# Convergence in measure #

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 results #

def MeasureTheory.TendstoInMeasure {α : Type u_1} {ι : Type u_2} {E : Type u_3} [Dist E] :
{x : } → (ιαE)(αE)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
Instances For
theorem MeasureTheory.tendstoInMeasure_iff_norm {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : } {μ : } {l : } {f : ιαE} {g : αE} :
∀ (ε : ), 0 < εFilter.Tendsto (fun (i : ι) => μ {x : α | ε f i x - g x}) l (nhds 0)
theorem MeasureTheory.TendstoInMeasure.congr' {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : } {μ : } [Dist E] {l : } {f : ιαE} {f' : ιαE} {g : αE} {g' : αE} (h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i) (h_right : g =ᵐ[μ] g') (h_tendsto : ) :
theorem MeasureTheory.TendstoInMeasure.congr {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : } {μ : } [Dist E] {l : } {f : ιαE} {f' : ιαE} {g : αE} {g' : αE} (h_left : ∀ (i : ι), f i =ᵐ[μ] f' i) (h_right : g =ᵐ[μ] g') (h_tendsto : ) :
theorem MeasureTheory.TendstoInMeasure.congr_left {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : } {μ : } [Dist E] {l : } {f : ιαE} {f' : ιαE} {g : αE} (h : ∀ (i : ι), f i =ᵐ[μ] f' i) (h_tendsto : ) :
theorem MeasureTheory.TendstoInMeasure.congr_right {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : } {μ : } [Dist E] {l : } {f : ιαE} {g : αE} {g' : αE} (h : g =ᵐ[μ] g') (h_tendsto : ) :
theorem MeasureTheory.tendstoInMeasure_of_tendsto_ae_of_stronglyMeasurable {α : Type u_1} {E : Type u_3} {m : } {μ : } [] {f : αE} {g : αE} (hf : ∀ (n : ), ) (hg : ) (hfg : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun (n : ) => f n x) Filter.atTop (nhds (g x))) :
MeasureTheory.TendstoInMeasure μ f Filter.atTop g

Auxiliary lemma for tendstoInMeasure_of_tendsto_ae.

theorem MeasureTheory.tendstoInMeasure_of_tendsto_ae {α : Type u_1} {E : Type u_3} {m : } {μ : } [] {f : αE} {g : αE} (hf : ∀ (n : ), ) (hfg : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun (n : ) => f n x) Filter.atTop (nhds (g x))) :
MeasureTheory.TendstoInMeasure μ f Filter.atTop g

Convergence a.e. implies convergence in measure in a finite measure space.

theorem MeasureTheory.ExistsSeqTendstoAe.exists_nat_measure_lt_two_inv {α : Type u_1} {E : Type u_3} {m : } {μ : } [] {f : αE} {g : αE} (hfg : MeasureTheory.TendstoInMeasure μ f Filter.atTop g) (n : ) :
∃ (N : ), m_1N, μ {x : α | 2⁻¹ ^ n dist (f m_1 x) (g x)} 2⁻¹ ^ n
noncomputable def MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeqAux {α : Type u_1} {E : Type u_3} {m : } {μ : } [] {f : αE} {g : αE} (hfg : MeasureTheory.TendstoInMeasure μ f Filter.atTop g) (n : ) :

Given a sequence of functions f which converges in measure to g, seqTendstoAeSeqAux is a sequence such that ∀ m ≥ seqTendstoAeSeqAux n, μ {x | 2⁻¹ ^ n ≤ dist (f m x) (g x)} ≤ 2⁻¹ ^ n.

Equations
Instances For
noncomputable def MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq {α : Type u_1} {E : Type u_3} {m : } {μ : } [] {f : αE} {g : αE} (hfg : MeasureTheory.TendstoInMeasure μ f Filter.atTop g) :

Transformation of seqTendstoAeSeqAux to makes sure it is strictly monotone.

Equations
Instances For
theorem MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_succ {α : Type u_1} {E : Type u_3} {m : } {μ : } [] {f : αE} {g : αE} (hfg : MeasureTheory.TendstoInMeasure μ f Filter.atTop g) {n : } :
=
theorem MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_spec {α : Type u_1} {E : Type u_3} {m : } {μ : } [] {f : αE} {g : αE} (hfg : MeasureTheory.TendstoInMeasure μ f Filter.atTop g) (n : ) (k : ) (hn : ) :
μ {x : α | 2⁻¹ ^ n dist (f k x) (g x)} 2⁻¹ ^ n
theorem MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_strictMono {α : Type u_1} {E : Type u_3} {m : } {μ : } [] {f : αE} {g : αE} (hfg : MeasureTheory.TendstoInMeasure μ f Filter.atTop g) :
theorem MeasureTheory.TendstoInMeasure.exists_seq_tendsto_ae {α : Type u_1} {E : Type u_3} {m : } {μ : } [] {f : αE} {g : αE} (hfg : MeasureTheory.TendstoInMeasure μ f Filter.atTop g) :
∃ (ns : ), ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun (i : ) => f (ns i) x) Filter.atTop (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 MeasureTheory.TendstoInMeasure.exists_seq_tendstoInMeasure_atTop {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : } {μ : } [] {u : } [u.NeBot] [u.IsCountablyGenerated] {f : ιαE} {g : αE} (hfg : ) :
∃ (ns : ι), MeasureTheory.TendstoInMeasure μ (fun (n : ) => f (ns n)) Filter.atTop g
theorem MeasureTheory.TendstoInMeasure.exists_seq_tendsto_ae' {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : } {μ : } [] {u : } [u.NeBot] [u.IsCountablyGenerated] {f : ιαE} {g : αE} (hfg : ) :
∃ (ns : ι), ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun (i : ) => f (ns i) x) Filter.atTop (nhds (g x))
theorem MeasureTheory.TendstoInMeasure.aemeasurable {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : } {μ : } [] [] {u : } [u.NeBot] [u.IsCountablyGenerated] {f : ιαE} {g : αE} (hf : ∀ (n : ι), AEMeasurable (f n) μ) (h_tendsto : ) :
theorem MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm_of_stronglyMeasurable {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : } {μ : } {p : ENNReal} {f : ιαE} {g : αE} (hp_ne_zero : p 0) (hp_ne_top : p ) (hf : ∀ (n : ι), ) (hg : ) {l : } (hfg : Filter.Tendsto (fun (n : ι) => MeasureTheory.eLpNorm (f n - g) p μ) l (nhds 0)) :

This lemma is superceded by MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm where we allow p = ∞ and only require AEStronglyMeasurable.

@[deprecated MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm_of_stronglyMeasurable]
theorem MeasureTheory.tendstoInMeasure_of_tendsto_snorm_of_stronglyMeasurable {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : } {μ : } {p : ENNReal} {f : ιαE} {g : αE} (hp_ne_zero : p 0) (hp_ne_top : p ) (hf : ∀ (n : ι), ) (hg : ) {l : } (hfg : Filter.Tendsto (fun (n : ι) => MeasureTheory.eLpNorm (f n - g) p μ) l (nhds 0)) :

This lemma is superceded by MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm where we allow p = ∞ and only require AEStronglyMeasurable.

theorem MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm_of_ne_top {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : } {μ : } {p : ENNReal} {f : ιαE} {g : αE} (hp_ne_zero : p 0) (hp_ne_top : p ) (hf : ∀ (n : ι), ) (hg : ) {l : } (hfg : Filter.Tendsto (fun (n : ι) => MeasureTheory.eLpNorm (f n - g) p μ) l (nhds 0)) :

This lemma is superceded by MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm where we allow p = ∞.

@[deprecated MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm_of_ne_top]
theorem MeasureTheory.tendstoInMeasure_of_tendsto_snorm_of_ne_top {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : } {μ : } {p : ENNReal} {f : ιαE} {g : αE} (hp_ne_zero : p 0) (hp_ne_top : p ) (hf : ∀ (n : ι), ) (hg : ) {l : } (hfg : Filter.Tendsto (fun (n : ι) => MeasureTheory.eLpNorm (f n - g) p μ) l (nhds 0)) :

This lemma is superceded by MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm where we allow p = ∞.

theorem MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm_top {α : Type u_1} {ι : Type u_2} {m : } {μ : } {E : Type u_4} {f : ιαE} {g : αE} {l : } (hfg : Filter.Tendsto (fun (n : ι) => MeasureTheory.eLpNorm (f n - g) μ) l (nhds 0)) :

See also MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm which work for general Lp-convergence for all p ≠ 0.

@[deprecated MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm_top]
theorem MeasureTheory.tendstoInMeasure_of_tendsto_snorm_top {α : Type u_1} {ι : Type u_2} {m : } {μ : } {E : Type u_4} {f : ιαE} {g : αE} {l : } (hfg : Filter.Tendsto (fun (n : ι) => MeasureTheory.eLpNorm (f n - g) μ) l (nhds 0)) :

See also MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm which work for general Lp-convergence for all p ≠ 0.

theorem MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : } {μ : } {p : ENNReal} {f : ιαE} {g : αE} {l : } (hp_ne_zero : p 0) (hf : ∀ (n : ι), ) (hg : ) (hfg : Filter.Tendsto (fun (n : ι) => MeasureTheory.eLpNorm (f n - g) p μ) l (nhds 0)) :

Convergence in Lp implies convergence in measure.

@[deprecated MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm]
theorem MeasureTheory.tendstoInMeasure_of_tendsto_snorm {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : } {μ : } {p : ENNReal} {f : ιαE} {g : αE} {l : } (hp_ne_zero : p 0) (hf : ∀ (n : ι), ) (hg : ) (hfg : Filter.Tendsto (fun (n : ι) => MeasureTheory.eLpNorm (f n - g) p μ) l (nhds 0)) :

Convergence in Lp implies convergence in measure.

theorem MeasureTheory.tendstoInMeasure_of_tendsto_Lp {α : Type u_1} {ι : Type u_2} {E : Type u_3} {m : } {μ : } {p : ENNReal} [hp : Fact (1 p)] {f : ι()} {g : ()} {l : } (hfg : Filter.Tendsto f l (nhds g)) :
MeasureTheory.TendstoInMeasure μ (fun (n : ι) => (f n)) l g

Convergence in Lp implies convergence in measure.