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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Ici_sup_hom : sup_hom α (upper_set α) | ⟨Ici, Ici_sup⟩ | def | upper_set.Ici_sup_hom | order.upper_lower | src/order/upper_lower/hom.lean | [
"order.upper_lower.basic",
"order.hom.complete_lattice"
] | [
"sup_hom",
"upper_set"
] | `upper_set.Ici` as a `sup_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_Ici_sup_hom : (Ici_sup_hom : α → upper_set α) = Ici | rfl | lemma | upper_set.coe_Ici_sup_hom | order.upper_lower | src/order/upper_lower/hom.lean | [
"order.upper_lower.basic",
"order.hom.complete_lattice"
] | [
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ici_sup_hom_apply (a : α) : Ici_sup_hom a = (Ici a) | rfl | lemma | upper_set.Ici_sup_hom_apply | order.upper_lower | src/order/upper_lower/hom.lean | [
"order.upper_lower.basic",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ici_Sup_hom : Sup_hom α (upper_set α) | ⟨Ici, λ s, (Ici_Sup s).trans Sup_image.symm⟩ | def | upper_set.Ici_Sup_hom | order.upper_lower | src/order/upper_lower/hom.lean | [
"order.upper_lower.basic",
"order.hom.complete_lattice"
] | [
"Sup_hom",
"upper_set"
] | `upper_set.Ici` as a `Sup_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_Ici_Sup_hom : (Ici_Sup_hom : α → upper_set α) = Ici | rfl | lemma | upper_set.coe_Ici_Sup_hom | order.upper_lower | src/order/upper_lower/hom.lean | [
"order.upper_lower.basic",
"order.hom.complete_lattice"
] | [
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ici_Sup_hom_apply (a : α) : Ici_Sup_hom a = Ici a | rfl | lemma | upper_set.Ici_Sup_hom_apply | order.upper_lower | src/order/upper_lower/hom.lean | [
"order.upper_lower.basic",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic_inf_hom : inf_hom α (lower_set α) | ⟨Iic, Iic_inf⟩ | def | lower_set.Iic_inf_hom | order.upper_lower | src/order/upper_lower/hom.lean | [
"order.upper_lower.basic",
"order.hom.complete_lattice"
] | [
"inf_hom",
"lower_set"
] | `lower_set.Iic` as an `inf_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_Iic_inf_hom : (Iic_inf_hom : α → lower_set α) = Iic | rfl | lemma | lower_set.coe_Iic_inf_hom | order.upper_lower | src/order/upper_lower/hom.lean | [
"order.upper_lower.basic",
"order.hom.complete_lattice"
] | [
"lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic_inf_hom_apply (a : α) : Iic_inf_hom a = Iic a | rfl | lemma | lower_set.Iic_inf_hom_apply | order.upper_lower | src/order/upper_lower/hom.lean | [
"order.upper_lower.basic",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic_Inf_hom : Inf_hom α (lower_set α) | ⟨Iic, λ s, (Iic_Inf s).trans Inf_image.symm⟩ | def | lower_set.Iic_Inf_hom | order.upper_lower | src/order/upper_lower/hom.lean | [
"order.upper_lower.basic",
"order.hom.complete_lattice"
] | [
"Inf_hom",
"lower_set"
] | `lower_set.Iic` as an `Inf_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_Iic_Inf_hom : (Iic_Inf_hom : α → lower_set α) = Iic | rfl | lemma | lower_set.coe_Iic_Inf_hom | order.upper_lower | src/order/upper_lower/hom.lean | [
"order.upper_lower.basic",
"order.hom.complete_lattice"
] | [
"lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic_Inf_hom_apply (a : α) : Iic_Inf_hom a = Iic a | rfl | lemma | lower_set.Iic_Inf_hom_apply | order.upper_lower | src/order/upper_lower/hom.lean | [
"order.upper_lower.basic",
"order.hom.complete_lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite.upper_closure [locally_finite_order_top α] (hs : s.finite) :
(upper_closure s : set α).finite | by { rw coe_upper_closure, exact hs.bUnion (λ _ _, finite_Ici _) } | lemma | set.finite.upper_closure | order.upper_lower | src/order/upper_lower/locally_finite.lean | [
"order.locally_finite",
"order.upper_lower.basic"
] | [
"coe_upper_closure",
"finite",
"locally_finite_order_top",
"upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite.lower_closure [locally_finite_order_bot α] (hs : s.finite) :
(lower_closure s : set α).finite | by { rw coe_lower_closure, exact hs.bUnion (λ _ _, finite_Iic _) } | lemma | set.finite.lower_closure | order.upper_lower | src/order/upper_lower/locally_finite.lean | [
"order.locally_finite",
"order.upper_lower.basic"
] | [
"coe_lower_closure",
"finite",
"locally_finite_order_bot",
"lower_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_fun.indep_comap_natural_of_lt (hf : ∀ i, strongly_measurable (f i))
(hfi : Indep_fun (λ i, mβ) f μ) (hij : i < j) :
indep (measurable_space.comap (f j) mβ) (filtration.natural f hf i) μ | begin
suffices : indep (⨆ k ∈ {j}, measurable_space.comap (f k) mβ)
(⨆ k ∈ {k | k ≤ i}, measurable_space.comap (f k) mβ) μ,
{ rwa supr_singleton at this },
exact indep_supr_of_disjoint (λ k, (hf k).measurable.comap_le) hfi (by simpa),
end | lemma | probability_theory.Indep_fun.indep_comap_natural_of_lt | probability | src/probability/borel_cantelli.lean | [
"probability.martingale.borel_cantelli",
"probability.conditional_expectation",
"probability.independence.basic"
] | [
"measurable_space.comap",
"supr_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_fun.condexp_natural_ae_eq_of_lt
[second_countable_topology β] [complete_space β] [normed_space ℝ β]
(hf : ∀ i, strongly_measurable (f i)) (hfi : Indep_fun (λ i, mβ) f μ) (hij : i < j) :
μ[f j | filtration.natural f hf i] =ᵐ[μ] λ ω, μ[f j] | condexp_indep_eq (hf j).measurable.comap_le (filtration.le _ _)
(comap_measurable $ f j).strongly_measurable
(hfi.indep_comap_natural_of_lt hf hij) | lemma | probability_theory.Indep_fun.condexp_natural_ae_eq_of_lt | probability | src/probability/borel_cantelli.lean | [
"probability.martingale.borel_cantelli",
"probability.conditional_expectation",
"probability.independence.basic"
] | [
"comap_measurable",
"complete_space",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_set.condexp_indicator_filtration_of_set_ae_eq
(hsm : ∀ n, measurable_set (s n)) (hs : Indep_set s μ) (hij : i < j) :
μ[(s j).indicator (λ ω, 1 : Ω → ℝ) | filtration_of_set hsm i] =ᵐ[μ] λ ω, (μ (s j)).to_real | begin
rw filtration.filtration_of_set_eq_natural hsm,
refine (Indep_fun.condexp_natural_ae_eq_of_lt _ hs.Indep_fun_indicator hij).trans _,
{ simp only [integral_indicator_const _ (hsm _), algebra.id.smul_eq_mul, mul_one] },
{ apply_instance }
end | lemma | probability_theory.Indep_set.condexp_indicator_filtration_of_set_ae_eq | probability | src/probability/borel_cantelli.lean | [
"probability.martingale.borel_cantelli",
"probability.conditional_expectation",
"probability.independence.basic"
] | [
"algebra.id.smul_eq_mul",
"measurable_set",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_limsup_eq_one {s : ℕ → set Ω}
(hsm : ∀ n, measurable_set (s n)) (hs : Indep_set s μ) (hs' : ∑' n, μ (s n) = ∞) :
μ (limsup s at_top) = 1 | begin
rw measure_congr (eventually_eq_set.2 (ae_mem_limsup_at_top_iff μ $
measurable_set_filtration_of_set' hsm) :
(limsup s at_top : set Ω) =ᵐ[μ] {ω | tendsto (λ n, ∑ k in finset.range n,
μ[(s (k + 1)).indicator (1 : Ω → ℝ) | filtration_of_set hsm k] ω) at_top at_top}),
suffices : {ω | tendsto (λ... | lemma | probability_theory.measure_limsup_eq_one | probability | src/probability/borel_cantelli.lean | [
"probability.martingale.borel_cantelli",
"probability.conditional_expectation",
"probability.independence.basic"
] | [
"ennreal.coe_ne_top",
"ennreal.sum_lt_top",
"ennreal.tendsto_nat_tsum",
"ennreal.tendsto_nhds_top_iff_nnreal",
"ennreal.to_real_le_to_real",
"ennreal.to_real_nonneg",
"ennreal.to_real_sum",
"ennreal.tsum_add_one_eq_top",
"finset.range",
"measurable_set",
"mem_upper_bounds",
"monotone_nat_of_le... | **The second Borel-Cantelli lemma**: Given a sequence of independent sets `(sₙ)` such that
`∑ n, μ sₙ = ∞`, `limsup sₙ` has measure 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
condexp_indep_eq
(hle₁ : m₁ ≤ m) (hle₂ : m₂ ≤ m) [sigma_finite (μ.trim hle₂)]
(hf : strongly_measurable[m₁] f) (hindp : indep m₁ m₂ μ) :
μ[f | m₂] =ᵐ[μ] λ x, μ[f] | begin
by_cases hfint : integrable f μ,
swap, { rw [condexp_undef hfint, integral_undef hfint], refl, },
have hfint₁ := hfint.trim hle₁ hf,
refine (ae_eq_condexp_of_forall_set_integral_eq hle₂ hfint
(λ s _ hs, integrable_on_const.2 (or.inr hs)) (λ s hms hs, _)
strongly_measurable_const.ae_strongly_measur... | lemma | measure_theory.condexp_indep_eq | probability | src/probability/conditional_expectation.lean | [
"probability.notation",
"probability.independence.basic",
"measure_theory.function.conditional_expectation.basic"
] | [
"coe_fn_coe_base",
"continuous.const_smul",
"continuous_linear_map.continuous",
"ennreal.one_ne_top",
"ennreal.to_real_mul",
"is_closed_eq",
"mul_comm",
"set.inter_comm",
"smul_add",
"smul_smul",
"submodule.coe_subtype",
"submodule.coe_subtypeL'",
"submodule.subtypeL"
] | If `m₁, m₂` are independent σ-algebras and `f` is `m₁`-measurable, then `𝔼[f | m₂] = 𝔼[f]`
almost everywhere. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cond (s : set Ω) : measure Ω | (μ s)⁻¹ • μ.restrict s | def | probability_theory.cond | probability | src/probability/conditional_probability.lean | [
"measure_theory.measure.measure_space"
] | [] | The conditional probability measure of measure `μ` on set `s` is `μ` restricted to `s`
and scaled by the inverse of `μ s` (to make it a probability measure):
`(μ s)⁻¹ • μ.restrict s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cond_is_probability_measure [is_finite_measure μ] (hcs : μ s ≠ 0) :
is_probability_measure $ μ[|s] | ⟨by { rw [cond, measure.smul_apply, measure.restrict_apply measurable_set.univ,
set.univ_inter], exact ennreal.inv_mul_cancel hcs (measure_ne_top _ s) }⟩ | lemma | probability_theory.cond_is_probability_measure | probability | src/probability/conditional_probability.lean | [
"measure_theory.measure.measure_space"
] | [
"ennreal.inv_mul_cancel",
"measurable_set.univ",
"set.univ_inter"
] | The conditional probability measure of any finite measure on any set of positive measure
is a probability measure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cond_empty : μ[|∅] = 0 | by simp [cond] | lemma | probability_theory.cond_empty | probability | src/probability/conditional_probability.lean | [
"measure_theory.measure.measure_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_univ [is_probability_measure μ] :
μ[|set.univ] = μ | by simp [cond, measure_univ, measure.restrict_univ] | lemma | probability_theory.cond_univ | probability | src/probability/conditional_probability.lean | [
"measure_theory.measure.measure_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_apply (hms : measurable_set s) (t : set Ω) :
μ[t|s] = (μ s)⁻¹ * μ (s ∩ t) | by { rw [cond, measure.smul_apply, measure.restrict_apply' hms, set.inter_comm], refl } | lemma | probability_theory.cond_apply | probability | src/probability/conditional_probability.lean | [
"measure_theory.measure.measure_space"
] | [
"measurable_set",
"set.inter_comm"
] | The axiomatic definition of conditional probability derived from a measure-theoretic one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cond_inter_self (hms : measurable_set s) (t : set Ω) :
μ[s ∩ t|s] = μ[t|s] | by rw [cond_apply _ hms, ← set.inter_assoc, set.inter_self, ← cond_apply _ hms] | lemma | probability_theory.cond_inter_self | probability | src/probability/conditional_probability.lean | [
"measure_theory.measure.measure_space"
] | [
"measurable_set",
"set.inter_assoc",
"set.inter_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inter_pos_of_cond_ne_zero (hms : measurable_set s) (hcst : μ[t|s] ≠ 0) :
0 < μ (s ∩ t) | begin
refine pos_iff_ne_zero.mpr (right_ne_zero_of_mul _),
{ exact (μ s)⁻¹ },
convert hcst,
simp [hms, set.inter_comm]
end | lemma | probability_theory.inter_pos_of_cond_ne_zero | probability | src/probability/conditional_probability.lean | [
"measure_theory.measure.measure_space"
] | [
"measurable_set",
"right_ne_zero_of_mul",
"set.inter_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_pos_of_inter_ne_zero [is_finite_measure μ]
(hms : measurable_set s) (hci : μ (s ∩ t) ≠ 0) :
0 < μ[|s] t | begin
rw cond_apply _ hms,
refine ennreal.mul_pos _ hci,
exact ennreal.inv_ne_zero.mpr (measure_ne_top _ _),
end | lemma | probability_theory.cond_pos_of_inter_ne_zero | probability | src/probability/conditional_probability.lean | [
"measure_theory.measure.measure_space"
] | [
"ennreal.mul_pos",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_cond_eq_cond_inter'
(hms : measurable_set s) (hmt : measurable_set t) (hcs : μ s ≠ ∞) (hci : μ (s ∩ t) ≠ 0) :
μ[|s][|t] = μ[|s ∩ t] | begin
have hcs : μ s ≠ 0 := (μ.to_outer_measure.pos_of_subset_ne_zero
(set.inter_subset_left _ _) hci).ne',
ext u,
simp [*, hms.inter hmt, cond_apply, ← mul_assoc, ← set.inter_assoc,
ennreal.mul_inv, mul_comm, ← mul_assoc, ennreal.inv_mul_cancel],
end | lemma | probability_theory.cond_cond_eq_cond_inter' | probability | src/probability/conditional_probability.lean | [
"measure_theory.measure.measure_space"
] | [
"ennreal.inv_mul_cancel",
"ennreal.mul_inv",
"measurable_set",
"mul_assoc",
"mul_comm",
"set.inter_assoc",
"set.inter_subset_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_cond_eq_cond_inter [is_finite_measure μ]
(hms : measurable_set s) (hmt : measurable_set t) (hci : μ (s ∩ t) ≠ 0) :
μ[|s][|t] = μ[|s ∩ t] | cond_cond_eq_cond_inter' μ hms hmt (measure_ne_top μ s) hci | lemma | probability_theory.cond_cond_eq_cond_inter | probability | src/probability/conditional_probability.lean | [
"measure_theory.measure.measure_space"
] | [
"measurable_set"
] | Conditioning first on `s` and then on `t` results in the same measure as conditioning
on `s ∩ t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cond_mul_eq_inter'
(hms : measurable_set s) (hcs : μ s ≠ 0) (hcs' : μ s ≠ ∞) (t : set Ω) :
μ[t|s] * μ s = μ (s ∩ t) | by rw [cond_apply μ hms t, mul_comm, ←mul_assoc,
ennreal.mul_inv_cancel hcs hcs', one_mul] | lemma | probability_theory.cond_mul_eq_inter' | probability | src/probability/conditional_probability.lean | [
"measure_theory.measure.measure_space"
] | [
"ennreal.mul_inv_cancel",
"measurable_set",
"mul_comm",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_mul_eq_inter [is_finite_measure μ]
(hms : measurable_set s) (hcs : μ s ≠ 0) (t : set Ω) :
μ[t|s] * μ s = μ (s ∩ t) | cond_mul_eq_inter' μ hms hcs (measure_ne_top _ s) t | lemma | probability_theory.cond_mul_eq_inter | probability | src/probability/conditional_probability.lean | [
"measure_theory.measure.measure_space"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_add_cond_compl_eq [is_finite_measure μ]
(hms : measurable_set s) (hcs : μ s ≠ 0) (hcs' : μ sᶜ ≠ 0) :
μ[t|s] * μ s + μ[t|sᶜ] * μ sᶜ = μ t | begin
rw [cond_mul_eq_inter μ hms hcs, cond_mul_eq_inter μ hms.compl hcs', set.inter_comm _ t,
set.inter_comm _ t],
exact measure_inter_add_diff t hms,
end | lemma | probability_theory.cond_add_cond_compl_eq | probability | src/probability/conditional_probability.lean | [
"measure_theory.measure.measure_space"
] | [
"measurable_set",
"set.inter_comm"
] | A version of the law of total probability. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cond_eq_inv_mul_cond_mul [is_finite_measure μ]
(hms : measurable_set s) (hmt : measurable_set t) :
μ[t|s] = (μ s)⁻¹ * μ[s|t] * (μ t) | begin
by_cases ht : μ t = 0,
{ simp [cond, ht, measure.restrict_apply hmt, or.inr (measure_inter_null_of_null_left s ht)] },
{ rw [mul_assoc, cond_mul_eq_inter μ hmt ht s, set.inter_comm, cond_apply _ hms] }
end | theorem | probability_theory.cond_eq_inv_mul_cond_mul | probability | src/probability/conditional_probability.lean | [
"measure_theory.measure.measure_space"
] | [
"measurable_set",
"mul_assoc",
"set.inter_comm"
] | **Bayes' Theorem** | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cond_count (s : set Ω) : measure Ω | measure.count[|s] | def | probability_theory.cond_count | probability | src/probability/cond_count.lean | [
"probability.conditional_probability"
] | [] | Given a set `s`, `cond_count s` is the counting measure conditioned on `s`. In particular,
`cond_count s t` is the proportion of `s` that is contained in `t`.
This is a probability measure when `s` is finite and nonempty and is given by
`probability_theory.cond_count_is_probability_measure`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cond_count_empty_meas : (cond_count ∅ : measure Ω) = 0 | by simp [cond_count] | lemma | probability_theory.cond_count_empty_meas | probability | src/probability/cond_count.lean | [
"probability.conditional_probability"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_count_empty {s : set Ω} : cond_count s ∅ = 0 | by simp | lemma | probability_theory.cond_count_empty | probability | src/probability/cond_count.lean | [
"probability.conditional_probability"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_of_cond_count_ne_zero {s t : set Ω} (h : cond_count s t ≠ 0) :
s.finite | begin
by_contra hs',
simpa [cond_count, cond, measure.count_apply_infinite hs'] using h,
end | lemma | probability_theory.finite_of_cond_count_ne_zero | probability | src/probability/cond_count.lean | [
"probability.conditional_probability"
] | [
"by_contra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_count_univ [fintype Ω] {s : set Ω} :
cond_count set.univ s = measure.count s / fintype.card Ω | begin
rw [cond_count, cond_apply _ measurable_set.univ, ←ennreal.div_eq_inv_mul, set.univ_inter],
congr',
rw [←finset.coe_univ, measure.count_apply, finset.univ.tsum_subtype' (λ _, (1 : ennreal))],
{ simp [finset.card_univ] },
{ exact (@finset.coe_univ Ω _).symm ▸ measurable_set.univ }
end | lemma | probability_theory.cond_count_univ | probability | src/probability/cond_count.lean | [
"probability.conditional_probability"
] | [
"ennreal",
"finset.card_univ",
"finset.coe_univ",
"fintype",
"fintype.card",
"measurable_set.univ",
"set.univ_inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_count_is_probability_measure {s : set Ω} (hs : s.finite) (hs' : s.nonempty) :
is_probability_measure (cond_count s) | { measure_univ :=
begin
rw [cond_count, cond_apply _ hs.measurable_set, set.inter_univ, ennreal.inv_mul_cancel],
{ exact λ h, hs'.ne_empty $ measure.empty_of_count_eq_zero h },
{ exact (measure.count_apply_lt_top.2 hs).ne }
end } | lemma | probability_theory.cond_count_is_probability_measure | probability | src/probability/cond_count.lean | [
"probability.conditional_probability"
] | [
"ennreal.inv_mul_cancel",
"set.inter_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_count_singleton (ω : Ω) (t : set Ω) [decidable (ω ∈ t)] :
cond_count {ω} t = if ω ∈ t then 1 else 0 | begin
rw [cond_count, cond_apply _ (measurable_set_singleton ω), measure.count_singleton,
inv_one, one_mul],
split_ifs,
{ rw [(by simpa : ({ω} : set Ω) ∩ t = {ω}), measure.count_singleton] },
{ rw [(by simpa : ({ω} : set Ω) ∩ t = ∅), measure.count_empty] },
end | lemma | probability_theory.cond_count_singleton | probability | src/probability/cond_count.lean | [
"probability.conditional_probability"
] | [
"inv_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_count_inter_self (hs : s.finite):
cond_count s (s ∩ t) = cond_count s t | by rw [cond_count, cond_inter_self _ hs.measurable_set] | lemma | probability_theory.cond_count_inter_self | probability | src/probability/cond_count.lean | [
"probability.conditional_probability"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_count_self (hs : s.finite) (hs' : s.nonempty) :
cond_count s s = 1 | begin
rw [cond_count, cond_apply _ hs.measurable_set, set.inter_self, ennreal.inv_mul_cancel],
{ exact λ h, hs'.ne_empty $ measure.empty_of_count_eq_zero h },
{ exact (measure.count_apply_lt_top.2 hs).ne }
end | lemma | probability_theory.cond_count_self | probability | src/probability/cond_count.lean | [
"probability.conditional_probability"
] | [
"ennreal.inv_mul_cancel",
"set.inter_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_count_eq_one_of (hs : s.finite) (hs' : s.nonempty) (ht : s ⊆ t) :
cond_count s t = 1 | begin
haveI := cond_count_is_probability_measure hs hs',
refine eq_of_le_of_not_lt prob_le_one _,
rw [not_lt, ← cond_count_self hs hs'],
exact measure_mono ht,
end | lemma | probability_theory.cond_count_eq_one_of | probability | src/probability/cond_count.lean | [
"probability.conditional_probability"
] | [
"eq_of_le_of_not_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pred_true_of_cond_count_eq_one (h : cond_count s t = 1) :
s ⊆ t | begin
have hsf := finite_of_cond_count_ne_zero (by { rw h, exact one_ne_zero }),
rw [cond_count, cond_apply _ hsf.measurable_set, mul_comm] at h,
replace h := ennreal.eq_inv_of_mul_eq_one_left h,
rw [inv_inv, measure.count_apply_finite _ hsf,
measure.count_apply_finite _ (hsf.inter_of_left _), nat.cast_inj]... | lemma | probability_theory.pred_true_of_cond_count_eq_one | probability | src/probability/cond_count.lean | [
"probability.conditional_probability"
] | [
"ennreal.eq_inv_of_mul_eq_one_left",
"finset.eq_of_subset_of_card_le",
"inv_inv",
"mul_comm",
"nat.cast_inj",
"one_ne_zero",
"set.finite.to_finset_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_count_eq_zero_iff (hs : s.finite) :
cond_count s t = 0 ↔ s ∩ t = ∅ | by simp [cond_count, cond_apply _ hs.measurable_set, measure.count_apply_eq_top,
set.not_infinite.2 hs, measure.count_apply_finite _ (hs.inter_of_left _)] | lemma | probability_theory.cond_count_eq_zero_iff | probability | src/probability/cond_count.lean | [
"probability.conditional_probability"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_count_of_univ (hs : s.finite) (hs' : s.nonempty) :
cond_count s set.univ = 1 | cond_count_eq_one_of hs hs' s.subset_univ | lemma | probability_theory.cond_count_of_univ | probability | src/probability/cond_count.lean | [
"probability.conditional_probability"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_count_inter (hs : s.finite) :
cond_count s (t ∩ u) = cond_count (s ∩ t) u * cond_count s t | begin
by_cases hst : s ∩ t = ∅,
{ rw [hst, cond_count_empty_meas, measure.coe_zero, pi.zero_apply, zero_mul,
cond_count_eq_zero_iff hs, ← set.inter_assoc, hst, set.empty_inter] },
rw [cond_count, cond_count, cond_apply _ hs.measurable_set, cond_apply _ hs.measurable_set,
cond_apply _ (hs.inter_of_left _... | lemma | probability_theory.cond_count_inter | probability | src/probability/cond_count.lean | [
"probability.conditional_probability"
] | [
"ennreal.mul_inv_cancel",
"measurable_set",
"mul_assoc",
"mul_comm",
"one_mul",
"set.empty_inter",
"set.inter_assoc",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_count_inter' (hs : s.finite) :
cond_count s (t ∩ u) = cond_count (s ∩ u) t * cond_count s u | begin
rw ← set.inter_comm,
exact cond_count_inter hs,
end | lemma | probability_theory.cond_count_inter' | probability | src/probability/cond_count.lean | [
"probability.conditional_probability"
] | [
"set.inter_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_count_union (hs : s.finite) (htu : disjoint t u) :
cond_count s (t ∪ u) = cond_count s t + cond_count s u | begin
rw [cond_count, cond_apply _ hs.measurable_set, cond_apply _ hs.measurable_set,
cond_apply _ hs.measurable_set, set.inter_union_distrib_left, measure_union, mul_add],
exacts [htu.mono inf_le_right inf_le_right, (hs.inter_of_left _).measurable_set],
end | lemma | probability_theory.cond_count_union | probability | src/probability/cond_count.lean | [
"probability.conditional_probability"
] | [
"disjoint",
"inf_le_right",
"measurable_set",
"set.inter_union_distrib_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_count_compl (t : set Ω) (hs : s.finite) (hs' : s.nonempty) :
cond_count s t + cond_count s tᶜ = 1 | begin
rw [← cond_count_union hs disjoint_compl_right, set.union_compl_self,
(cond_count_is_probability_measure hs hs').measure_univ],
end | lemma | probability_theory.cond_count_compl | probability | src/probability/cond_count.lean | [
"probability.conditional_probability"
] | [
"disjoint_compl_right",
"set.union_compl_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_count_disjoint_union (hs : s.finite) (ht : t.finite) (hst : disjoint s t) :
cond_count s u * cond_count (s ∪ t) s + cond_count t u * cond_count (s ∪ t) t =
cond_count (s ∪ t) u | begin
rcases s.eq_empty_or_nonempty with (rfl | hs');
rcases t.eq_empty_or_nonempty with (rfl | ht'),
{ simp },
{ simp [cond_count_self ht ht'] },
{ simp [cond_count_self hs hs'] },
rw [cond_count, cond_count, cond_count, cond_apply _ hs.measurable_set,
cond_apply _ ht.measurable_set, cond_apply _ (hs.u... | lemma | probability_theory.cond_count_disjoint_union | probability | src/probability/cond_count.lean | [
"probability.conditional_probability"
] | [
"disjoint",
"ennreal.mul_inv_cancel",
"inf_le_left",
"measurable_set",
"mul_assoc",
"mul_comm",
"one_mul",
"set.union_inter_cancel_left",
"set.union_inter_cancel_right",
"set.union_inter_distrib_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cond_count_add_compl_eq (u t : set Ω) (hs : s.finite) :
cond_count (s ∩ u) t * cond_count s u + cond_count (s ∩ uᶜ) t * cond_count s uᶜ =
cond_count s t | begin
conv_rhs { rw [(by simp : s = s ∩ u ∪ s ∩ uᶜ),
← cond_count_disjoint_union (hs.inter_of_left _) (hs.inter_of_left _)
(disjoint_compl_right.mono inf_le_right inf_le_right)] },
simp [cond_count_inter_self hs],
end | lemma | probability_theory.cond_count_add_compl_eq | probability | src/probability/cond_count.lean | [
"probability.conditional_probability"
] | [
"inf_le_right"
] | A version of the law of total probability for counting probabilites. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pdf {m : measurable_space Ω} (X : Ω → E)
(ℙ : measure Ω) (μ : measure E . volume_tac) : Prop | (pdf' : measurable X ∧ ∃ (f : E → ℝ≥0∞), measurable f ∧ map X ℙ = μ.with_density f) | class | measure_theory.has_pdf | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"measurable",
"measurable_space"
] | A random variable `X : Ω → E` is said to `has_pdf` with respect to the measure `ℙ` on `Ω` and
`μ` on `E` if there exists a measurable function `f` such that the push-forward measure of `ℙ`
along `X` equals `μ.with_density f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pdf.measurable {m : measurable_space Ω}
(X : Ω → E) (ℙ : measure Ω) (μ : measure E . volume_tac) [hX : has_pdf X ℙ μ] :
measurable X | hX.pdf'.1 | lemma | measure_theory.has_pdf.measurable | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"measurable",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pdf {m : measurable_space Ω} (X : Ω → E) (ℙ : measure Ω) (μ : measure E . volume_tac) | if hX : has_pdf X ℙ μ then classical.some hX.pdf'.2 else 0 | def | measure_theory.pdf | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"measurable_space"
] | If `X` is a random variable that `has_pdf X ℙ μ`, then `pdf X` is the measurable function `f`
such that the push-forward measure of `ℙ` along `X` equals `μ.with_density f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pdf_undef {m : measurable_space Ω} {ℙ : measure Ω} {μ : measure E} {X : Ω → E}
(h : ¬ has_pdf X ℙ μ) :
pdf X ℙ μ = 0 | by simp only [pdf, dif_neg h] | lemma | measure_theory.pdf_undef | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_pdf_of_pdf_ne_zero {m : measurable_space Ω} {ℙ : measure Ω} {μ : measure E} {X : Ω → E}
(h : pdf X ℙ μ ≠ 0) : has_pdf X ℙ μ | begin
by_contra hpdf,
rw [pdf, dif_neg hpdf] at h,
exact hpdf (false.rec (has_pdf X ℙ μ) (h rfl))
end | lemma | measure_theory.has_pdf_of_pdf_ne_zero | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"by_contra",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pdf_eq_zero_of_not_measurable {m : measurable_space Ω}
{ℙ : measure Ω} {μ : measure E} {X : Ω → E} (hX : ¬ measurable X) :
pdf X ℙ μ = 0 | pdf_undef (λ hpdf, hX hpdf.pdf'.1) | lemma | measure_theory.pdf_eq_zero_of_not_measurable | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"measurable",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_of_pdf_ne_zero {m : measurable_space Ω}
{ℙ : measure Ω} {μ : measure E} (X : Ω → E) (h : pdf X ℙ μ ≠ 0) :
measurable X | by { by_contra hX, exact h (pdf_eq_zero_of_not_measurable hX) } | lemma | measure_theory.measurable_of_pdf_ne_zero | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"by_contra",
"measurable",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_pdf {m : measurable_space Ω}
(X : Ω → E) (ℙ : measure Ω) (μ : measure E . volume_tac) :
measurable (pdf X ℙ μ) | begin
by_cases hX : has_pdf X ℙ μ,
{ rw [pdf, dif_pos hX],
exact (classical.some_spec hX.pdf'.2).1 },
{ rw [pdf, dif_neg hX],
exact measurable_zero }
end | lemma | measure_theory.measurable_pdf | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"measurable",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_with_density_pdf {m : measurable_space Ω}
(X : Ω → E) (ℙ : measure Ω) (μ : measure E . volume_tac) [hX : has_pdf X ℙ μ] :
measure.map X ℙ = μ.with_density (pdf X ℙ μ) | begin
rw [pdf, dif_pos hX],
exact (classical.some_spec hX.pdf'.2).2
end | lemma | measure_theory.map_eq_with_density_pdf | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_set_lintegral_pdf {m : measurable_space Ω}
(X : Ω → E) (ℙ : measure Ω) (μ : measure E . volume_tac) [hX : has_pdf X ℙ μ]
{s : set E} (hs : measurable_set s) :
measure.map X ℙ s = ∫⁻ x in s, pdf X ℙ μ x ∂μ | by rw [← with_density_apply _ hs, map_eq_with_density_pdf X ℙ μ] | lemma | measure_theory.map_eq_set_lintegral_pdf | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"measurable_set",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_eq_measure_univ {X : Ω → E} [has_pdf X ℙ μ] :
∫⁻ x, pdf X ℙ μ x ∂μ = ℙ set.univ | begin
rw [← set_lintegral_univ, ← map_eq_set_lintegral_pdf X ℙ μ measurable_set.univ,
measure.map_apply (has_pdf.measurable X ℙ μ) measurable_set.univ, set.preimage_univ],
end | lemma | measure_theory.pdf.lintegral_eq_measure_univ | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"measurable_set.univ",
"set.preimage_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ae_lt_top [is_finite_measure ℙ] {μ : measure E} {X : Ω → E} :
∀ᵐ x ∂μ, pdf X ℙ μ x < ∞ | begin
by_cases hpdf : has_pdf X ℙ μ,
{ haveI := hpdf,
refine ae_lt_top (measurable_pdf X ℙ μ) _,
rw lintegral_eq_measure_univ,
exact (measure_lt_top _ _).ne },
{ rw [pdf, dif_neg hpdf],
exact filter.eventually_of_forall (λ x, with_top.zero_lt_top) }
end | lemma | measure_theory.pdf.ae_lt_top | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"filter.eventually_of_forall",
"with_top.zero_lt_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_real_to_real_ae_eq [is_finite_measure ℙ] {X : Ω → E} :
(λ x, ennreal.of_real (pdf X ℙ μ x).to_real) =ᵐ[μ] pdf X ℙ μ | of_real_to_real_ae_eq ae_lt_top | lemma | measure_theory.pdf.of_real_to_real_ae_eq | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"ennreal.of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_iff_integrable_mul_pdf [is_finite_measure ℙ] {X : Ω → E} [has_pdf X ℙ μ]
{f : E → ℝ} (hf : measurable f) :
integrable (λ x, f (X x)) ℙ ↔ integrable (λ x, f x * (pdf X ℙ μ x).to_real) μ | begin
rw [← integrable_map_measure hf.ae_strongly_measurable (has_pdf.measurable X ℙ μ).ae_measurable,
map_eq_with_density_pdf X ℙ μ,
integrable_with_density_iff (measurable_pdf _ _ _) ae_lt_top],
apply_instance
end | lemma | measure_theory.pdf.integrable_iff_integrable_mul_pdf | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"ae_measurable",
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_fun_mul_eq_integral [is_finite_measure ℙ]
{X : Ω → E} [has_pdf X ℙ μ] {f : E → ℝ} (hf : measurable f) :
∫ x, f x * (pdf X ℙ μ x).to_real ∂μ = ∫ x, f (X x) ∂ℙ | begin
by_cases hpdf : integrable (λ x, f x * (pdf X ℙ μ x).to_real) μ,
{ rw [← integral_map (has_pdf.measurable X ℙ μ).ae_measurable hf.ae_strongly_measurable,
map_eq_with_density_pdf X ℙ μ,
integral_eq_lintegral_pos_part_sub_lintegral_neg_part hpdf,
integral_eq_lintegral_pos_part_sub_linteg... | lemma | measure_theory.pdf.integral_fun_mul_eq_integral | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"ae_measurable",
"ennreal.coe_mul",
"ennreal.of_real",
"ennreal.of_real_mul",
"ennreal.to_real_nonneg",
"filter.eventually_eq.mul",
"measurable",
"mul_comm",
"neg_mul_eq_neg_mul",
"nnnorm_smul",
"real.ennnorm_eq_of_real",
"smul_eq_mul"
] | **The Law of the Unconscious Statistician**: Given a random variable `X` and a measurable
function `f`, `f ∘ X` is a random variable with expectation `∫ x, f x * pdf X ∂μ`
where `μ` is a measure on the codomain of `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_absolutely_continuous {X : Ω → E} [has_pdf X ℙ μ] : map X ℙ ≪ μ | by { rw map_eq_with_density_pdf X ℙ μ, exact with_density_absolutely_continuous _ _, } | lemma | measure_theory.pdf.map_absolutely_continuous | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_quasi_measure_preserving {X : Ω → E} [has_pdf X ℙ μ] : quasi_measure_preserving X ℙ μ | { measurable := has_pdf.measurable X ℙ μ,
absolutely_continuous := map_absolutely_continuous, } | lemma | measure_theory.pdf.to_quasi_measure_preserving | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"measurable"
] | A random variable that `has_pdf` is quasi-measure preserving. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
have_lebesgue_decomposition_of_has_pdf {X : Ω → E} [hX' : has_pdf X ℙ μ] :
(map X ℙ).have_lebesgue_decomposition μ | ⟨⟨⟨0, pdf X ℙ μ⟩,
by simp only [zero_add, measurable_pdf X ℙ μ, true_and, mutually_singular.zero_left,
map_eq_with_density_pdf X ℙ μ] ⟩⟩ | lemma | measure_theory.pdf.have_lebesgue_decomposition_of_has_pdf | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_pdf_iff {X : Ω → E} :
has_pdf X ℙ μ ↔ measurable X ∧ (map X ℙ).have_lebesgue_decomposition μ ∧ map X ℙ ≪ μ | begin
split,
{ intro hX',
exactI ⟨hX'.pdf'.1, have_lebesgue_decomposition_of_has_pdf, map_absolutely_continuous⟩ },
{ rintros ⟨hX, h_decomp, h⟩,
haveI := h_decomp,
refine ⟨⟨hX, (measure.map X ℙ).rn_deriv μ, measurable_rn_deriv _ _, _⟩⟩,
rwa with_density_rn_deriv_eq }
end | lemma | measure_theory.pdf.has_pdf_iff | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_pdf_iff_of_measurable {X : Ω → E} (hX : measurable X) :
has_pdf X ℙ μ ↔ (map X ℙ).have_lebesgue_decomposition μ ∧ map X ℙ ≪ μ | by { rw has_pdf_iff, simp only [hX, true_and], } | lemma | measure_theory.pdf.has_pdf_iff_of_measurable | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_measure_preserving_has_pdf {X : Ω → E} [has_pdf X ℙ μ]
{g : E → F} (hg : quasi_measure_preserving g μ ν)
(hmap : (map g (map X ℙ)).have_lebesgue_decomposition ν) :
has_pdf (g ∘ X) ℙ ν | begin
rw [has_pdf_iff, ← map_map hg.measurable (has_pdf.measurable X ℙ μ)],
refine ⟨hg.measurable.comp (has_pdf.measurable X ℙ μ), hmap, _⟩,
rw [map_eq_with_density_pdf X ℙ μ],
refine absolutely_continuous.mk (λ s hsm hs, _),
rw [map_apply hg.measurable hsm, with_density_apply _ (hg.measurable hsm)],
have :... | lemma | measure_theory.pdf.quasi_measure_preserving_has_pdf | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [] | A random variable that `has_pdf` transformed under a `quasi_measure_preserving`
map also `has_pdf` if `(map g (map X ℙ)).have_lebesgue_decomposition μ`.
`quasi_measure_preserving_has_pdf'` is more useful in the case we are working with a
probability measure and a real-valued random variable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quasi_measure_preserving_has_pdf' [is_finite_measure ℙ] [sigma_finite ν]
{X : Ω → E} [has_pdf X ℙ μ] {g : E → F} (hg : quasi_measure_preserving g μ ν) :
has_pdf (g ∘ X) ℙ ν | quasi_measure_preserving_has_pdf hg infer_instance | lemma | measure_theory.pdf.quasi_measure_preserving_has_pdf' | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.has_pdf_iff_of_measurable (hX : measurable X) : has_pdf X ℙ ↔ map X ℙ ≪ volume | begin
rw [has_pdf_iff_of_measurable hX, and_iff_right_iff_imp],
exact λ h, infer_instance,
end | lemma | measure_theory.pdf.real.has_pdf_iff_of_measurable | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"and_iff_right_iff_imp",
"measurable"
] | A real-valued random variable `X` `has_pdf X ℙ λ` (where `λ` is the Lebesgue measure) if and
only if the push-forward measure of `ℙ` along `X` is absolutely continuous with respect to `λ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.has_pdf_iff : has_pdf X ℙ ↔ measurable X ∧ map X ℙ ≪ volume | begin
by_cases hX : measurable X,
{ rw [real.has_pdf_iff_of_measurable hX, iff_and_self],
exact λ h, hX,
apply_instance },
{ exact ⟨λ h, false.elim (hX h.pdf'.1), λ h, false.elim (hX h.1)⟩, }
end | lemma | measure_theory.pdf.real.has_pdf_iff | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"iff_and_self",
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_mul_eq_integral [has_pdf X ℙ] :
∫ x, x * (pdf X ℙ volume x).to_real = ∫ x, X x ∂ℙ | integral_fun_mul_eq_integral measurable_id | lemma | measure_theory.pdf.integral_mul_eq_integral | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"measurable_id"
] | If `X` is a real-valued random variable that has pdf `f`, then the expectation of `X` equals
`∫ x, x * f x ∂λ` where `λ` is the Lebesgue measure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_finite_integral_mul {f : ℝ → ℝ} {g : ℝ → ℝ≥0∞}
(hg : pdf X ℙ =ᵐ[volume] g) (hgi : ∫⁻ x, ‖f x‖₊ * g x ≠ ∞) :
has_finite_integral (λ x, f x * (pdf X ℙ volume x).to_real) | begin
rw has_finite_integral,
have : (λ x, ↑‖f x‖₊ * g x) =ᵐ[volume] (λ x, ‖f x * (pdf X ℙ volume x).to_real‖₊),
{ refine ae_eq_trans (filter.eventually_eq.mul (ae_eq_refl (λ x, ‖f x‖₊))
(ae_eq_trans hg.symm of_real_to_real_ae_eq.symm)) _,
simp_rw [← smul_eq_mul, nnnorm_smul, ennreal.coe_mul, smul_eq_mu... | lemma | measure_theory.pdf.has_finite_integral_mul | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"ennreal.coe_mul",
"ennreal.to_real_nonneg",
"filter.eventually_eq.mul",
"lt_top_iff_ne_top",
"nnnorm_smul",
"real.ennnorm_eq_of_real",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_uniform {m : measurable_space Ω} (X : Ω → E) (support : set E)
(ℙ : measure Ω) (μ : measure E . volume_tac) | pdf X ℙ μ =ᵐ[μ] support.indicator ((μ support)⁻¹ • 1) | def | measure_theory.pdf.is_uniform | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"measurable_space"
] | A random variable `X` has uniform distribution if it has a probability density function `f`
with support `s` such that `f = (μ s)⁻¹ 1ₛ` a.e. where `1ₛ` is the indicator function for `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_pdf {m : measurable_space Ω} {X : Ω → E} {ℙ : measure Ω} {μ : measure E}
{s : set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞) (hu : is_uniform X s ℙ μ) :
has_pdf X ℙ μ | has_pdf_of_pdf_ne_zero
begin
intro hpdf,
rw [is_uniform, hpdf] at hu,
suffices : μ (s ∩ function.support ((μ s)⁻¹ • 1)) = 0,
{ have heq : function.support ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) = set.univ,
{ ext x,
rw [function.mem_support],
simp [hnt] },
rw [heq, set.inter_univ] at this,
exact hns ... | lemma | measure_theory.pdf.is_uniform.has_pdf | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"function.support",
"measurable_space",
"set.inter_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pdf_to_real_ae_eq {m : measurable_space Ω}
{X : Ω → E} {ℙ : measure Ω} {μ : measure E} {s : set E} (hX : is_uniform X s ℙ μ) :
(λ x, (pdf X ℙ μ x).to_real) =ᵐ[μ]
(λ x, (s.indicator ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) x).to_real) | filter.eventually_eq.fun_comp hX ennreal.to_real | lemma | measure_theory.pdf.is_uniform.pdf_to_real_ae_eq | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"ennreal.to_real",
"filter.eventually_eq.fun_comp",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_preimage {m : measurable_space Ω} {X : Ω → E} {ℙ : measure Ω} {μ : measure E}
{s : set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞) (hms : measurable_set s)
(hu : is_uniform X s ℙ μ)
{A : set E} (hA : measurable_set A) :
ℙ (X ⁻¹' A) = μ (s ∩ A) / μ s | begin
haveI := hu.has_pdf hns hnt,
rw [←measure.map_apply (has_pdf.measurable X ℙ μ) hA, map_eq_set_lintegral_pdf X ℙ μ hA,
lintegral_congr_ae hu.restrict],
simp only [hms, hA, lintegral_indicator, pi.smul_apply, pi.one_apply, algebra.id.smul_eq_mul,
mul_one, lintegral_const, restrict_apply', set.univ_int... | lemma | measure_theory.pdf.is_uniform.measure_preimage | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"algebra.id.smul_eq_mul",
"ennreal.div_eq_inv_mul",
"measurable_set",
"measurable_space",
"mul_one",
"pi.one_apply",
"pi.smul_apply",
"set.univ_inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_probability_measure {m : measurable_space Ω} {X : Ω → E} {ℙ : measure Ω} {μ : measure E}
{s : set E} (hns : μ s ≠ 0) (hnt : μ s ≠ ∞) (hms : measurable_set s)
(hu : is_uniform X s ℙ μ) :
is_probability_measure ℙ | ⟨begin
have : X ⁻¹' set.univ = set.univ, { simp only [set.preimage_univ] },
rw [←this, hu.measure_preimage hns hnt hms measurable_set.univ, set.inter_univ,
ennreal.div_self hns hnt],
end⟩ | lemma | measure_theory.pdf.is_uniform.is_probability_measure | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"ennreal.div_self",
"measurable_set",
"measurable_set.univ",
"measurable_space",
"set.inter_univ",
"set.preimage_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pdf_integrable [is_finite_measure ℙ] (hcs : is_compact s) (huX : is_uniform X s ℙ) :
integrable (λ x : ℝ, x * (pdf X ℙ volume x).to_real) | begin
by_cases hsupp : volume s = ∞,
{ have : pdf X ℙ =ᵐ[volume] 0,
{ refine ae_eq_trans huX _,
simp [hsupp] },
refine integrable.congr (integrable_zero _ _ _) _,
rw [(by simp : (λ x, 0 : ℝ → ℝ) = (λ x, x * (0 : ℝ≥0∞).to_real))],
refine filter.eventually_eq.mul (ae_eq_refl _)
(filter.eve... | lemma | measure_theory.pdf.is_uniform.mul_pdf_integrable | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"algebra.id.smul_eq_mul",
"ennreal.mul_lt_top",
"ennreal.to_real",
"filter.eventually_eq.fun_comp",
"filter.eventually_eq.mul",
"is_compact",
"mul_one",
"pi.one_apply",
"pi.smul_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_eq (hnt : volume s ≠ ∞) (huX : is_uniform X s ℙ) :
∫ x, X x ∂ℙ = (volume s)⁻¹.to_real * ∫ x in s, x | begin
haveI := has_pdf hns hnt huX,
haveI := huX.is_probability_measure hns hnt hms,
rw ← integral_mul_eq_integral,
rw integral_congr_ae (filter.eventually_eq.mul (ae_eq_refl _) (pdf_to_real_ae_eq huX)),
have : ∀ x, x * (s.indicator ((volume s)⁻¹ • (1 : ℝ → ℝ≥0∞)) x).to_real =
x * (s.indicator ((volume s)... | lemma | measure_theory.pdf.is_uniform.integral_eq | probability | src/probability/density.lean | [
"measure_theory.decomposition.radon_nikodym",
"measure_theory.measure.haar.of_basis"
] | [
"algebra.id.smul_eq_mul",
"filter.eventually_eq.mul",
"mul_comm",
"mul_one"
] | A real uniform random variable `X` with support `s` has expectation
`(λ s)⁻¹ * ∫ x in s, x ∂λ` where `λ` is the Lebesgue measure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ident_distrib
(f : α → γ) (g : β → γ) (μ : measure α . volume_tac) (ν : measure β . volume_tac) : Prop | (ae_measurable_fst : ae_measurable f μ)
(ae_measurable_snd : ae_measurable g ν)
(map_eq : measure.map f μ = measure.map g ν) | structure | probability_theory.ident_distrib | probability | src/probability/ident_distrib.lean | [
"probability.variance",
"measure_theory.function.uniform_integrable"
] | [
"ae_measurable",
"map_eq"
] | Two functions defined on two (possibly different) measure spaces are identically distributed if
their image measures coincide. This only makes sense when the functions are ae measurable
(as otherwise the image measures are not defined), so we require this as well in the definition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl (hf : ae_measurable f μ) :
ident_distrib f f μ μ | { ae_measurable_fst := hf,
ae_measurable_snd := hf,
map_eq := rfl } | lemma | probability_theory.ident_distrib.refl | probability | src/probability/ident_distrib.lean | [
"probability.variance",
"measure_theory.function.uniform_integrable"
] | [
"ae_measurable",
"map_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (h : ident_distrib f g μ ν) : ident_distrib g f ν μ | { ae_measurable_fst := h.ae_measurable_snd,
ae_measurable_snd := h.ae_measurable_fst,
map_eq := h.map_eq.symm } | lemma | probability_theory.ident_distrib.symm | probability | src/probability/ident_distrib.lean | [
"probability.variance",
"measure_theory.function.uniform_integrable"
] | [
"map_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans {ρ : measure δ} {h : δ → γ}
(h₁ : ident_distrib f g μ ν) (h₂ : ident_distrib g h ν ρ) : ident_distrib f h μ ρ | { ae_measurable_fst := h₁.ae_measurable_fst,
ae_measurable_snd := h₂.ae_measurable_snd,
map_eq := h₁.map_eq.trans h₂.map_eq } | lemma | probability_theory.ident_distrib.trans | probability | src/probability/ident_distrib.lean | [
"probability.variance",
"measure_theory.function.uniform_integrable"
] | [
"map_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_of_ae_measurable {u : γ → δ} (h : ident_distrib f g μ ν)
(hu : ae_measurable u (measure.map f μ)) :
ident_distrib (u ∘ f) (u ∘ g) μ ν | { ae_measurable_fst := hu.comp_ae_measurable h.ae_measurable_fst,
ae_measurable_snd :=
by { rw h.map_eq at hu, exact hu.comp_ae_measurable h.ae_measurable_snd },
map_eq :=
begin
rw [← ae_measurable.map_map_of_ae_measurable hu h.ae_measurable_fst,
← ae_measurable.map_map_of_ae_measurable _ h.ae_measu... | lemma | probability_theory.ident_distrib.comp_of_ae_measurable | probability | src/probability/ident_distrib.lean | [
"probability.variance",
"measure_theory.function.uniform_integrable"
] | [
"ae_measurable",
"ae_measurable.map_map_of_ae_measurable",
"map_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {u : γ → δ} (h : ident_distrib f g μ ν) (hu : measurable u) :
ident_distrib (u ∘ f) (u ∘ g) μ ν | h.comp_of_ae_measurable hu.ae_measurable | lemma | probability_theory.ident_distrib.comp | probability | src/probability/ident_distrib.lean | [
"probability.variance",
"measure_theory.function.uniform_integrable"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_ae_eq {g : α → γ} (hf : ae_measurable f μ) (heq : f =ᵐ[μ] g) :
ident_distrib f g μ μ | { ae_measurable_fst := hf,
ae_measurable_snd := hf.congr heq,
map_eq := measure.map_congr heq } | lemma | probability_theory.ident_distrib.of_ae_eq | probability | src/probability/ident_distrib.lean | [
"probability.variance",
"measure_theory.function.uniform_integrable"
] | [
"ae_measurable",
"map_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_mem_eq (h : ident_distrib f g μ ν) {s : set γ} (hs : measurable_set s) :
μ (f ⁻¹' s) = ν (g ⁻¹' s) | by rw [← measure.map_apply_of_ae_measurable h.ae_measurable_fst hs,
← measure.map_apply_of_ae_measurable h.ae_measurable_snd hs, h.map_eq] | lemma | probability_theory.ident_distrib.measure_mem_eq | probability | src/probability/ident_distrib.lean | [
"probability.variance",
"measure_theory.function.uniform_integrable"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ae_snd (h : ident_distrib f g μ ν) {p : γ → Prop}
(pmeas : measurable_set {x | p x}) (hp : ∀ᵐ x ∂μ, p (f x)) :
∀ᵐ x ∂ν, p (g x) | begin
apply (ae_map_iff h.ae_measurable_snd pmeas).1,
rw ← h.map_eq,
exact (ae_map_iff h.ae_measurable_fst pmeas).2 hp,
end | lemma | probability_theory.ident_distrib.ae_snd | probability | src/probability/ident_distrib.lean | [
"probability.variance",
"measure_theory.function.uniform_integrable"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ae_mem_snd (h : ident_distrib f g μ ν) {t : set γ}
(tmeas : measurable_set t) (ht : ∀ᵐ x ∂μ, f x ∈ t) :
∀ᵐ x ∂ν, g x ∈ t | h.ae_snd tmeas ht | lemma | probability_theory.ident_distrib.ae_mem_snd | probability | src/probability/ident_distrib.lean | [
"probability.variance",
"measure_theory.function.uniform_integrable"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ae_strongly_measurable_fst [topological_space γ]
[metrizable_space γ] [opens_measurable_space γ] [second_countable_topology γ]
(h : ident_distrib f g μ ν) :
ae_strongly_measurable f μ | h.ae_measurable_fst.ae_strongly_measurable | lemma | probability_theory.ident_distrib.ae_strongly_measurable_fst | probability | src/probability/ident_distrib.lean | [
"probability.variance",
"measure_theory.function.uniform_integrable"
] | [
"opens_measurable_space",
"topological_space"
] | In a second countable topology, the first function in an identically distributed pair is a.e.
strongly measurable. So is the second function, but use `h.symm.ae_strongly_measurable_fst` as
`h.ae_strongly_measurable_snd` has a different meaning. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ae_strongly_measurable_snd [topological_space γ] [metrizable_space γ] [borel_space γ]
(h : ident_distrib f g μ ν) (hf : ae_strongly_measurable f μ) :
ae_strongly_measurable g ν | begin
refine ae_strongly_measurable_iff_ae_measurable_separable.2 ⟨h.ae_measurable_snd, _⟩,
rcases (ae_strongly_measurable_iff_ae_measurable_separable.1 hf).2 with ⟨t, t_sep, ht⟩,
refine ⟨closure t, t_sep.closure, _⟩,
apply h.ae_mem_snd is_closed_closure.measurable_set,
filter_upwards [ht] with x hx using sub... | lemma | probability_theory.ident_distrib.ae_strongly_measurable_snd | probability | src/probability/ident_distrib.lean | [
"probability.variance",
"measure_theory.function.uniform_integrable"
] | [
"borel_space",
"subset_closure",
"topological_space"
] | If `f` and `g` are identically distributed and `f` is a.e. strongly measurable, so is `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ae_strongly_measurable_iff [topological_space γ] [metrizable_space γ] [borel_space γ]
(h : ident_distrib f g μ ν) :
ae_strongly_measurable f μ ↔ ae_strongly_measurable g ν | ⟨λ hf, h.ae_strongly_measurable_snd hf, λ hg, h.symm.ae_strongly_measurable_snd hg⟩ | lemma | probability_theory.ident_distrib.ae_strongly_measurable_iff | probability | src/probability/ident_distrib.lean | [
"probability.variance",
"measure_theory.function.uniform_integrable"
] | [
"borel_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ess_sup_eq [conditionally_complete_linear_order γ] [topological_space γ]
[opens_measurable_space γ] [order_closed_topology γ] (h : ident_distrib f g μ ν) :
ess_sup f μ = ess_sup g ν | begin
have I : ∀ a, μ {x : α | a < f x} = ν {x : β | a < g x} :=
λ a, h.measure_mem_eq measurable_set_Ioi,
simp_rw [ess_sup_eq_Inf, I],
end | lemma | probability_theory.ident_distrib.ess_sup_eq | probability | src/probability/ident_distrib.lean | [
"probability.variance",
"measure_theory.function.uniform_integrable"
] | [
"conditionally_complete_linear_order",
"ess_sup",
"ess_sup_eq_Inf",
"measurable_set_Ioi",
"opens_measurable_space",
"order_closed_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_eq {f : α → ℝ≥0∞} {g : β → ℝ≥0∞} (h : ident_distrib f g μ ν) :
∫⁻ x, f x ∂μ = ∫⁻ x, g x ∂ν | begin
change ∫⁻ x, id (f x) ∂μ = ∫⁻ x, id (g x) ∂ν,
rw [← lintegral_map' ae_measurable_id h.ae_measurable_fst,
← lintegral_map' ae_measurable_id h.ae_measurable_snd, h.map_eq],
end | lemma | probability_theory.ident_distrib.lintegral_eq | probability | src/probability/ident_distrib.lean | [
"probability.variance",
"measure_theory.function.uniform_integrable"
] | [
"ae_measurable_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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