Convergence in measure #
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We define convergence in measure which is one of the many notions of convergence in probability.
A sequence of functions f is said to converge in measure to some function g
if for all ε > 0, the measure of the set {x | ε ≤ dist (f i x) (g x)} tends to 0 as i
converges along some given filter l.
Convergence in measure is most notably used in the formulation of the weak law of large numbers and is also useful in theorems such as the Vitali convergence theorem. This file provides some basic lemmas for working with convergence in measure and establishes some relations between convergence in measure and other notions of convergence.
Main definitions #
measure_theory.tendsto_in_measure (μ : measure α) (f : ι → α → E) (g : α → E):fconverges inμ-measure tog.
Main results #
measure_theory.tendsto_in_measure_of_tendsto_ae: convergence almost everywhere in a finite measure space implies convergence in measure.measure_theory.tendsto_in_measure.exists_seq_tendsto_ae: iffis a sequence of functions which converges in measure tog, thenfhas a subsequence which convergence almost everywhere tog.measure_theory.tendsto_in_measure_of_tendsto_snorm: convergence in Lp implies convergence in measure.
def
measure_theory.tendsto_in_measure
{α : Type u_1}
{ι : Type u_2}
{E : Type u_3}
[has_dist E]
{m : measurable_space α}
(μ : measure_theory.measure α)
(f : ι → α → E)
(l : filter ι)
(g : α → E) :
Prop
A sequence of functions f is said to converge in measure to some function g if for all
ε > 0, the measure of the set {x | ε ≤ dist (f i x) (g x)} tends to 0 as i converges along
some given filter l.
Equations
- measure_theory.tendsto_in_measure μ f l g = ∀ (ε : ℝ), 0 < ε → filter.tendsto (λ (i : ι), ⇑μ {x : α | ε ≤ has_dist.dist (f i x) (g x)}) l (nhds 0)
theorem
measure_theory.tendsto_in_measure_iff_norm
{α : Type u_1}
{ι : Type u_2}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[seminormed_add_comm_group E]
{l : filter ι}
{f : ι → α → E}
{g : α → E} :
@[protected]
theorem
measure_theory.tendsto_in_measure.congr'
{α : Type u_1}
{ι : Type u_2}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[has_dist E]
{l : filter ι}
{f f' : ι → α → E}
{g g' : α → E}
(h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i)
(h_right : g =ᵐ[μ] g')
(h_tendsto : measure_theory.tendsto_in_measure μ f l g) :
measure_theory.tendsto_in_measure μ f' l g'
@[protected]
theorem
measure_theory.tendsto_in_measure.congr
{α : Type u_1}
{ι : Type u_2}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[has_dist E]
{l : filter ι}
{f f' : ι → α → E}
{g g' : α → E}
(h_left : ∀ (i : ι), f i =ᵐ[μ] f' i)
(h_right : g =ᵐ[μ] g')
(h_tendsto : measure_theory.tendsto_in_measure μ f l g) :
measure_theory.tendsto_in_measure μ f' l g'
theorem
measure_theory.tendsto_in_measure.congr_left
{α : Type u_1}
{ι : Type u_2}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[has_dist E]
{l : filter ι}
{f f' : ι → α → E}
{g : α → E}
(h : ∀ (i : ι), f i =ᵐ[μ] f' i)
(h_tendsto : measure_theory.tendsto_in_measure μ f l g) :
measure_theory.tendsto_in_measure μ f' l g
theorem
measure_theory.tendsto_in_measure.congr_right
{α : Type u_1}
{ι : Type u_2}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[has_dist E]
{l : filter ι}
{f : ι → α → E}
{g g' : α → E}
(h : g =ᵐ[μ] g')
(h_tendsto : measure_theory.tendsto_in_measure μ f l g) :
measure_theory.tendsto_in_measure μ f l g'
theorem
measure_theory.tendsto_in_measure_of_tendsto_ae_of_strongly_measurable
{α : Type u_1}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[metric_space E]
{f : ℕ → α → E}
{g : α → E}
[measure_theory.is_finite_measure μ]
(hf : ∀ (n : ℕ), measure_theory.strongly_measurable (f n))
(hg : measure_theory.strongly_measurable g)
(hfg : ∀ᵐ (x : α) ∂μ, filter.tendsto (λ (n : ℕ), f n x) filter.at_top (nhds (g x))) :
Auxiliary lemma for tendsto_in_measure_of_tendsto_ae.
theorem
measure_theory.tendsto_in_measure_of_tendsto_ae
{α : Type u_1}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[metric_space E]
{f : ℕ → α → E}
{g : α → E}
[measure_theory.is_finite_measure μ]
(hf : ∀ (n : ℕ), measure_theory.ae_strongly_measurable (f n) μ)
(hfg : ∀ᵐ (x : α) ∂μ, filter.tendsto (λ (n : ℕ), f n x) filter.at_top (nhds (g x))) :
Convergence a.e. implies convergence in measure in a finite measure space.
theorem
measure_theory.exists_seq_tendsto_ae.exists_nat_measure_lt_two_inv
{α : Type u_1}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[metric_space E]
{f : ℕ → α → E}
{g : α → E}
(hfg : measure_theory.tendsto_in_measure μ f filter.at_top g)
(n : ℕ) :
noncomputable
def
measure_theory.exists_seq_tendsto_ae.seq_tendsto_ae_seq_aux
{α : Type u_1}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[metric_space E]
{f : ℕ → α → E}
{g : α → E}
(hfg : measure_theory.tendsto_in_measure μ f filter.at_top g)
(n : ℕ) :
Given a sequence of functions f which converges in measure to g,
seq_tendsto_ae_seq_aux is a sequence such that
∀ m ≥ seq_tendsto_ae_seq_aux n, μ {x | 2⁻¹ ^ n ≤ dist (f m x) (g x)} ≤ 2⁻¹ ^ n.
Equations
noncomputable
def
measure_theory.exists_seq_tendsto_ae.seq_tendsto_ae_seq
{α : Type u_1}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[metric_space E]
{f : ℕ → α → E}
{g : α → E}
(hfg : measure_theory.tendsto_in_measure μ f filter.at_top g) :
Transformation of seq_tendsto_ae_seq_aux to makes sure it is strictly monotone.
Equations
- measure_theory.exists_seq_tendsto_ae.seq_tendsto_ae_seq hfg (n + 1) = linear_order.max (measure_theory.exists_seq_tendsto_ae.seq_tendsto_ae_seq_aux hfg (n + 1)) (measure_theory.exists_seq_tendsto_ae.seq_tendsto_ae_seq hfg n + 1)
- measure_theory.exists_seq_tendsto_ae.seq_tendsto_ae_seq hfg 0 = measure_theory.exists_seq_tendsto_ae.seq_tendsto_ae_seq_aux hfg 0
theorem
measure_theory.exists_seq_tendsto_ae.seq_tendsto_ae_seq_succ
{α : Type u_1}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[metric_space E]
{f : ℕ → α → E}
{g : α → E}
(hfg : measure_theory.tendsto_in_measure μ f filter.at_top g)
{n : ℕ} :
theorem
measure_theory.exists_seq_tendsto_ae.seq_tendsto_ae_seq_spec
{α : Type u_1}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[metric_space E]
{f : ℕ → α → E}
{g : α → E}
(hfg : measure_theory.tendsto_in_measure μ f filter.at_top g)
(n k : ℕ)
(hn : measure_theory.exists_seq_tendsto_ae.seq_tendsto_ae_seq hfg n ≤ k) :
theorem
measure_theory.exists_seq_tendsto_ae.seq_tendsto_ae_seq_strict_mono
{α : Type u_1}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[metric_space E]
{f : ℕ → α → E}
{g : α → E}
(hfg : measure_theory.tendsto_in_measure μ f filter.at_top g) :
theorem
measure_theory.tendsto_in_measure.exists_seq_tendsto_ae
{α : Type u_1}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[metric_space E]
{f : ℕ → α → E}
{g : α → E}
(hfg : measure_theory.tendsto_in_measure μ f filter.at_top g) :
∃ (ns : ℕ → ℕ), strict_mono ns ∧ ∀ᵐ (x : α) ∂μ, filter.tendsto (λ (i : ℕ), f (ns i) x) filter.at_top (nhds (g x))
If f is a sequence of functions which converges in measure to g, then there exists a
subsequence of f which converges a.e. to g.
theorem
measure_theory.tendsto_in_measure.exists_seq_tendsto_in_measure_at_top
{α : Type u_1}
{ι : Type u_2}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[metric_space E]
{u : filter ι}
[u.ne_bot]
[u.is_countably_generated]
{f : ι → α → E}
{g : α → E}
(hfg : measure_theory.tendsto_in_measure μ f u g) :
∃ (ns : ℕ → ι), measure_theory.tendsto_in_measure μ (λ (n : ℕ), f (ns n)) filter.at_top g
theorem
measure_theory.tendsto_in_measure.exists_seq_tendsto_ae'
{α : Type u_1}
{ι : Type u_2}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[metric_space E]
{u : filter ι}
[u.ne_bot]
[u.is_countably_generated]
{f : ι → α → E}
{g : α → E}
(hfg : measure_theory.tendsto_in_measure μ f u g) :
theorem
measure_theory.tendsto_in_measure.ae_measurable
{α : Type u_1}
{ι : Type u_2}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[measurable_space E]
[normed_add_comm_group E]
[borel_space E]
{u : filter ι}
[u.ne_bot]
[u.is_countably_generated]
{f : ι → α → E}
{g : α → E}
(hf : ∀ (n : ι), ae_measurable (f n) μ)
(h_tendsto : measure_theory.tendsto_in_measure μ f u g) :
ae_measurable g μ
theorem
measure_theory.tendsto_in_measure_of_tendsto_snorm_of_strongly_measurable
{α : Type u_1}
{ι : Type u_2}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[normed_add_comm_group E]
{p : ennreal}
{f : ι → α → E}
{g : α → E}
(hp_ne_zero : p ≠ 0)
(hp_ne_top : p ≠ ⊤)
(hf : ∀ (n : ι), measure_theory.strongly_measurable (f n))
(hg : measure_theory.strongly_measurable g)
{l : filter ι}
(hfg : filter.tendsto (λ (n : ι), measure_theory.snorm (f n - g) p μ) l (nhds 0)) :
This lemma is superceded by measure_theory.tendsto_in_measure_of_tendsto_snorm where we
allow p = ∞ and only require ae_strongly_measurable.
theorem
measure_theory.tendsto_in_measure_of_tendsto_snorm_of_ne_top
{α : Type u_1}
{ι : Type u_2}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[normed_add_comm_group E]
{p : ennreal}
{f : ι → α → E}
{g : α → E}
(hp_ne_zero : p ≠ 0)
(hp_ne_top : p ≠ ⊤)
(hf : ∀ (n : ι), measure_theory.ae_strongly_measurable (f n) μ)
(hg : measure_theory.ae_strongly_measurable g μ)
{l : filter ι}
(hfg : filter.tendsto (λ (n : ι), measure_theory.snorm (f n - g) p μ) l (nhds 0)) :
This lemma is superceded by measure_theory.tendsto_in_measure_of_tendsto_snorm where we
allow p = ∞.
theorem
measure_theory.tendsto_in_measure_of_tendsto_snorm_top
{α : Type u_1}
{ι : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
{E : Type u_3}
[normed_add_comm_group E]
{f : ι → α → E}
{g : α → E}
{l : filter ι}
(hfg : filter.tendsto (λ (n : ι), measure_theory.snorm (f n - g) ⊤ μ) l (nhds 0)) :
See also measure_theory.tendsto_in_measure_of_tendsto_snorm which work for general
Lp-convergence for all p ≠ 0.
theorem
measure_theory.tendsto_in_measure_of_tendsto_snorm
{α : Type u_1}
{ι : Type u_2}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[normed_add_comm_group E]
{p : ennreal}
{f : ι → α → E}
{g : α → E}
{l : filter ι}
(hp_ne_zero : p ≠ 0)
(hf : ∀ (n : ι), measure_theory.ae_strongly_measurable (f n) μ)
(hg : measure_theory.ae_strongly_measurable g μ)
(hfg : filter.tendsto (λ (n : ι), measure_theory.snorm (f n - g) p μ) l (nhds 0)) :
Convergence in Lp implies convergence in measure.
theorem
measure_theory.tendsto_in_measure_of_tendsto_Lp
{α : Type u_1}
{ι : Type u_2}
{E : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
[normed_add_comm_group E]
{p : ennreal}
[hp : fact (1 ≤ p)]
{f : ι → ↥(measure_theory.Lp E p μ)}
{g : ↥(measure_theory.Lp E p μ)}
{l : filter ι}
(hfg : filter.tendsto f l (nhds g)) :
measure_theory.tendsto_in_measure μ (λ (n : ι), ⇑(f n)) l ⇑g
Convergence in Lp implies convergence in measure.