File: Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean

full_name: MeasureTheory.TendstoInMeasure
code: /-- 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`. -/
def TendstoInMeasure [Dist E] {_ : MeasurableSpace α} (μ : Measure α) (f : ι → α → E) (l : Filter ι)
    (g : α → E) : Prop 
goal: {x : MeasurableSpace α} → MeasureTheory.Measure α → (ι → α → E) → Filter ι → (α → E) → Prop

full_name: MeasureTheory.tendstoInMeasure_iff_norm
code: theorem tendstoInMeasure_iff_norm [SeminormedAddCommGroup E] {l : Filter ι} {f : ι → α → E}
    {g : α → E} :
    TendstoInMeasure μ f l g ↔
      ∀ ε, 0 < ε → Tendsto (fun i => μ { x | ε ≤ ‖f i x - g x‖ }) l (𝓝 0) 
goal: MeasureTheory.TendstoInMeasure μ f l g ↔ ∀ (ε : ℝ), 0 < ε → Filter.Tendsto (fun i => μ {x | ε ≤ ‖f i x - g x‖}) l (𝓝 0)

full_name: MeasureTheory.TendstoInMeasure.congr'
code: protected theorem congr' (h_left : ∀ᶠ i in l, f i =ᵐ[μ] f' i) (h_right : g =ᵐ[μ] g')
    (h_tendsto : TendstoInMeasure μ f l g) : TendstoInMeasure μ f' l g' 
goal: MeasureTheory.TendstoInMeasure μ f' l g'

full_name: MeasureTheory.TendstoInMeasure.congr
code: protected theorem congr (h_left : ∀ i, f i =ᵐ[μ] f' i) (h_right : g =ᵐ[μ] g')
    (h_tendsto : TendstoInMeasure μ f l g) : TendstoInMeasure μ f' l g' 
goal: MeasureTheory.TendstoInMeasure μ f' l g'

full_name: MeasureTheory.TendstoInMeasure.congr_left
code: theorem congr_left (h : ∀ i, f i =ᵐ[μ] f' i) (h_tendsto : TendstoInMeasure μ f l g) :
    TendstoInMeasure μ f' l g 
goal: MeasureTheory.TendstoInMeasure μ f' l g

full_name: MeasureTheory.TendstoInMeasure.congr_right
code: theorem congr_right (h : g =ᵐ[μ] g') (h_tendsto : TendstoInMeasure μ f l g) :
    TendstoInMeasure μ f l g' 
goal: MeasureTheory.TendstoInMeasure μ f l g'

full_name: MeasureTheory.tendstoInMeasure_of_tendsto_ae_of_stronglyMeasurable
code: /-- Auxiliary lemma for `tendstoInMeasure_of_tendsto_ae`. -/
theorem tendstoInMeasure_of_tendsto_ae_of_stronglyMeasurable [IsFiniteMeasure μ]
    (hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g)
    (hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) : TendstoInMeasure μ f atTop g 
goal: MeasureTheory.TendstoInMeasure μ f Filter.atTop g

full_name: MeasureTheory.tendstoInMeasure_of_tendsto_ae
code: /-- Convergence a.e. implies convergence in measure in a finite measure space. -/
theorem tendstoInMeasure_of_tendsto_ae [IsFiniteMeasure μ] (hf : ∀ n, AEStronglyMeasurable (f n) μ)
    (hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) : TendstoInMeasure μ f atTop g 
goal: MeasureTheory.TendstoInMeasure μ f Filter.atTop g

full_name: MeasureTheory.ExistsSeqTendstoAe.exists_nat_measure_lt_two_inv
code: theorem exists_nat_measure_lt_two_inv (hfg : TendstoInMeasure μ f atTop g) (n : ℕ) :
    ∃ N, ∀ m ≥ N, μ { x | (2 : ℝ)⁻¹ ^ n ≤ dist (f m x) (g x) } ≤ (2⁻¹ : ℝ≥0∞) ^ n 
goal: ∃ N, ∀ m_1 ≥ N, μ {x | 2⁻¹ ^ n ≤ dist (f m_1 x) (g x)} ≤ 2⁻¹ ^ n

full_name: MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeqAux
code: /-- 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`. -/
noncomputable def seqTendstoAeSeqAux (hfg : TendstoInMeasure μ f atTop g) (n : ℕ) 
goal: ℕ

full_name: MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq
code: /-- Transformation of `seqTendstoAeSeqAux` to makes sure it is strictly monotone. -/
noncomputable def seqTendstoAeSeq (hfg : TendstoInMeasure μ f atTop g) : ℕ → ℕ
  
goal: ℕ → ℕ

full_name: MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_succ
code: theorem seqTendstoAeSeq_succ (hfg : TendstoInMeasure μ f atTop g) {n : ℕ} :
    seqTendstoAeSeq hfg (n + 1) =
      max (seqTendstoAeSeqAux hfg (n + 1)) (seqTendstoAeSeq hfg n + 1) 
goal: MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq hfg (n + 1) =
  max (MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeqAux hfg (n + 1))
    (MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq hfg n + 1)

full_name: MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_spec
code: theorem seqTendstoAeSeq_spec (hfg : TendstoInMeasure μ f atTop g) (n k : ℕ)
    (hn : seqTendstoAeSeq hfg n ≤ k) :
    μ { x | (2 : ℝ)⁻¹ ^ n ≤ dist (f k x) (g x) } ≤ (2 : ℝ≥0∞)⁻¹ ^ n 
goal: μ {x | 2⁻¹ ^ n ≤ dist (f k x) (g x)} ≤ 2⁻¹ ^ n

full_name: MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_strictMono
code: theorem seqTendstoAeSeq_strictMono (hfg : TendstoInMeasure μ f atTop g) :
    StrictMono (seqTendstoAeSeq hfg) 
goal: StrictMono (MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq hfg)

full_name: MeasureTheory.TendstoInMeasure.exists_seq_tendsto_ae
code: /-- 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 TendstoInMeasure.exists_seq_tendsto_ae (hfg : TendstoInMeasure μ f atTop g) :
    ∃ ns : ℕ → ℕ, StrictMono ns ∧ ∀ᵐ x ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x)) 
goal: ∃ ns, StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun i => f (ns i) x) Filter.atTop (𝓝 (g x))

full_name: MeasureTheory.TendstoInMeasure.exists_seq_tendstoInMeasure_atTop
code: theorem TendstoInMeasure.exists_seq_tendstoInMeasure_atTop {u : Filter ι} [NeBot u]
    [IsCountablyGenerated u] {f : ι → α → E} {g : α → E} (hfg : TendstoInMeasure μ f u g) :
    ∃ ns : ℕ → ι, TendstoInMeasure μ (fun n => f (ns n)) atTop g 
goal: ∃ ns, MeasureTheory.TendstoInMeasure μ (fun n => f (ns n)) Filter.atTop g

full_name: MeasureTheory.TendstoInMeasure.exists_seq_tendsto_ae'
code: theorem TendstoInMeasure.exists_seq_tendsto_ae' {u : Filter ι} [NeBot u] [IsCountablyGenerated u]
    {f : ι → α → E} {g : α → E} (hfg : TendstoInMeasure μ f u g) :
    ∃ ns : ℕ → ι, ∀ᵐ x ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x)) 
goal: ∃ ns, ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun i => f (ns i) x) Filter.atTop (𝓝 (g x))

full_name: MeasureTheory.TendstoInMeasure.aemeasurable
code: theorem TendstoInMeasure.aemeasurable {u : Filter ι} [NeBot u] [IsCountablyGenerated u]
    {f : ι → α → E} {g : α → E} (hf : ∀ n, AEMeasurable (f n) μ)
    (h_tendsto : TendstoInMeasure μ f u g) : AEMeasurable g μ 
goal: AEMeasurable g μ

full_name: MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm_of_stronglyMeasurable
code: /-- This lemma is superceded by `MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm` where we
allow `p = ∞` and only require `AEStronglyMeasurable`. -/
theorem tendstoInMeasure_of_tendsto_eLpNorm_of_stronglyMeasurable (hp_ne_zero : p ≠ 0)
    (hp_ne_top : p ≠ ∞) (hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g)
    {l : Filter ι} (hfg : Tendsto (fun n => eLpNorm (f n - g) p μ) l (𝓝 0)) :
    TendstoInMeasure μ f l g 
goal: MeasureTheory.TendstoInMeasure μ f l g

full_name: MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm_of_ne_top
code: /-- This lemma is superceded by `MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm` where we
allow `p = ∞`. -/
theorem tendstoInMeasure_of_tendsto_eLpNorm_of_ne_top (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hg : AEStronglyMeasurable g μ) {l : Filter ι}
    (hfg : Tendsto (fun n => eLpNorm (f n - g) p μ) l (𝓝 0)) : TendstoInMeasure μ f l g 
goal: MeasureTheory.TendstoInMeasure μ f l g

full_name: MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm_top
code: /-- See also `MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm` which work for general
Lp-convergence for all `p ≠ 0`. -/
theorem tendstoInMeasure_of_tendsto_eLpNorm_top {E} [NormedAddCommGroup E] {f : ι → α → E}
    {g : α → E} {l : Filter ι} (hfg : Tendsto (fun n => eLpNorm (f n - g) ∞ μ) l (𝓝 0)) :
    TendstoInMeasure μ f l g 
goal: MeasureTheory.TendstoInMeasure μ f l g

full_name: MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm
code: /-- Convergence in Lp implies convergence in measure. -/
theorem tendstoInMeasure_of_tendsto_eLpNorm {l : Filter ι} (hp_ne_zero : p ≠ 0)
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hg : AEStronglyMeasurable g μ)
    (hfg : Tendsto (fun n => eLpNorm (f n - g) p μ) l (𝓝 0)) : TendstoInMeasure μ f l g 
goal: MeasureTheory.TendstoInMeasure μ f l g

full_name: MeasureTheory.tendstoInMeasure_of_tendsto_Lp
code: /-- Convergence in Lp implies convergence in measure. -/
theorem tendstoInMeasure_of_tendsto_Lp [hp : Fact (1 ≤ p)] {f : ι → Lp E p μ} {g : Lp E p μ}
    {l : Filter ι} (hfg : Tendsto f l (𝓝 g)) : TendstoInMeasure μ (fun n => f n) l g 
goal: MeasureTheory.TendstoInMeasure μ (fun n => ↑↑(f n)) l ↑↑g