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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|---|---|---|---|---|---|---|---|---|---|---|
strong_law_aux7 :
∀ᵐ ω, tendsto (λ (n : ℕ), (∑ i in range n, X i ω) / n) at_top (𝓝 (𝔼[X 0])) | begin
obtain ⟨c, -, cone, clim⟩ :
∃ (c : ℕ → ℝ), strict_anti c ∧ (∀ (n : ℕ), 1 < c n) ∧ tendsto c at_top (𝓝 1) :=
exists_seq_strict_anti_tendsto (1 : ℝ),
have : ∀ k, ∀ᵐ ω, tendsto (λ (n : ℕ), (∑ i in range ⌊c k ^ n⌋₊, X i ω) / ⌊c k ^ n⌋₊)
at_top (𝓝 (𝔼[X 0])) := λ k, strong_law_aux6 X hint hindep hi... | lemma | strong_law_aux7 | probability | src/probability/strong_law.lean | [
"probability.ident_distrib",
"measure_theory.integral.interval_integral",
"analysis.specific_limits.floor_pow",
"analysis.p_series",
"analysis.asymptotics.specific_asymptotics"
] | [
"exists_seq_strict_anti_tendsto",
"strict_anti",
"strong_law_aux6",
"tendsto_div_of_monotone_of_tendsto_div_floor_pow"
] | `Xᵢ` satisfies the strong law of large numbers along all integers. This follows from the
corresponding fact along the sequences `c^n`, and the fact that any integer can be sandwiched
between `c^n` and `c^(n+1)` with comparably small error if `c` is close enough to `1`
(which is formalized in `tendsto_div_of_monotone_of... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_law_ae
(X : ℕ → Ω → ℝ) (hint : integrable (X 0))
(hindep : pairwise (λ i j, indep_fun (X i) (X j)))
(hident : ∀ i, ident_distrib (X i) (X 0)) :
∀ᵐ ω, tendsto (λ (n : ℕ), (∑ i in range n, X i ω) / n) at_top (𝓝 (𝔼[X 0])) | begin
let pos : ℝ → ℝ := (λ x, max x 0),
let neg : ℝ → ℝ := (λ x, max (-x) 0),
have posm : measurable pos := measurable_id'.max measurable_const,
have negm : measurable neg := measurable_id'.neg.max measurable_const,
have A : ∀ᵐ ω, tendsto (λ (n : ℕ), (∑ i in range n, (pos ∘ (X i)) ω) / n)
at_top (𝓝 (𝔼[... | theorem | strong_law_ae | probability | src/probability/strong_law.lean | [
"probability.ident_distrib",
"measure_theory.integral.interval_integral",
"analysis.specific_limits.floor_pow",
"analysis.p_series",
"analysis.asymptotics.specific_asymptotics"
] | [
"measurable",
"measurable_const",
"pairwise",
"strong_law_aux7",
"sub_div"
] | *Strong law of large numbers*, almost sure version: if `X n` is a sequence of independent
identically distributed integrable real-valued random variables, then `∑ i in range n, X i / n`
converges almost surely to `𝔼[X 0]`. We give here the strong version, due to Etemadi, that only
requires pairwise independence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strong_law_Lp
{p : ℝ≥0∞} (hp : 1 ≤ p) (hp' : p ≠ ∞)
(X : ℕ → Ω → ℝ) (hℒp : mem_ℒp (X 0) p)
(hindep : pairwise (λ i j, indep_fun (X i) (X j)))
(hident : ∀ i, ident_distrib (X i) (X 0)) :
tendsto (λ n, snorm (λ ω, (∑ i in range n, X i ω) / n - 𝔼[X 0]) p ℙ) at_top (𝓝 0) | begin
have hmeas : ∀ i, ae_strongly_measurable (X i) ℙ :=
λ i, (hident i).ae_strongly_measurable_iff.2 hℒp.1,
have hint : integrable (X 0) ℙ := hℒp.integrable hp,
have havg : ∀ n, ae_strongly_measurable (λ ω, (∑ i in range n, X i ω) / n) ℙ,
{ intro n,
simp_rw div_eq_mul_inv,
exact ae_strongly_measur... | theorem | strong_law_Lp | probability | src/probability/strong_law.lean | [
"probability.ident_distrib",
"measure_theory.integral.interval_integral",
"analysis.specific_limits.floor_pow",
"analysis.p_series",
"analysis.asymptotics.specific_asymptotics"
] | [
"div_eq_mul_inv",
"pairwise",
"pi.coe_nat",
"pi.div_apply",
"strong_law_ae"
] | *Strong law of large numbers*, Lᵖ version: if `X n` is a sequence of independent
identically distributed real-valued random variables in Lᵖ, then `∑ i in range n, X i / n`
converges in Lᵖ to `𝔼[X 0]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
evariance {Ω : Type*} {m : measurable_space Ω} (X : Ω → ℝ) (μ : measure Ω) : ℝ≥0∞ | ∫⁻ ω, ‖X ω - μ[X]‖₊^2 ∂μ | def | probability_theory.evariance | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"measurable_space"
] | The `ℝ≥0∞`-valued variance of a real-valued random variable defined as the Lebesgue integral of
`(X - 𝔼[X])^2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
variance {Ω : Type*} {m : measurable_space Ω} (X : Ω → ℝ) (μ : measure Ω) : ℝ | (evariance X μ).to_real | def | probability_theory.variance | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"measurable_space"
] | The `ℝ`-valued variance of a real-valued random variable defined by applying `ennreal.to_real`
to `evariance`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.measure_theory.mem_ℒp.evariance_lt_top [is_finite_measure μ] (hX : mem_ℒp X 2 μ) :
evariance X μ < ∞ | begin
have := ennreal.pow_lt_top (hX.sub $ mem_ℒp_const $ μ[X]).2 2,
rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ennreal.two_ne_top,
← ennreal.rpow_two] at this,
simp only [pi.sub_apply, ennreal.to_real_bit0, ennreal.one_to_real, one_div] at this,
rw [← ennreal.rpow_mul, inv_mul_cancel (two_ne_zero : (2 ... | lemma | measure_theory.mem_ℒp.evariance_lt_top | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"ennreal.one_to_real",
"ennreal.pow_lt_top",
"ennreal.rpow_mul",
"ennreal.rpow_one",
"ennreal.rpow_two",
"ennreal.to_real_bit0",
"ennreal.two_ne_top",
"inv_mul_cancel",
"one_div",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
evariance_eq_top [is_finite_measure μ]
(hXm : ae_strongly_measurable X μ) (hX : ¬ mem_ℒp X 2 μ) :
evariance X μ = ∞ | begin
by_contra h,
rw [← ne.def, ← lt_top_iff_ne_top] at h,
have : mem_ℒp (λ ω, X ω - μ[X]) 2 μ,
{ refine ⟨hXm.sub ae_strongly_measurable_const, _⟩,
rw snorm_eq_lintegral_rpow_nnnorm two_ne_zero ennreal.two_ne_top,
simp only [ennreal.to_real_bit0, ennreal.one_to_real, ennreal.rpow_two, ne.def],
exac... | lemma | probability_theory.evariance_eq_top | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"by_contra",
"ennreal.one_to_real",
"ennreal.rpow_lt_top_of_nonneg",
"ennreal.rpow_two",
"ennreal.to_real_bit0",
"ennreal.two_ne_top",
"lt_top_iff_ne_top",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
evariance_lt_top_iff_mem_ℒp [is_finite_measure μ]
(hX : ae_strongly_measurable X μ) :
evariance X μ < ∞ ↔ mem_ℒp X 2 μ | begin
refine ⟨_, measure_theory.mem_ℒp.evariance_lt_top⟩,
contrapose,
rw [not_lt, top_le_iff],
exact evariance_eq_top hX
end | lemma | probability_theory.evariance_lt_top_iff_mem_ℒp | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"top_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.measure_theory.mem_ℒp.of_real_variance_eq [is_finite_measure μ]
(hX : mem_ℒp X 2 μ) :
ennreal.of_real (variance X μ) = evariance X μ | by { rw [variance, ennreal.of_real_to_real], exact hX.evariance_lt_top.ne, } | lemma | measure_theory.mem_ℒp.of_real_variance_eq | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"ennreal.of_real",
"ennreal.of_real_to_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
evariance_eq_lintegral_of_real (X : Ω → ℝ) (μ : measure Ω) :
evariance X μ = ∫⁻ ω, ennreal.of_real ((X ω - μ[X])^2) ∂μ | begin
rw evariance,
congr,
ext1 ω,
rw [pow_two, ← ennreal.coe_mul, ← nnnorm_mul, ← pow_two],
congr,
exact (real.to_nnreal_eq_nnnorm_of_nonneg $ sq_nonneg _).symm,
end | lemma | probability_theory.evariance_eq_lintegral_of_real | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"ennreal.coe_mul",
"ennreal.of_real",
"nnnorm_mul",
"pow_two",
"real.to_nnreal_eq_nnnorm_of_nonneg",
"sq_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.measure_theory.mem_ℒp.variance_eq_of_integral_eq_zero
(hX : mem_ℒp X 2 μ) (hXint : μ[X] = 0) :
variance X μ = μ[X^2] | begin
rw [variance, evariance_eq_lintegral_of_real, ← of_real_integral_eq_lintegral_of_real,
ennreal.to_real_of_real];
simp_rw [hXint, sub_zero],
{ refl },
{ exact integral_nonneg (λ ω, pow_two_nonneg _) },
{ convert hX.integrable_norm_rpow two_ne_zero ennreal.two_ne_top,
ext ω,
simp only [pi.sub_... | lemma | measure_theory.mem_ℒp.variance_eq_of_integral_eq_zero | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"ennreal.one_to_real",
"ennreal.to_real_bit0",
"ennreal.to_real_of_real",
"ennreal.two_ne_top",
"pow_bit0_abs",
"real.norm_eq_abs",
"real.rpow_two",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.measure_theory.mem_ℒp.variance_eq [is_finite_measure μ]
(hX : mem_ℒp X 2 μ) :
variance X μ = μ[(X - (λ ω, μ[X]))^2] | begin
rw [variance, evariance_eq_lintegral_of_real, ← of_real_integral_eq_lintegral_of_real,
ennreal.to_real_of_real],
{ refl },
{ exact integral_nonneg (λ ω, pow_two_nonneg _) },
{ convert (hX.sub $ mem_ℒp_const (μ[X])).integrable_norm_rpow
two_ne_zero ennreal.two_ne_top,
ext ω,
simp only [pi... | lemma | measure_theory.mem_ℒp.variance_eq | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"ennreal.one_to_real",
"ennreal.to_real_bit0",
"ennreal.to_real_of_real",
"ennreal.two_ne_top",
"pow_bit0_abs",
"real.norm_eq_abs",
"real.rpow_two",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
evariance_zero : evariance 0 μ = 0 | by simp [evariance] | lemma | probability_theory.evariance_zero | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
evariance_eq_zero_iff (hX : ae_measurable X μ) :
evariance X μ = 0 ↔ X =ᵐ[μ] λ ω, μ[X] | begin
rw [evariance, lintegral_eq_zero_iff'],
split; intro hX; filter_upwards [hX] with ω hω,
{ simp only [pi.zero_apply, pow_eq_zero_iff, nat.succ_pos', ennreal.coe_eq_zero,
nnnorm_eq_zero, sub_eq_zero] at hω,
exact hω },
{ rw hω,
simp },
{ measurability }
end | lemma | probability_theory.evariance_eq_zero_iff | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"ae_measurable",
"ennreal.coe_eq_zero",
"measurability",
"nat.succ_pos'",
"pow_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
evariance_mul (c : ℝ) (X : Ω → ℝ) (μ : measure Ω) :
evariance (λ ω, c * X ω) μ = ennreal.of_real (c^2) * evariance X μ | begin
rw [evariance, evariance, ← lintegral_const_mul' _ _ ennreal.of_real_lt_top.ne],
congr,
ext1 ω,
rw [ennreal.of_real, ← ennreal.coe_pow, ← ennreal.coe_pow, ← ennreal.coe_mul],
congr,
rw [← sq_abs, ← real.rpow_two, real.to_nnreal_rpow_of_nonneg (abs_nonneg _), nnreal.rpow_two,
← mul_pow, real.to_nnr... | lemma | probability_theory.evariance_mul | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"abs_nonneg",
"ennreal.coe_mul",
"ennreal.coe_pow",
"ennreal.of_real",
"integral_smul_const",
"mul_comm",
"mul_pow",
"nnnorm_norm",
"nnreal.rpow_two",
"norm_abs_eq_norm",
"norm_mul",
"real.rpow_two",
"real.to_nnreal_mul_nnnorm",
"real.to_nnreal_rpow_of_nonneg",
"smul_eq_mul",
"sq_abs"
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
variance_zero (μ : measure Ω) : variance 0 μ = 0 | by simp only [variance, evariance_zero, ennreal.zero_to_real] | lemma | probability_theory.variance_zero | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"ennreal.zero_to_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
variance_nonneg (X : Ω → ℝ) (μ : measure Ω) :
0 ≤ variance X μ | ennreal.to_real_nonneg | lemma | probability_theory.variance_nonneg | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"ennreal.to_real_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
variance_mul (c : ℝ) (X : Ω → ℝ) (μ : measure Ω) :
variance (λ ω, c * X ω) μ = c^2 * variance X μ | begin
rw [variance, evariance_mul, ennreal.to_real_mul, ennreal.to_real_of_real (sq_nonneg _)],
refl,
end | lemma | probability_theory.variance_mul | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"ennreal.to_real_mul",
"ennreal.to_real_of_real",
"sq_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
variance_smul (c : ℝ) (X : Ω → ℝ) (μ : measure Ω) :
variance (c • X) μ = c^2 * variance X μ | variance_mul c X μ | lemma | probability_theory.variance_smul | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
variance_smul' {A : Type*} [comm_semiring A] [algebra A ℝ]
(c : A) (X : Ω → ℝ) (μ : measure Ω) :
variance (c • X) μ = c^2 • variance X μ | begin
convert variance_smul (algebra_map A ℝ c) X μ,
{ ext1 x, simp only [algebra_map_smul], },
{ simp only [algebra.smul_def, map_pow], }
end | lemma | probability_theory.variance_smul' | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"algebra",
"algebra.smul_def",
"algebra_map",
"algebra_map_smul",
"comm_semiring",
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
variance_def' [is_probability_measure (ℙ : measure Ω)]
{X : Ω → ℝ} (hX : mem_ℒp X 2) :
Var[X] = 𝔼[X^2] - 𝔼[X]^2 | begin
rw [hX.variance_eq, sub_sq', integral_sub', integral_add'], rotate,
{ exact hX.integrable_sq },
{ convert integrable_const (𝔼[X] ^ 2),
apply_instance },
{ apply hX.integrable_sq.add,
convert integrable_const (𝔼[X] ^ 2),
apply_instance },
{ exact ((hX.integrable one_le_two).const_mul 2).mul... | lemma | probability_theory.variance_def' | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"algebra.id.smul_eq_mul",
"ennreal.one_to_real",
"one_le_two",
"one_mul",
"pi.bit0_apply",
"pi.mul_apply",
"pi.one_apply",
"pi.pow_apply",
"ring",
"sub_sq'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
variance_le_expectation_sq [is_probability_measure (ℙ : measure Ω)]
{X : Ω → ℝ} (hm : ae_strongly_measurable X ℙ) :
Var[X] ≤ 𝔼[X^2] | begin
by_cases hX : mem_ℒp X 2,
{ rw variance_def' hX,
simp only [sq_nonneg, sub_le_self_iff] },
rw [variance, evariance_eq_lintegral_of_real, ← integral_eq_lintegral_of_nonneg_ae],
by_cases hint : integrable X, swap,
{ simp only [integral_undef hint, pi.pow_apply, pi.sub_apply, sub_zero] },
{ rw integr... | lemma | probability_theory.variance_le_expectation_sq | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"ae_measurable.pow_const",
"pi.pow_apply",
"sq_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
evariance_def' [is_probability_measure (ℙ : measure Ω)]
{X : Ω → ℝ} (hX : ae_strongly_measurable X ℙ) :
eVar[X] = (∫⁻ ω, ‖X ω‖₊^2) - ennreal.of_real (𝔼[X]^2) | begin
by_cases hℒ : mem_ℒp X 2,
{ rw [← hℒ.of_real_variance_eq, variance_def' hℒ, ennreal.of_real_sub _ (sq_nonneg _)],
congr,
simp_rw ← ennreal.coe_pow,
rw lintegral_coe_eq_integral,
{ congr' 2 with ω,
simp only [pi.pow_apply, nnreal.coe_pow, coe_nnnorm, real.norm_eq_abs, pow_bit0_abs] },
... | lemma | probability_theory.evariance_def' | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"ennreal.coe_pow",
"ennreal.of_real",
"ennreal.of_real_sub",
"ennreal.one_to_real",
"ennreal.rpow_eq_top_iff",
"ennreal.rpow_two",
"ennreal.sub_eq_top_iff",
"ennreal.to_real_bit0",
"ennreal.two_ne_top",
"inv_lt_zero",
"inv_pos",
"nnreal.coe_pow",
"not_and",
"not_and_distrib",
"one_div",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
meas_ge_le_evariance_div_sq {X : Ω → ℝ}
(hX : ae_strongly_measurable X ℙ) {c : ℝ≥0} (hc : c ≠ 0) :
ℙ {ω | ↑c ≤ |X ω - 𝔼[X]|} ≤ eVar[X] / c ^ 2 | begin
have A : (c : ℝ≥0∞) ≠ 0, { rwa [ne.def, ennreal.coe_eq_zero] },
have B : ae_strongly_measurable (λ (ω : Ω), 𝔼[X]) ℙ := ae_strongly_measurable_const,
convert meas_ge_le_mul_pow_snorm ℙ two_ne_zero ennreal.two_ne_top (hX.sub B) A,
{ ext ω,
simp only [pi.sub_apply, ennreal.coe_le_coe, ← real.norm_eq_abs... | theorem | probability_theory.meas_ge_le_evariance_div_sq | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"div_eq_mul_inv",
"ennreal.coe_eq_zero",
"ennreal.coe_le_coe",
"ennreal.inv_pow",
"ennreal.of_real_coe_nnreal",
"ennreal.one_to_real",
"ennreal.rpow_mul",
"ennreal.rpow_one",
"ennreal.rpow_two",
"ennreal.to_real_bit0",
"ennreal.two_ne_top",
"inv_mul_cancel",
"mul_comm",
"nnreal.coe_le_coe"... | *Chebyshev's inequality* for `ℝ≥0∞`-valued variance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
meas_ge_le_variance_div_sq [is_finite_measure (ℙ : measure Ω)]
{X : Ω → ℝ} (hX : mem_ℒp X 2) {c : ℝ} (hc : 0 < c) :
ℙ {ω | c ≤ |X ω - 𝔼[X]|} ≤ ennreal.of_real (Var[X] / c ^ 2) | begin
rw [ennreal.of_real_div_of_pos (sq_pos_of_ne_zero _ hc.ne.symm), hX.of_real_variance_eq],
convert @meas_ge_le_evariance_div_sq _ _ _ hX.1 (c.to_nnreal) (by simp [hc]),
{ simp only [real.coe_to_nnreal', max_le_iff, abs_nonneg, and_true] },
{ rw ennreal.of_real_pow hc.le,
refl }
end | theorem | probability_theory.meas_ge_le_variance_div_sq | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"abs_nonneg",
"ennreal.of_real",
"ennreal.of_real_div_of_pos",
"ennreal.of_real_pow",
"max_le_iff",
"real.coe_to_nnreal'",
"sq_pos_of_ne_zero"
] | *Chebyshev's inequality* : one can control the deviation probability of a real random variable
from its expectation in terms of the variance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indep_fun.variance_add [is_probability_measure (ℙ : measure Ω)]
{X Y : Ω → ℝ} (hX : mem_ℒp X 2) (hY : mem_ℒp Y 2) (h : indep_fun X Y) :
Var[X + Y] = Var[X] + Var[Y] | calc
Var[X + Y] = 𝔼[λ a, (X a)^2 + (Y a)^2 + 2 * X a * Y a] - 𝔼[X+Y]^2 :
by simp [variance_def' (hX.add hY), add_sq']
... = (𝔼[X^2] + 𝔼[Y^2] + 2 * 𝔼[X * Y]) - (𝔼[X] + 𝔼[Y])^2 :
begin
simp only [pi.add_apply, pi.pow_apply, pi.mul_apply, mul_assoc],
rw [integral_add, integral_add, integral_add, integral_mul_... | theorem | probability_theory.indep_fun.variance_add | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"add_sq'",
"mul_assoc",
"one_le_two",
"pi.mul_apply",
"pi.pow_apply",
"ring"
] | The variance of the sum of two independent random variables is the sum of the variances. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indep_fun.variance_sum [is_probability_measure (ℙ : measure Ω)]
{ι : Type*} {X : ι → Ω → ℝ} {s : finset ι}
(hs : ∀ i ∈ s, mem_ℒp (X i) 2) (h : set.pairwise ↑s (λ i j, indep_fun (X i) (X j))) :
Var[∑ i in s, X i] = ∑ i in s, Var[X i] | begin
classical,
induction s using finset.induction_on with k s ks IH,
{ simp only [finset.sum_empty, variance_zero] },
rw [variance_def' (mem_ℒp_finset_sum' _ hs), sum_insert ks, sum_insert ks],
simp only [add_sq'],
calc 𝔼[X k ^ 2 + (∑ i in s, X i) ^ 2 + 2 * X k * ∑ i in s, X i] - 𝔼[X k + ∑ i in s, X i] ... | theorem | probability_theory.indep_fun.variance_sum | probability | src/probability/variance.lean | [
"probability.notation",
"probability.integration",
"measure_theory.function.l2_space"
] | [
"add_sq'",
"finset",
"finset.induction_on",
"mul_assoc",
"one_le_two",
"pi.bit0_apply",
"pi.mul_apply",
"pi.one_apply",
"ring",
"set.pairwise",
"set.subset_insert"
] | The variance of a finite sum of pairwise independent random variables is the sum of the
variances. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Indep_sets [measurable_space Ω] (π : ι → set (set Ω)) (μ : measure Ω . volume_tac) :
Prop | ∀ (s : finset ι) {f : ι → set Ω} (H : ∀ i, i ∈ s → f i ∈ π i), μ (⋂ i ∈ s, f i) = ∏ i in s, μ (f i) | def | probability_theory.Indep_sets | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"finset",
"measurable_space"
] | A family of sets of sets `π : ι → set (set Ω)` is independent with respect to a measure `μ` if
for any finite set of indices `s = {i_1, ..., i_n}`, for any sets
`f i_1 ∈ π i_1, ..., f i_n ∈ π i_n`, then `μ (⋂ i in s, f i) = ∏ i in s, μ (f i) `.
It will be used for families of pi_systems. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indep_sets [measurable_space Ω] (s1 s2 : set (set Ω)) (μ : measure Ω . volume_tac) : Prop | ∀ t1 t2 : set Ω, t1 ∈ s1 → t2 ∈ s2 → μ (t1 ∩ t2) = μ t1 * μ t2 | def | probability_theory.indep_sets | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space"
] | Two sets of sets `s₁, s₂` are independent with respect to a measure `μ` if for any sets
`t₁ ∈ p₁, t₂ ∈ s₂`, then `μ (t₁ ∩ t₂) = μ (t₁) * μ (t₂)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Indep (m : ι → measurable_space Ω) [measurable_space Ω] (μ : measure Ω . volume_tac) :
Prop | Indep_sets (λ x, {s | measurable_set[m x] s}) μ | def | probability_theory.Indep | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_set",
"measurable_space"
] | A family of measurable space structures (i.e. of σ-algebras) is independent with respect to a
measure `μ` (typically defined on a finer σ-algebra) if the family of sets of measurable sets they
define is independent. `m : ι → measurable_space Ω` is independent with respect to measure `μ` if
for any finite set of indices... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indep (m₁ m₂ : measurable_space Ω) [measurable_space Ω] (μ : measure Ω . volume_tac) :
Prop | indep_sets {s | measurable_set[m₁] s} {s | measurable_set[m₂] s} μ | def | probability_theory.indep | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_set",
"measurable_space"
] | Two measurable space structures (or σ-algebras) `m₁, m₂` are independent with respect to a
measure `μ` (defined on a third σ-algebra) if for any sets `t₁ ∈ m₁, t₂ ∈ m₂`,
`μ (t₁ ∩ t₂) = μ (t₁) * μ (t₂)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Indep_set [measurable_space Ω] (s : ι → set Ω) (μ : measure Ω . volume_tac) : Prop | Indep (λ i, generate_from {s i}) μ | def | probability_theory.Indep_set | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space"
] | A family of sets is independent if the family of measurable space structures they generate is
independent. For a set `s`, the generated measurable space has measurable sets `∅, s, sᶜ, univ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indep_set [measurable_space Ω] (s t : set Ω) (μ : measure Ω . volume_tac) : Prop | indep (generate_from {s}) (generate_from {t}) μ | def | probability_theory.indep_set | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space"
] | Two sets are independent if the two measurable space structures they generate are independent.
For a set `s`, the generated measurable space structure has measurable sets `∅, s, sᶜ, univ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Indep_fun [measurable_space Ω] {β : ι → Type*} (m : Π (x : ι), measurable_space (β x))
(f : Π (x : ι), Ω → β x) (μ : measure Ω . volume_tac) : Prop | Indep (λ x, measurable_space.comap (f x) (m x)) μ | def | probability_theory.Indep_fun | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space",
"measurable_space.comap"
] | A family of functions defined on the same space `Ω` and taking values in possibly different
spaces, each with a measurable space structure, is independent if the family of measurable space
structures they generate on `Ω` is independent. For a function `g` with codomain having measurable
space structure `m`, the generat... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indep_fun {β γ} [measurable_space Ω] [mβ : measurable_space β] [mγ : measurable_space γ]
(f : Ω → β) (g : Ω → γ) (μ : measure Ω . volume_tac) : Prop | indep (measurable_space.comap f mβ) (measurable_space.comap g mγ) μ | def | probability_theory.indep_fun | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space",
"measurable_space.comap"
] | Two functions are independent if the two measurable space structures they generate are
independent. For a function `f` with codomain having measurable space structure `m`, the generated
measurable space structure is `measurable_space.comap f m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indep_sets.symm {s₁ s₂ : set (set Ω)} [measurable_space Ω] {μ : measure Ω}
(h : indep_sets s₁ s₂ μ) :
indep_sets s₂ s₁ μ | by { intros t1 t2 ht1 ht2, rw [set.inter_comm, mul_comm], exact h t2 t1 ht2 ht1, } | lemma | probability_theory.indep_sets.symm | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space",
"mul_comm",
"set.inter_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep.symm {m₁ m₂ : measurable_space Ω} [measurable_space Ω] {μ : measure Ω}
(h : indep m₁ m₂ μ) :
indep m₂ m₁ μ | indep_sets.symm h | lemma | probability_theory.indep.symm | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_bot_right (m' : measurable_space Ω) {m : measurable_space Ω}
{μ : measure Ω} [is_probability_measure μ] :
indep m' ⊥ μ | begin
intros s t hs ht,
rw [set.mem_set_of_eq, measurable_space.measurable_set_bot_iff] at ht,
cases ht,
{ rw [ht, set.inter_empty, measure_empty, mul_zero], },
{ rw [ht, set.inter_univ, measure_univ, mul_one], },
end | lemma | probability_theory.indep_bot_right | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space",
"measurable_space.measurable_set_bot_iff",
"mul_one",
"mul_zero",
"set.inter_empty",
"set.inter_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_bot_left (m' : measurable_space Ω) {m : measurable_space Ω}
{μ : measure Ω} [is_probability_measure μ] :
indep ⊥ m' μ | (indep_bot_right m').symm | lemma | probability_theory.indep_bot_left | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_set_empty_right {m : measurable_space Ω} {μ : measure Ω} [is_probability_measure μ]
(s : set Ω) :
indep_set s ∅ μ | by { simp only [indep_set, generate_from_singleton_empty], exact indep_bot_right _, } | lemma | probability_theory.indep_set_empty_right | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_set_empty_left {m : measurable_space Ω} {μ : measure Ω} [is_probability_measure μ]
(s : set Ω) :
indep_set ∅ s μ | (indep_set_empty_right s).symm | lemma | probability_theory.indep_set_empty_left | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_sets_of_indep_sets_of_le_left {s₁ s₂ s₃: set (set Ω)} [measurable_space Ω]
{μ : measure Ω} (h_indep : indep_sets s₁ s₂ μ) (h31 : s₃ ⊆ s₁) :
indep_sets s₃ s₂ μ | λ t1 t2 ht1 ht2, h_indep t1 t2 (set.mem_of_subset_of_mem h31 ht1) ht2 | lemma | probability_theory.indep_sets_of_indep_sets_of_le_left | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space",
"set.mem_of_subset_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_sets_of_indep_sets_of_le_right {s₁ s₂ s₃: set (set Ω)} [measurable_space Ω]
{μ : measure Ω} (h_indep : indep_sets s₁ s₂ μ) (h32 : s₃ ⊆ s₂) :
indep_sets s₁ s₃ μ | λ t1 t2 ht1 ht2, h_indep t1 t2 ht1 (set.mem_of_subset_of_mem h32 ht2) | lemma | probability_theory.indep_sets_of_indep_sets_of_le_right | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space",
"set.mem_of_subset_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_of_indep_of_le_left {m₁ m₂ m₃: measurable_space Ω} [measurable_space Ω]
{μ : measure Ω} (h_indep : indep m₁ m₂ μ) (h31 : m₃ ≤ m₁) :
indep m₃ m₂ μ | λ t1 t2 ht1 ht2, h_indep t1 t2 (h31 _ ht1) ht2 | lemma | probability_theory.indep_of_indep_of_le_left | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_of_indep_of_le_right {m₁ m₂ m₃: measurable_space Ω} [measurable_space Ω]
{μ : measure Ω} (h_indep : indep m₁ m₂ μ) (h32 : m₃ ≤ m₂) :
indep m₁ m₃ μ | λ t1 t2 ht1 ht2, h_indep t1 t2 ht1 (h32 _ ht2) | lemma | probability_theory.indep_of_indep_of_le_right | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_sets.union [measurable_space Ω] {s₁ s₂ s' : set (set Ω)} {μ : measure Ω}
(h₁ : indep_sets s₁ s' μ) (h₂ : indep_sets s₂ s' μ) :
indep_sets (s₁ ∪ s₂) s' μ | begin
intros t1 t2 ht1 ht2,
cases (set.mem_union _ _ _).mp ht1 with ht1₁ ht1₂,
{ exact h₁ t1 t2 ht1₁ ht2, },
{ exact h₂ t1 t2 ht1₂ ht2, },
end | lemma | probability_theory.indep_sets.union | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space",
"set.mem_union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_sets.union_iff [measurable_space Ω] {s₁ s₂ s' : set (set Ω)}
{μ : measure Ω} :
indep_sets (s₁ ∪ s₂) s' μ ↔ indep_sets s₁ s' μ ∧ indep_sets s₂ s' μ | ⟨λ h, ⟨indep_sets_of_indep_sets_of_le_left h (set.subset_union_left s₁ s₂),
indep_sets_of_indep_sets_of_le_left h (set.subset_union_right s₁ s₂)⟩,
λ h, indep_sets.union h.left h.right⟩ | lemma | probability_theory.indep_sets.union_iff | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space",
"set.subset_union_left",
"set.subset_union_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_sets.Union [measurable_space Ω] {s : ι → set (set Ω)} {s' : set (set Ω)}
{μ : measure Ω} (hyp : ∀ n, indep_sets (s n) s' μ) :
indep_sets (⋃ n, s n) s' μ | begin
intros t1 t2 ht1 ht2,
rw set.mem_Union at ht1,
cases ht1 with n ht1,
exact hyp n t1 t2 ht1 ht2,
end | lemma | probability_theory.indep_sets.Union | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space",
"set.mem_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_sets.bUnion [measurable_space Ω] {s : ι → set (set Ω)} {s' : set (set Ω)}
{μ : measure Ω} {u : set ι} (hyp : ∀ n ∈ u, indep_sets (s n) s' μ) :
indep_sets (⋃ n ∈ u, s n) s' μ | begin
intros t1 t2 ht1 ht2,
simp_rw set.mem_Union at ht1,
rcases ht1 with ⟨n, hpn, ht1⟩,
exact hyp n hpn t1 t2 ht1 ht2,
end | lemma | probability_theory.indep_sets.bUnion | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space",
"set.mem_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_sets.inter [measurable_space Ω] {s₁ s' : set (set Ω)} (s₂ : set (set Ω))
{μ : measure Ω} (h₁ : indep_sets s₁ s' μ) :
indep_sets (s₁ ∩ s₂) s' μ | λ t1 t2 ht1 ht2, h₁ t1 t2 ((set.mem_inter_iff _ _ _).mp ht1).left ht2 | lemma | probability_theory.indep_sets.inter | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space",
"set.mem_inter_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_sets.Inter [measurable_space Ω] {s : ι → set (set Ω)} {s' : set (set Ω)}
{μ : measure Ω} (h : ∃ n, indep_sets (s n) s' μ) :
indep_sets (⋂ n, s n) s' μ | by {intros t1 t2 ht1 ht2, cases h with n h, exact h t1 t2 (set.mem_Inter.mp ht1 n) ht2 } | lemma | probability_theory.indep_sets.Inter | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_sets.bInter [measurable_space Ω] {s : ι → set (set Ω)} {s' : set (set Ω)}
{μ : measure Ω} {u : set ι} (h : ∃ n ∈ u, indep_sets (s n) s' μ) :
indep_sets (⋂ n ∈ u, s n) s' μ | begin
intros t1 t2 ht1 ht2,
rcases h with ⟨n, hn, h⟩,
exact h t1 t2 (set.bInter_subset_of_mem hn ht1) ht2,
end | lemma | probability_theory.indep_sets.bInter | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space",
"set.bInter_subset_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_sets_singleton_iff [measurable_space Ω] {s t : set Ω} {μ : measure Ω} :
indep_sets {s} {t} μ ↔ μ (s ∩ t) = μ s * μ t | ⟨λ h, h s t rfl rfl,
λ h s1 t1 hs1 ht1, by rwa [set.mem_singleton_iff.mp hs1, set.mem_singleton_iff.mp ht1]⟩ | lemma | probability_theory.indep_sets_singleton_iff | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_sets.indep_sets {s : ι → set (set Ω)} [measurable_space Ω] {μ : measure Ω}
(h_indep : Indep_sets s μ) {i j : ι} (hij : i ≠ j) :
indep_sets (s i) (s j) μ | begin
classical,
intros t₁ t₂ ht₁ ht₂,
have hf_m : ∀ (x : ι), x ∈ {i, j} → (ite (x=i) t₁ t₂) ∈ s x,
{ intros x hx,
cases finset.mem_insert.mp hx with hx hx,
{ simp [hx, ht₁], },
{ simp [finset.mem_singleton.mp hx, hij.symm, ht₂], }, },
have h1 : t₁ = ite (i = i) t₁ t₂, by simp only [if_true, eq_se... | lemma | probability_theory.Indep_sets.indep_sets | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"finset",
"finset.mem_singleton",
"finset.prod_insert",
"finset.prod_singleton",
"finset.set_bInter_insert",
"finset.set_bInter_singleton",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep.indep {m : ι → measurable_space Ω} [measurable_space Ω] {μ : measure Ω}
(h_indep : Indep m μ) {i j : ι} (hij : i ≠ j) :
indep (m i) (m j) μ | begin
change indep_sets ((λ x, measurable_set[m x]) i) ((λ x, measurable_set[m x]) j) μ,
exact Indep_sets.indep_sets h_indep hij,
end | lemma | probability_theory.Indep.indep | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_set",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_fun.indep_fun {m₀ : measurable_space Ω} {μ : measure Ω} {β : ι → Type*}
{m : Π x, measurable_space (β x)} {f : Π i, Ω → β i} (hf_Indep : Indep_fun m f μ)
{i j : ι} (hij : i ≠ j) :
indep_fun (f i) (f j) μ | hf_Indep.indep hij | lemma | probability_theory.Indep_fun.indep_fun | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep.Indep_sets [measurable_space Ω] {μ : measure Ω} {m : ι → measurable_space Ω}
{s : ι → set (set Ω)} (hms : ∀ n, m n = generate_from (s n))
(h_indep : Indep m μ) :
Indep_sets s μ | λ S f hfs, h_indep S $ λ x hxS,
((hms x).symm ▸ measurable_set_generate_from (hfs x hxS) : measurable_set[m x] (f x)) | lemma | probability_theory.Indep.Indep_sets | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_set",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep.indep_sets [measurable_space Ω] {μ : measure Ω} {s1 s2 : set (set Ω)}
(h_indep : indep (generate_from s1) (generate_from s2) μ) :
indep_sets s1 s2 μ | λ t1 t2 ht1 ht2, h_indep t1 t2 (measurable_set_generate_from ht1) (measurable_set_generate_from ht2) | lemma | probability_theory.indep.indep_sets | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_sets.indep_aux {m2 : measurable_space Ω}
{m : measurable_space Ω} {μ : measure Ω} [is_probability_measure μ] {p1 p2 : set (set Ω)}
(h2 : m2 ≤ m) (hp2 : is_pi_system p2) (hpm2 : m2 = generate_from p2)
(hyp : indep_sets p1 p2 μ) {t1 t2 : set Ω} (ht1 : t1 ∈ p1) (ht2m : measurable_set[m2] t2) :
μ (t1 ∩ t2) = ... | begin
let μ_inter := μ.restrict t1,
let ν := (μ t1) • μ,
have h_univ : μ_inter set.univ = ν set.univ,
by rw [measure.restrict_apply_univ, measure.smul_apply, smul_eq_mul, measure_univ, mul_one],
haveI : is_finite_measure μ_inter := @restrict.is_finite_measure Ω _ t1 μ ⟨measure_lt_top μ t1⟩,
rw [set.inter_co... | lemma | probability_theory.indep_sets.indep_aux | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"is_pi_system",
"measurable_set",
"measurable_space",
"mul_one",
"set.inter_comm",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_sets.indep {m1 m2 : measurable_space Ω} {m : measurable_space Ω}
{μ : measure Ω} [is_probability_measure μ] {p1 p2 : set (set Ω)} (h1 : m1 ≤ m) (h2 : m2 ≤ m)
(hp1 : is_pi_system p1) (hp2 : is_pi_system p2) (hpm1 : m1 = generate_from p1)
(hpm2 : m2 = generate_from p2) (hyp : indep_sets p1 p2 μ) :
indep m1 ... | begin
intros t1 t2 ht1 ht2,
let μ_inter := μ.restrict t2,
let ν := (μ t2) • μ,
have h_univ : μ_inter set.univ = ν set.univ,
by rw [measure.restrict_apply_univ, measure.smul_apply, smul_eq_mul, measure_univ, mul_one],
haveI : is_finite_measure μ_inter := @restrict.is_finite_measure Ω _ t2 μ ⟨measure_lt_top μ... | lemma | probability_theory.indep_sets.indep | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"is_pi_system",
"measurable_set",
"measurable_space",
"mul_comm",
"mul_one",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_sets.indep' {m : measurable_space Ω}
{μ : measure Ω} [is_probability_measure μ] {p1 p2 : set (set Ω)}
(hp1m : ∀ s ∈ p1, measurable_set s) (hp2m : ∀ s ∈ p2, measurable_set s)
(hp1 : is_pi_system p1) (hp2 : is_pi_system p2) (hyp : indep_sets p1 p2 μ) :
indep (generate_from p1) (generate_from p2) μ | hyp.indep (generate_from_le hp1m) (generate_from_le hp2m) hp1 hp2 rfl rfl | lemma | probability_theory.indep_sets.indep' | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"is_pi_system",
"measurable_set",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_sets_pi_Union_Inter_of_disjoint [is_probability_measure μ]
{s : ι → set (set Ω)} {S T : set ι}
(h_indep : Indep_sets s μ) (hST : disjoint S T) :
indep_sets (pi_Union_Inter s S) (pi_Union_Inter s T) μ | begin
rintros t1 t2 ⟨p1, hp1, f1, ht1_m, ht1_eq⟩ ⟨p2, hp2, f2, ht2_m, ht2_eq⟩,
classical,
let g := λ i, ite (i ∈ p1) (f1 i) set.univ ∩ ite (i ∈ p2) (f2 i) set.univ,
have h_P_inter : μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, μ (g n),
{ have hgm : ∀ i ∈ p1 ∪ p2, g i ∈ s i,
{ intros i hi_mem_union,
rw finset.mem_u... | lemma | probability_theory.indep_sets_pi_Union_Inter_of_disjoint | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"disjoint",
"finset.mem_union",
"finset.prod_ite_mem",
"finset.prod_mul_distrib",
"finset.union_inter_cancel_left",
"finset.union_inter_cancel_right",
"mul_one",
"one_mul",
"pi_Union_Inter",
"set.inter_univ",
"set.mem_Inter",
"set.mem_inter_iff",
"set.mem_ite_univ_right",
"set.univ_inter"
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_set.indep_generate_from_of_disjoint [is_probability_measure μ] {s : ι → set Ω}
(hsm : ∀ n, measurable_set (s n)) (hs : Indep_set s μ) (S T : set ι) (hST : disjoint S T) :
indep (generate_from {t | ∃ n ∈ S, s n = t}) (generate_from {t | ∃ k ∈ T, s k = t}) μ | begin
rw [← generate_from_pi_Union_Inter_singleton_left,
← generate_from_pi_Union_Inter_singleton_left],
refine indep_sets.indep'
(λ t ht, generate_from_pi_Union_Inter_le _ _ _ _ (measurable_set_generate_from ht))
(λ t ht, generate_from_pi_Union_Inter_le _ _ _ _ (measurable_set_generate_from ht))
_ ... | lemma | probability_theory.Indep_set.indep_generate_from_of_disjoint | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"disjoint",
"generate_from_pi_Union_Inter_le",
"generate_from_pi_Union_Inter_singleton_left",
"is_pi_system.singleton",
"is_pi_system_pi_Union_Inter",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_supr_of_disjoint [is_probability_measure μ] {m : ι → measurable_space Ω}
(h_le : ∀ i, m i ≤ m0) (h_indep : Indep m μ) {S T : set ι} (hST : disjoint S T) :
indep (⨆ i ∈ S, m i) (⨆ i ∈ T, m i) μ | begin
refine indep_sets.indep (supr₂_le (λ i _, h_le i)) (supr₂_le (λ i _, h_le i)) _ _
(generate_from_pi_Union_Inter_measurable_set m S).symm
(generate_from_pi_Union_Inter_measurable_set m T).symm _,
{ exact is_pi_system_pi_Union_Inter _ (λ n, @is_pi_system_measurable_set Ω (m n)) _, },
{ exact is_pi_sys... | lemma | probability_theory.indep_supr_of_disjoint | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"disjoint",
"generate_from_pi_Union_Inter_measurable_set",
"is_pi_system_pi_Union_Inter",
"measurable_space",
"supr₂_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_supr_of_directed_le {Ω} {m : ι → measurable_space Ω}
{m' m0 : measurable_space Ω} {μ : measure Ω} [is_probability_measure μ]
(h_indep : ∀ i, indep (m i) m' μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0)
(hm : directed (≤) m) :
indep (⨆ i, m i) m' μ | begin
let p : ι → set (set Ω) := λ n, {t | measurable_set[m n] t},
have hp : ∀ n, is_pi_system (p n) := λ n, @is_pi_system_measurable_set Ω (m n),
have h_gen_n : ∀ n, m n = generate_from (p n),
from λ n, (@generate_from_measurable_set Ω (m n)).symm,
have hp_supr_pi : is_pi_system (⋃ n, p n) := is_pi_system_... | lemma | probability_theory.indep_supr_of_directed_le | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"directed",
"is_pi_system",
"is_pi_system_Union_of_directed_le",
"measurable_set",
"measurable_space",
"supr_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_set.indep_generate_from_lt [preorder ι] [is_probability_measure μ]
{s : ι → set Ω} (hsm : ∀ n, measurable_set (s n)) (hs : Indep_set s μ) (i : ι) :
indep (generate_from {s i}) (generate_from {t | ∃ j < i, s j = t}) μ | begin
convert hs.indep_generate_from_of_disjoint hsm {i} {j | j < i}
(set.disjoint_singleton_left.mpr (lt_irrefl _)),
simp only [set.mem_singleton_iff, exists_prop, exists_eq_left, set.set_of_eq_eq_singleton'],
end | lemma | probability_theory.Indep_set.indep_generate_from_lt | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"exists_eq_left",
"exists_prop",
"measurable_set",
"set.mem_singleton_iff",
"set.set_of_eq_eq_singleton'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_set.indep_generate_from_le [linear_order ι] [is_probability_measure μ]
{s : ι → set Ω} (hsm : ∀ n, measurable_set (s n)) (hs : Indep_set s μ)
(i : ι) {k : ι} (hk : i < k) :
indep (generate_from {s k}) (generate_from {t | ∃ j ≤ i, s j = t}) μ | begin
convert hs.indep_generate_from_of_disjoint hsm {k} {j | j ≤ i}
(set.disjoint_singleton_left.mpr hk.not_le),
simp only [set.mem_singleton_iff, exists_prop, exists_eq_left, set.set_of_eq_eq_singleton'],
end | lemma | probability_theory.Indep_set.indep_generate_from_le | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"exists_eq_left",
"exists_prop",
"measurable_set",
"set.mem_singleton_iff",
"set.set_of_eq_eq_singleton'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_set.indep_generate_from_le_nat [is_probability_measure μ]
{s : ℕ → set Ω} (hsm : ∀ n, measurable_set (s n)) (hs : Indep_set s μ) (n : ℕ):
indep (generate_from {s (n + 1)}) (generate_from {t | ∃ k ≤ n, s k = t}) μ | hs.indep_generate_from_le hsm _ n.lt_succ_self | lemma | probability_theory.Indep_set.indep_generate_from_le_nat | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_supr_of_monotone [semilattice_sup ι] {Ω} {m : ι → measurable_space Ω}
{m' m0 : measurable_space Ω} {μ : measure Ω} [is_probability_measure μ]
(h_indep : ∀ i, indep (m i) m' μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0) (hm : monotone m) :
indep (⨆ i, m i) m' μ | indep_supr_of_directed_le h_indep h_le h_le' (monotone.directed_le hm) | lemma | probability_theory.indep_supr_of_monotone | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space",
"monotone",
"monotone.directed_le",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_supr_of_antitone [semilattice_inf ι] {Ω} {m : ι → measurable_space Ω}
{m' m0 : measurable_space Ω} {μ : measure Ω} [is_probability_measure μ]
(h_indep : ∀ i, indep (m i) m' μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0) (hm : antitone m) :
indep (⨆ i, m i) m' μ | indep_supr_of_directed_le h_indep h_le h_le' (directed_of_inf hm) | lemma | probability_theory.indep_supr_of_antitone | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"antitone",
"directed_of_inf",
"measurable_space",
"semilattice_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_sets.pi_Union_Inter_of_not_mem {π : ι → set (set Ω)} {a : ι} {S : finset ι}
(hp_ind : Indep_sets π μ) (haS : a ∉ S) :
indep_sets (pi_Union_Inter π S) (π a) μ | begin
rintros t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia,
rw [finset.coe_subset] at hs_mem,
classical,
let f := λ n, ite (n = a) t2 (ite (n ∈ s) (ft1 n) set.univ),
have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n,
{ intros n hn_mem_insert,
simp_rw f,
cases (finset.mem_insert.mp hn_mem_insert) wi... | lemma | probability_theory.Indep_sets.pi_Union_Inter_of_not_mem | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"finset",
"finset.coe_subset",
"finset.prod_insert",
"finset.set_bInter_insert",
"mul_comm",
"pi_Union_Inter",
"set.inter_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_sets.Indep [is_probability_measure μ] (m : ι → measurable_space Ω)
(h_le : ∀ i, m i ≤ m0) (π : ι → set (set Ω)) (h_pi : ∀ n, is_pi_system (π n))
(h_generate : ∀ i, m i = generate_from (π i)) (h_ind : Indep_sets π μ) :
Indep m μ | begin
classical,
refine finset.induction _ _,
{ simp only [measure_univ, implies_true_iff, set.Inter_false, set.Inter_univ, finset.prod_empty,
eq_self_iff_true], },
intros a S ha_notin_S h_rec f hf_m,
have hf_m_S : ∀ x ∈ S, measurable_set[m x] (f x) := λ x hx, hf_m x (by simp [hx]),
rw [finset.set_bIn... | theorem | probability_theory.Indep_sets.Indep | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"finset.induction",
"finset.mem_insert_self",
"finset.prod_empty",
"finset.prod_insert",
"finset.set_bInter_insert",
"generate_from_pi_Union_Inter_le",
"is_pi_system",
"is_pi_system_pi_Union_Inter",
"le_generate_from_pi_Union_Inter",
"measurable_set",
"measurable_space",
"pi_Union_Inter",
"s... | The measurable space structures generated by independent pi-systems are independent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indep_set_iff_indep_sets_singleton {m0 : measurable_space Ω}
(hs_meas : measurable_set s) (ht_meas : measurable_set t)
(μ : measure Ω . volume_tac) [is_probability_measure μ] :
indep_set s t μ ↔ indep_sets {s} {t} μ | ⟨indep.indep_sets, λ h, indep_sets.indep
(generate_from_le (λ u hu, by rwa set.mem_singleton_iff.mp hu))
(generate_from_le (λ u hu, by rwa set.mem_singleton_iff.mp hu)) (is_pi_system.singleton s)
(is_pi_system.singleton t) rfl rfl h⟩ | lemma | probability_theory.indep_set_iff_indep_sets_singleton | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"is_pi_system.singleton",
"measurable_set",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_set_iff_measure_inter_eq_mul {m0 : measurable_space Ω}
(hs_meas : measurable_set s) (ht_meas : measurable_set t)
(μ : measure Ω . volume_tac) [is_probability_measure μ] :
indep_set s t μ ↔ μ (s ∩ t) = μ s * μ t | (indep_set_iff_indep_sets_singleton hs_meas ht_meas μ).trans indep_sets_singleton_iff | lemma | probability_theory.indep_set_iff_measure_inter_eq_mul | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_set",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_sets.indep_set_of_mem {m0 : measurable_space Ω} (hs : s ∈ S) (ht : t ∈ T)
(hs_meas : measurable_set s) (ht_meas : measurable_set t) (μ : measure Ω . volume_tac)
[is_probability_measure μ] (h_indep : indep_sets S T μ) :
indep_set s t μ | (indep_set_iff_measure_inter_eq_mul hs_meas ht_meas μ).mpr (h_indep s t hs ht) | lemma | probability_theory.indep_sets.indep_set_of_mem | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_set",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep.indep_set_of_measurable_set {m₁ m₂ m0 : measurable_space Ω} {μ : measure Ω}
(h_indep : indep m₁ m₂ μ) {s t : set Ω} (hs : measurable_set[m₁] s) (ht : measurable_set[m₂] t) :
indep_set s t μ | begin
refine λ s' t' hs' ht', h_indep s' t' _ _,
{ refine generate_from_induction (λ u, measurable_set[m₁] u) {s} _ _ _ _ hs',
{ simp only [hs, set.mem_singleton_iff, set.mem_set_of_eq, forall_eq], },
{ exact @measurable_set.empty _ m₁, },
{ exact λ u hu, hu.compl, },
{ exact λ f hf, measurable_set.... | lemma | probability_theory.indep.indep_set_of_measurable_set | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"forall_eq",
"measurable_set",
"measurable_set.Union",
"measurable_set.empty",
"measurable_space",
"set.mem_singleton_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_iff_forall_indep_set (m₁ m₂ : measurable_space Ω) {m0 : measurable_space Ω}
(μ : measure Ω) :
indep m₁ m₂ μ ↔ ∀ s t, measurable_set[m₁] s → measurable_set[m₂] t → indep_set s t μ | ⟨λ h, λ s t hs ht, h.indep_set_of_measurable_set hs ht,
λ h s t hs ht, h s t hs ht s t (measurable_set_generate_from (set.mem_singleton s))
(measurable_set_generate_from (set.mem_singleton t))⟩ | lemma | probability_theory.indep_iff_forall_indep_set | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_set",
"measurable_space",
"set.mem_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_sets.meas_Inter [fintype ι] (h : Indep_sets π μ) (hf : ∀ i, f i ∈ π i) :
μ (⋂ i, f i) = ∏ i, μ (f i) | by simp [← h _ (λ i _, hf _)] | lemma | probability_theory.Indep_sets.meas_Inter | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_comap_mem_iff : Indep (λ i, measurable_space.comap (∈ f i) ⊤) μ ↔ Indep_set f μ | by { simp_rw ←generate_from_singleton, refl } | lemma | probability_theory.Indep_comap_mem_iff | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_sets_singleton_iff :
Indep_sets (λ i, {f i}) μ ↔ ∀ t, μ (⋂ i ∈ t, f i) = ∏ i in t, μ (f i) | forall_congr $ λ t,
⟨λ h, h $ λ _ _, mem_singleton _,
λ h f hf, begin
refine eq.trans _ (h.trans $ finset.prod_congr rfl $ λ i hi, congr_arg _ $ (hf i hi).symm),
rw Inter₂_congr hf,
end⟩ | lemma | probability_theory.Indep_sets_singleton_iff | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"finset.prod_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_set_iff_Indep_sets_singleton (hf : ∀ i, measurable_set (f i)) :
Indep_set f μ ↔ Indep_sets (λ i, {f i}) μ | ⟨Indep.Indep_sets $ λ _, rfl, Indep_sets.Indep _
(λ i, generate_from_le $ by { rintro t (rfl : t = _), exact hf _}) _
(λ _, is_pi_system.singleton _) $ λ _, rfl⟩ | lemma | probability_theory.Indep_set_iff_Indep_sets_singleton | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"is_pi_system.singleton",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_set_iff_measure_Inter_eq_prod (hf : ∀ i, measurable_set (f i)) :
Indep_set f μ ↔ ∀ s, μ (⋂ i ∈ s, f i) = ∏ i in s, μ (f i) | (Indep_set_iff_Indep_sets_singleton hf).trans Indep_sets_singleton_iff | lemma | probability_theory.Indep_set_iff_measure_Inter_eq_prod | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_sets.Indep_set_of_mem (hfπ : ∀ i, f i ∈ π i) (hf : ∀ i, measurable_set (f i))
(hπ : Indep_sets π μ) : Indep_set f μ | (Indep_set_iff_measure_Inter_eq_prod hf).2 $ λ t, hπ _ $ λ i _, hfπ _ | lemma | probability_theory.Indep_sets.Indep_set_of_mem | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_fun_iff_measure_inter_preimage_eq_mul
{mβ : measurable_space β} {mβ' : measurable_space β'} :
indep_fun f g μ
↔ ∀ s t, measurable_set s → measurable_set t
→ μ (f ⁻¹' s ∩ g ⁻¹' t) = μ (f ⁻¹' s) * μ (g ⁻¹' t) | begin
split; intro h,
{ refine λ s t hs ht, h (f ⁻¹' s) (g ⁻¹' t) ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩, },
{ rintros _ _ ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩, exact h s t hs ht, },
end | lemma | probability_theory.indep_fun_iff_measure_inter_preimage_eq_mul | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_set",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_fun_iff_measure_inter_preimage_eq_mul {ι : Type*} {β : ι → Type*}
(m : Π x, measurable_space (β x)) (f : Π i, Ω → β i) :
Indep_fun m f μ
↔ ∀ (S : finset ι) {sets : Π i : ι, set (β i)} (H : ∀ i, i ∈ S → measurable_set[m i] (sets i)),
μ (⋂ i ∈ S, (f i) ⁻¹' (sets i)) = ∏ i in S, μ ((f i) ⁻¹' (sets i)) | begin
refine ⟨λ h S sets h_meas, h _ (λ i hi_mem, ⟨sets i, h_meas i hi_mem, rfl⟩), _⟩,
intros h S setsΩ h_meas,
classical,
let setsβ : (Π i : ι, set (β i)) := λ i,
dite (i ∈ S) (λ hi_mem, (h_meas i hi_mem).some) (λ _, set.univ),
have h_measβ : ∀ i ∈ S, measurable_set[m i] (setsβ i),
{ intros i hi_mem,
... | lemma | probability_theory.Indep_fun_iff_measure_inter_preimage_eq_mul | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"finset",
"finset.prod_congr",
"measurable_set",
"measurable_space",
"set.mem_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_fun_iff_indep_set_preimage {mβ : measurable_space β} {mβ' : measurable_space β'}
[is_probability_measure μ] (hf : measurable f) (hg : measurable g) :
indep_fun f g μ ↔ ∀ s t, measurable_set s → measurable_set t → indep_set (f ⁻¹' s) (g ⁻¹' t) μ | begin
refine indep_fun_iff_measure_inter_preimage_eq_mul.trans _,
split; intros h s t hs ht; specialize h s t hs ht,
{ rwa indep_set_iff_measure_inter_eq_mul (hf hs) (hg ht) μ, },
{ rwa ← indep_set_iff_measure_inter_eq_mul (hf hs) (hg ht) μ, },
end | lemma | probability_theory.indep_fun_iff_indep_set_preimage | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable",
"measurable_set",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_fun.symm {mβ : measurable_space β} {f g : Ω → β} (hfg : indep_fun f g μ) :
indep_fun g f μ | hfg.symm | lemma | probability_theory.indep_fun.symm | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_fun.ae_eq {mβ : measurable_space β} {f g f' g' : Ω → β}
(hfg : indep_fun f g μ) (hf : f =ᵐ[μ] f') (hg : g =ᵐ[μ] g') :
indep_fun f' g' μ | begin
rintro _ _ ⟨A, hA, rfl⟩ ⟨B, hB, rfl⟩,
have h1 : f ⁻¹' A =ᵐ[μ] f' ⁻¹' A := hf.fun_comp A,
have h2 : g ⁻¹' B =ᵐ[μ] g' ⁻¹' B := hg.fun_comp B,
rw [← measure_congr h1, ← measure_congr h2, ← measure_congr (h1.inter h2)],
exact hfg _ _ ⟨_, hA, rfl⟩ ⟨_, hB, rfl⟩
end | lemma | probability_theory.indep_fun.ae_eq | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_fun.comp {mβ : measurable_space β} {mβ' : measurable_space β'}
{mγ : measurable_space γ} {mγ' : measurable_space γ'} {φ : β → γ} {ψ : β' → γ'}
(hfg : indep_fun f g μ) (hφ : measurable φ) (hψ : measurable ψ) :
indep_fun (φ ∘ f) (ψ ∘ g) μ | begin
rintro _ _ ⟨A, hA, rfl⟩ ⟨B, hB, rfl⟩,
apply hfg,
{ exact ⟨φ ⁻¹' A, hφ hA, set.preimage_comp.symm⟩ },
{ exact ⟨ψ ⁻¹' B, hψ hB, set.preimage_comp.symm⟩ }
end | lemma | probability_theory.indep_fun.comp | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_fun.indep_fun_finset [is_probability_measure μ]
{ι : Type*} {β : ι → Type*} {m : Π i, measurable_space (β i)}
{f : Π i, Ω → β i} (S T : finset ι) (hST : disjoint S T) (hf_Indep : Indep_fun m f μ)
(hf_meas : ∀ i, measurable (f i)) :
indep_fun (λ a (i : S), f i a) (λ a (i : T), f i a) μ | begin
-- We introduce π-systems, build from the π-system of boxes which generates `measurable_space.pi`.
let πSβ := (set.pi (set.univ : set S) ''
(set.pi (set.univ : set S) (λ i, {s : set (β i) | measurable_set[m i] s}))),
let πS := {s : set Ω | ∃ t ∈ πSβ, (λ a (i : S), f i a) ⁻¹' t = s},
have hπS_pi : is_p... | lemma | probability_theory.Indep_fun.indep_fun_finset | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"disjoint",
"exists_prop",
"finset",
"finset.mem_union",
"finset.prod_congr",
"finset.prod_union",
"is_pi_system",
"measurable",
"measurable_set",
"measurable_space",
"set.inter_univ",
"set.mem_Inter",
"set.mem_image",
"set.mem_inter_iff",
"set.mem_preimage",
"set.mem_univ",
"set.mem... | If `f` is a family of mutually independent random variables (`Indep_fun m f μ`) and `S, T` are
two disjoint finite index sets, then the tuple formed by `f i` for `i ∈ S` is independent of the
tuple `(f i)_i` for `i ∈ T`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Indep_fun.indep_fun_prod [is_probability_measure μ]
{ι : Type*} {β : ι → Type*} {m : Π i, measurable_space (β i)}
{f : Π i, Ω → β i} (hf_Indep : Indep_fun m f μ) (hf_meas : ∀ i, measurable (f i))
(i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
indep_fun (λ a, (f i a, f j a)) (f k) μ | begin
classical,
have h_right : f k = (λ p : (Π j : ({k} : finset ι), β j), p ⟨k, finset.mem_singleton_self k⟩)
∘ (λ a (j : ({k} : finset ι)), f j a) := rfl,
have h_meas_right : measurable
(λ p : (Π j : ({k} : finset ι), β j), p ⟨k, finset.mem_singleton_self k⟩),
from measurable_pi_apply ⟨k, finset.... | lemma | probability_theory.Indep_fun.indep_fun_prod | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"finset",
"finset.disjoint_singleton_right",
"finset.mem_insert",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"finset.mem_singleton",
"finset.mem_singleton_self",
"measurable",
"measurable.prod",
"measurable_pi_apply",
"measurable_space",
"not_or_distrib",
"prod.mk.inj_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_fun.mul [is_probability_measure μ]
{ι : Type*} {β : Type*} {m : measurable_space β} [has_mul β] [has_measurable_mul₂ β]
{f : ι → Ω → β} (hf_Indep : Indep_fun (λ _, m) f μ) (hf_meas : ∀ i, measurable (f i))
(i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
indep_fun (f i * f j) (f k) μ | begin
have : indep_fun (λ ω, (f i ω, f j ω)) (f k) μ := hf_Indep.indep_fun_prod hf_meas i j k hik hjk,
change indep_fun ((λ p : β × β, p.fst * p.snd) ∘ (λ ω, (f i ω, f j ω))) (id ∘ (f k)) μ,
exact indep_fun.comp this (measurable_fst.mul measurable_snd) measurable_id,
end | lemma | probability_theory.Indep_fun.mul | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"has_measurable_mul₂",
"measurable",
"measurable_id",
"measurable_snd",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_fun.indep_fun_finset_prod_of_not_mem [is_probability_measure μ]
{ι : Type*} {β : Type*} {m : measurable_space β} [comm_monoid β] [has_measurable_mul₂ β]
{f : ι → Ω → β} (hf_Indep : Indep_fun (λ _, m) f μ) (hf_meas : ∀ i, measurable (f i))
{s : finset ι} {i : ι} (hi : i ∉ s) :
indep_fun (∏ j in s, f j) (f ... | begin
classical,
have h_right : f i = (λ p : (Π j : ({i} : finset ι), β), p ⟨i, finset.mem_singleton_self i⟩)
∘ (λ a (j : ({i} : finset ι)), f j a) := rfl,
have h_meas_right : measurable
(λ p : (Π j : ({i} : finset ι), β), p ⟨i, finset.mem_singleton_self i⟩),
from measurable_pi_apply ⟨i, finset.mem_... | lemma | probability_theory.Indep_fun.indep_fun_finset_prod_of_not_mem | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"comm_monoid",
"finset",
"finset.mem_singleton_self",
"finset.prod_apply",
"finset.prod_coe_sort",
"finset.univ",
"has_measurable_mul₂",
"measurable",
"measurable_pi_apply",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_fun.indep_fun_prod_range_succ [is_probability_measure μ]
{β : Type*} {m : measurable_space β} [comm_monoid β] [has_measurable_mul₂ β]
{f : ℕ → Ω → β} (hf_Indep : Indep_fun (λ _, m) f μ) (hf_meas : ∀ i, measurable (f i))
(n : ℕ) :
indep_fun (∏ j in finset.range n, f j) (f n) μ | hf_Indep.indep_fun_finset_prod_of_not_mem hf_meas finset.not_mem_range_self | lemma | probability_theory.Indep_fun.indep_fun_prod_range_succ | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"comm_monoid",
"finset.not_mem_range_self",
"finset.range",
"has_measurable_mul₂",
"measurable",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Indep_set.Indep_fun_indicator [has_zero β] [has_one β] {m : measurable_space β}
{s : ι → set Ω} (hs : Indep_set s μ) :
Indep_fun (λ n, m) (λ n, (s n).indicator (λ ω, 1)) μ | begin
classical,
rw Indep_fun_iff_measure_inter_preimage_eq_mul,
rintro S π hπ,
simp_rw set.indicator_const_preimage_eq_union,
refine @hs S (λ i, ite (1 ∈ π i) (s i) ∅ ∪ ite ((0 : β) ∈ π i) (s i)ᶜ ∅) (λ i hi, _),
have hsi : measurable_set[generate_from {s i}] (s i),
from measurable_set_generate_from (se... | lemma | probability_theory.Indep_set.Indep_fun_indicator | probability.independence | src/probability/independence/basic.lean | [
"measure_theory.constructions.pi"
] | [
"measurable_set",
"measurable_set.empty",
"measurable_set.ite'",
"measurable_set.union",
"measurable_space",
"set.mem_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_eq_zero_or_one_or_top_of_indep_set_self {t : set Ω} (h_indep : indep_set t t μ) :
μ t = 0 ∨ μ t = 1 ∨ μ t = ∞ | begin
specialize h_indep t t (measurable_set_generate_from (set.mem_singleton t))
(measurable_set_generate_from (set.mem_singleton t)),
by_cases h0 : μ t = 0,
{ exact or.inl h0, },
by_cases h_top : μ t = ∞,
{ exact or.inr (or.inr h_top), },
rw [← one_mul (μ (t ∩ t)), set.inter_self, ennreal.mul_eq_mul_r... | lemma | probability_theory.measure_eq_zero_or_one_or_top_of_indep_set_self | probability.independence | src/probability/independence/zero_one.lean | [
"probability.independence.basic"
] | [
"ennreal.mul_eq_mul_right",
"one_mul",
"set.inter_self",
"set.mem_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measure_eq_zero_or_one_of_indep_set_self [is_finite_measure μ] {t : set Ω}
(h_indep : indep_set t t μ) :
μ t = 0 ∨ μ t = 1 | begin
have h_0_1_top := measure_eq_zero_or_one_or_top_of_indep_set_self h_indep,
simpa [measure_ne_top μ] using h_0_1_top,
end | lemma | probability_theory.measure_eq_zero_or_one_of_indep_set_self | probability.independence | src/probability/independence/zero_one.lean | [
"probability.independence.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_bsupr_compl (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ) (t : set ι) :
indep (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) μ | indep_supr_of_disjoint h_le h_indep disjoint_compl_right | lemma | probability_theory.indep_bsupr_compl | probability.independence | src/probability/independence/zero_one.lean | [
"probability.independence.basic"
] | [
"disjoint_compl_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_bsupr_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : Indep s μ)
(hf : ∀ t, p t → tᶜ ∈ f) {t : set ι} (ht : p t) :
indep (⨆ n ∈ t, s n) (limsup s f) μ | begin
refine indep_of_indep_of_le_right (indep_bsupr_compl h_le h_indep t) _,
refine Limsup_le_of_le (by is_bounded_default) _,
simp only [set.mem_compl_iff, eventually_map],
exact eventually_of_mem (hf t ht) le_supr₂,
end | lemma | probability_theory.indep_bsupr_limsup | probability.independence | src/probability/independence/zero_one.lean | [
"probability.independence.basic"
] | [
"le_supr₂",
"set.mem_compl_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indep_supr_directed_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)) :
indep (⨆ a, ⨆ n ∈ (ns a), s n) (limsup s f) μ | begin
refine indep_supr_of_directed_le _ _ _ _,
{ exact λ a, indep_bsupr_limsup h_le h_indep hf (hnsp a), },
{ exact λ a, supr₂_le (λ n hn, h_le n), },
{ exact limsup_le_supr.trans (supr_le h_le), },
{ intros a b,
obtain ⟨c, hc⟩ := hns a b,
refine ⟨c, _, _⟩; refine supr_mono (λ n, supr_mono' (λ hn, ⟨_... | lemma | probability_theory.indep_supr_directed_limsup | probability.independence | src/probability/independence/zero_one.lean | [
"probability.independence.basic"
] | [
"directed",
"supr_le",
"supr_mono",
"supr_mono'",
"supr₂_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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