegral_restrict | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_integral_restrict {E : Type*} [normed_add_comm_group E] [normed_space ℝ E]
[complete_space E] {f : β → E} {a : α} (hs : measurable_set s) (t : set β) :
∫ x in t, f x ∂(kernel.restrict κ hs a) = ∫ x in (t ∩ s), f x ∂(κ a) | by rw [restrict_apply, measure.restrict_restrict' hs] | lemma | probability_theory.kernel.set_integral_restrict | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"complete_space",
"measurable_set",
"normed_add_comm_group",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel.restrict (κ : kernel α β) [is_finite_kernel κ] (hs : measurable_set s) :
is_finite_kernel (kernel.restrict κ hs) | begin
refine ⟨⟨is_finite_kernel.bound κ, is_finite_kernel.bound_lt_top κ, λ a, _⟩⟩,
rw restrict_apply' κ hs a measurable_set.univ,
exact measure_le_bound κ a _,
end | instance | probability_theory.kernel.is_finite_kernel.restrict | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable_set",
"measurable_set.univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_s_finite_kernel.restrict (κ : kernel α β) [is_s_finite_kernel κ]
(hs : measurable_set s) :
is_s_finite_kernel (kernel.restrict κ hs) | begin
refine ⟨⟨λ n, kernel.restrict (seq κ n) hs, infer_instance, _⟩⟩,
ext1 a,
simp_rw [sum_apply, restrict_apply, ← measure.restrict_sum _ hs, ← sum_apply, kernel_sum_seq],
end | instance | probability_theory.kernel.is_s_finite_kernel.restrict | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_right (κ : kernel α β) (hf : measurable_embedding f) :
kernel α γ | { val := λ a, (κ a).comap f,
property :=
begin
refine measure.measurable_measure.mpr (λ t ht, _),
have : (λ a, measure.comap f (κ a) t) = λ a, κ a (f '' t),
{ ext1 a,
rw measure.comap_apply _ hf.injective (λ s' hs', _) _ ht,
exact hf.measurable_set_image.mpr hs', },
rw this,
exact ke... | def | probability_theory.kernel.comap_right | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable_embedding"
] | Kernel with value `(κ a).comap f`, for a measurable embedding `f`. That is, for a measurable set
`t : set β`, `comap_right κ hf a t = κ a (f '' t)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_right_apply (κ : kernel α β) (hf : measurable_embedding f) (a : α) :
comap_right κ hf a = measure.comap f (κ a) | rfl | lemma | probability_theory.kernel.comap_right_apply | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_right_apply' (κ : kernel α β) (hf : measurable_embedding f)
(a : α) {t : set γ} (ht : measurable_set t) :
comap_right κ hf a t = κ a (f '' t) | by rw [comap_right_apply,
measure.comap_apply _ hf.injective (λ s, hf.measurable_set_image.mpr) _ ht] | lemma | probability_theory.kernel.comap_right_apply' | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable_embedding",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_markov_kernel.comap_right (κ : kernel α β) (hf : measurable_embedding f)
(hκ : ∀ a, κ a (set.range f) = 1) :
is_markov_kernel (comap_right κ hf) | begin
refine ⟨λ a, ⟨_⟩⟩,
rw comap_right_apply' κ hf a measurable_set.univ,
simp only [set.image_univ, subtype.range_coe_subtype, set.set_of_mem_eq],
exact hκ a,
end | lemma | probability_theory.kernel.is_markov_kernel.comap_right | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable_embedding",
"measurable_set.univ",
"set.image_univ",
"set.range",
"set.set_of_mem_eq",
"subtype.range_coe_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel.comap_right (κ : kernel α β) [is_finite_kernel κ]
(hf : measurable_embedding f) :
is_finite_kernel (comap_right κ hf) | begin
refine ⟨⟨is_finite_kernel.bound κ, is_finite_kernel.bound_lt_top κ, λ a, _⟩⟩,
rw comap_right_apply' κ hf a measurable_set.univ,
exact measure_le_bound κ a _,
end | instance | probability_theory.kernel.is_finite_kernel.comap_right | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable_embedding",
"measurable_set.univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_s_finite_kernel.comap_right (κ : kernel α β) [is_s_finite_kernel κ]
(hf : measurable_embedding f) :
is_s_finite_kernel (comap_right κ hf) | begin
refine ⟨⟨λ n, comap_right (seq κ n) hf, infer_instance, _⟩⟩,
ext1 a,
rw sum_apply,
simp_rw comap_right_apply _ hf,
have : measure.sum (λ n, measure.comap f (seq κ n a))
= measure.comap f (measure.sum (λ n, seq κ n a)),
{ ext1 t ht,
rw [measure.comap_apply _ hf.injective (λ s', hf.measurable_se... | instance | probability_theory.kernel.is_s_finite_kernel.comap_right | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
piecewise (hs : measurable_set s) (κ η : kernel α β) :
kernel α β | { val := λ a, if a ∈ s then κ a else η a,
property := measurable.piecewise hs (kernel.measurable _) (kernel.measurable _) } | def | probability_theory.kernel.piecewise | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"measurable.piecewise",
"measurable_set"
] | `piecewise hs κ η` is the kernel equal to `κ` on the measurable set `s` and to `η` on its
complement. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
piecewise_apply (a : α) :
piecewise hs κ η a = if a ∈ s then κ a else η a | rfl | lemma | probability_theory.kernel.piecewise_apply | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
piecewise_apply' (a : α) (t : set β) :
piecewise hs κ η a t = if a ∈ s then κ a t else η a t | by { rw piecewise_apply, split_ifs; refl, } | lemma | probability_theory.kernel.piecewise_apply' | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_markov_kernel.piecewise [is_markov_kernel κ] [is_markov_kernel η] :
is_markov_kernel (piecewise hs κ η) | by { refine ⟨λ a, ⟨_⟩⟩, rw [piecewise_apply', measure_univ, measure_univ, if_t_t], } | instance | probability_theory.kernel.is_markov_kernel.piecewise | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel.piecewise [is_finite_kernel κ] [is_finite_kernel η] :
is_finite_kernel (piecewise hs κ η) | begin
refine ⟨⟨max (is_finite_kernel.bound κ) (is_finite_kernel.bound η), _, λ a, _⟩⟩,
{ exact max_lt (is_finite_kernel.bound_lt_top κ) (is_finite_kernel.bound_lt_top η), },
rw [piecewise_apply'],
exact (ite_le_sup _ _ _).trans (sup_le_sup (measure_le_bound _ _ _) (measure_le_bound _ _ _)),
end | instance | probability_theory.kernel.is_finite_kernel.piecewise | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"ite_le_sup",
"sup_le_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_s_finite_kernel.piecewise [is_s_finite_kernel κ] [is_s_finite_kernel η] :
is_s_finite_kernel (piecewise hs κ η) | begin
refine ⟨⟨λ n, piecewise hs (seq κ n) (seq η n), infer_instance, _⟩⟩,
ext1 a,
simp_rw [sum_apply, kernel.piecewise_apply],
split_ifs; exact (measure_sum_seq _ a).symm,
end | instance | probability_theory.kernel.is_s_finite_kernel.piecewise | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_piecewise (a : α) (g : β → ℝ≥0∞) :
∫⁻ b, g b ∂(piecewise hs κ η a) = if a ∈ s then ∫⁻ b, g b ∂(κ a) else ∫⁻ b, g b ∂(η a) | by { simp_rw piecewise_apply, split_ifs; refl, } | lemma | probability_theory.kernel.lintegral_piecewise | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_lintegral_piecewise (a : α) (g : β → ℝ≥0∞) (t : set β) :
∫⁻ b in t, g b ∂(piecewise hs κ η a)
= if a ∈ s then ∫⁻ b in t, g b ∂(κ a) else ∫⁻ b in t, g b ∂(η a) | by { simp_rw piecewise_apply, split_ifs; refl, } | lemma | probability_theory.kernel.set_lintegral_piecewise | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_piecewise {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [complete_space E]
(a : α) (g : β → E) :
∫ b, g b ∂(piecewise hs κ η a) = if a ∈ s then ∫ b, g b ∂(κ a) else ∫ b, g b ∂(η a) | by { simp_rw piecewise_apply, split_ifs; refl, } | lemma | probability_theory.kernel.integral_piecewise | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"complete_space",
"normed_add_comm_group",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_integral_piecewise {E : Type*} [normed_add_comm_group E] [normed_space ℝ E]
[complete_space E] (a : α) (g : β → E) (t : set β) :
∫ b in t, g b ∂(piecewise hs κ η a)
= if a ∈ s then ∫ b in t, g b ∂(κ a) else ∫ b in t, g b ∂(η a) | by { simp_rw piecewise_apply, split_ifs; refl, } | lemma | probability_theory.kernel.set_integral_piecewise | probability.kernel | src/probability/kernel/basic.lean | [
"measure_theory.integral.bochner",
"measure_theory.constructions.prod.basic"
] | [
"complete_space",
"normed_add_comm_group",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_prod_fun (κ : kernel α β) (η : kernel (α × β) γ) (a : α) (s : set (β × γ)) : ℝ≥0∞ | ∫⁻ b, η (a, b) {c | (b, c) ∈ s} ∂(κ a) | def | probability_theory.kernel.comp_prod_fun | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [] | Auxiliary function for the definition of the composition-product of two kernels.
For all `a : α`, `comp_prod_fun κ η a` is a countably additive function with value zero on the empty
set, and the composition-product of kernels is defined in `kernel.comp_prod` through
`measure.of_measurable`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_prod_fun_empty (κ : kernel α β) (η : kernel (α × β) γ) (a : α) :
comp_prod_fun κ η a ∅ = 0 | by simp only [comp_prod_fun, set.mem_empty_iff_false, set.set_of_false, measure_empty,
measure_theory.lintegral_const, zero_mul] | lemma | probability_theory.kernel.comp_prod_fun_empty | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"measure_theory.lintegral_const",
"set.mem_empty_iff_false",
"set.set_of_false",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_prod_fun_Union (κ : kernel α β) (η : kernel (α × β) γ) [is_s_finite_kernel η] (a : α)
(f : ℕ → set (β × γ)) (hf_meas : ∀ i, measurable_set (f i)) (hf_disj : pairwise (disjoint on f)) :
comp_prod_fun κ η a (⋃ i, f i) = ∑' i, comp_prod_fun κ η a (f i) | begin
have h_Union : (λ b, η (a, b) {c : γ | (b, c) ∈ ⋃ i, f i})
= λ b, η (a,b) (⋃ i, {c : γ | (b, c) ∈ f i}),
{ ext1 b,
congr' with c,
simp only [set.mem_Union, set.supr_eq_Union, set.mem_set_of_eq],
refl, },
rw [comp_prod_fun, h_Union],
have h_tsum : (λ b, η (a, b) (⋃ i, {c : γ | (b, c) ∈ f i}... | lemma | probability_theory.kernel.comp_prod_fun_Union | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"ae_measurable",
"disjoint",
"measurable_prod_mk_left",
"measurable_set",
"measurable_snd",
"pairwise",
"set.bot_eq_empty",
"set.le_eq_subset",
"set.mem_Union",
"set.mem_empty_iff_false",
"set.singleton_subset_iff",
"set.supr_eq_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_prod_fun_tsum_right (κ : kernel α β) (η : kernel (α × β) γ) [is_s_finite_kernel η]
(a : α) (hs : measurable_set s) :
comp_prod_fun κ η a s = ∑' n, comp_prod_fun κ (seq η n) a s | begin
simp_rw [comp_prod_fun, (measure_sum_seq η _).symm],
have : ∫⁻ b, measure.sum (λ n, seq η n (a, b)) {c : γ | (b, c) ∈ s} ∂(κ a)
= ∫⁻ b, ∑' n, seq η n (a, b) {c : γ | (b, c) ∈ s} ∂(κ a),
{ congr',
ext1 b,
rw measure.sum_apply,
exact measurable_prod_mk_left hs, },
rw [this, lintegral_tsum (λ... | lemma | probability_theory.kernel.comp_prod_fun_tsum_right | probability.kernel | src/probability/kernel/composition.lean | [
"probability.kernel.measurable_integral"
] | [
"ae_measurable",
"measurable_prod_mk_left",
"measurable_set",
"measurable_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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