File: Mathlib/MeasureTheory/Function/UniformIntegrable.lean

full_name: MeasureTheory.UnifIntegrable
code: /-- Uniform integrability in the measure theory sense.

A sequence of functions `f` is said to be uniformly integrable if for all `ε > 0`, there exists
some `δ > 0` such that for all sets `s` with measure less than `δ`, the Lp-norm of `f i`
restricted on `s` is less than `ε`.

Uniform integrability is also known as uniformly absolutely continuous integrals. -/
def UnifIntegrable {_ : MeasurableSpace α} (f : ι → α → β) (p : ℝ≥0∞) (μ : Measure α) : Prop 
goal: {x : MeasurableSpace α} → (ι → α → β) → ℝ≥0∞ → MeasureTheory.Measure α → Prop

full_name: MeasureTheory.UniformIntegrable
code: /-- In probability theory, a family of measurable functions is uniformly integrable if it is
uniformly integrable in the measure theory sense and is uniformly bounded. -/
def UniformIntegrable {_ : MeasurableSpace α} (f : ι → α → β) (p : ℝ≥0∞) (μ : Measure α) : Prop 
goal: {x : MeasurableSpace α} → (ι → α → β) → ℝ≥0∞ → MeasureTheory.Measure α → Prop

full_name: MeasureTheory.UniformIntegrable.aeStronglyMeasurable
code: protected theorem aeStronglyMeasurable {f : ι → α → β} {p : ℝ≥0∞} (hf : UniformIntegrable f p μ)
    (i : ι) : AEStronglyMeasurable (f i) μ 
goal: MeasureTheory.AEStronglyMeasurable (f i) μ

full_name: MeasureTheory.UniformIntegrable.unifIntegrable
code: protected theorem unifIntegrable {f : ι → α → β} {p : ℝ≥0∞} (hf : UniformIntegrable f p μ) :
    UnifIntegrable f p μ 
goal: MeasureTheory.UnifIntegrable f p μ

full_name: MeasureTheory.UniformIntegrable.memℒp
code: protected theorem memℒp {f : ι → α → β} {p : ℝ≥0∞} (hf : UniformIntegrable f p μ) (i : ι) :
    Memℒp (f i) p μ 
goal: MeasureTheory.Memℒp (f i) p μ

full_name: MeasureTheory.UnifIntegrable.add
code: protected theorem add (hf : UnifIntegrable f p μ) (hg : UnifIntegrable g p μ) (hp : 1 ≤ p)
    (hf_meas : ∀ i, AEStronglyMeasurable (f i) μ) (hg_meas : ∀ i, AEStronglyMeasurable (g i) μ) :
    UnifIntegrable (f + g) p μ 
goal: MeasureTheory.UnifIntegrable (f + g) p μ

full_name: MeasureTheory.UnifIntegrable.neg
code: protected theorem neg (hf : UnifIntegrable f p μ) : UnifIntegrable (-f) p μ 
goal: MeasureTheory.UnifIntegrable (-f) p μ

full_name: MeasureTheory.UnifIntegrable.sub
code: protected theorem sub (hf : UnifIntegrable f p μ) (hg : UnifIntegrable g p μ) (hp : 1 ≤ p)
    (hf_meas : ∀ i, AEStronglyMeasurable (f i) μ) (hg_meas : ∀ i, AEStronglyMeasurable (g i) μ) :
    UnifIntegrable (f - g) p μ 
goal: MeasureTheory.UnifIntegrable (f - g) p μ

full_name: MeasureTheory.UnifIntegrable.ae_eq
code: protected theorem ae_eq (hf : UnifIntegrable f p μ) (hfg : ∀ n, f n =ᵐ[μ] g n) :
    UnifIntegrable g p μ 
goal: MeasureTheory.UnifIntegrable g p μ

full_name: MeasureTheory.UnifIntegrable.indicator
code: /-- Uniform integrability is preserved by restriction of the functions to a set. -/
protected theorem indicator (hf : UnifIntegrable f p μ) (E : Set α) :
    UnifIntegrable (fun i => E.indicator (f i)) p μ 
goal: MeasureTheory.UnifIntegrable (fun i => E.indicator (f i)) p μ

full_name: MeasureTheory.UnifIntegrable.restrict
code: /-- Uniform integrability is preserved by restriction of the measure to a set. -/
protected theorem restrict (hf : UnifIntegrable f p μ) (E : Set α) :
    UnifIntegrable f p (μ.restrict E) 
goal: MeasureTheory.UnifIntegrable f p (μ.restrict E)

full_name: MeasureTheory.unifIntegrable_zero_meas
code: theorem unifIntegrable_zero_meas [MeasurableSpace α] {p : ℝ≥0∞} {f : ι → α → β} :
    UnifIntegrable f p (0 : Measure α) 
goal: MeasureTheory.UnifIntegrable f p 0

full_name: MeasureTheory.unifIntegrable_congr_ae
code: theorem unifIntegrable_congr_ae {p : ℝ≥0∞} {f g : ι → α → β} (hfg : ∀ n, f n =ᵐ[μ] g n) :
    UnifIntegrable f p μ ↔ UnifIntegrable g p μ 
goal: MeasureTheory.UnifIntegrable f p μ ↔ MeasureTheory.UnifIntegrable g p μ

full_name: MeasureTheory.tendsto_indicator_ge
code: theorem tendsto_indicator_ge (f : α → β) (x : α) :
    Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) 
goal: Filter.Tendsto (fun M => {x | ↑M ≤ ↑‖f x‖₊}.indicator f x) Filter.atTop (𝓝 0)

full_name: MeasureTheory.Memℒp.integral_indicator_norm_ge_le
code: /-- This lemma is weaker than `MeasureTheory.Memℒp.integral_indicator_norm_ge_nonneg_le`
as the latter provides `0 ≤ M` and does not require the measurability of `f`. -/
theorem Memℒp.integral_indicator_norm_ge_le (hf : Memℒp f 1 μ) (hmeas : StronglyMeasurable f)
    {ε : ℝ} (hε : 0 < ε) :
    ∃ M : ℝ, (∫⁻ x, ‖{ x | M ≤ ‖f x‖₊ }.indicator f x‖₊ ∂μ) ≤ ENNReal.ofReal ε 
goal: ∃ M, ∫⁻ (x : α), ↑‖{x | M ≤ ↑‖f x‖₊}.indicator f x‖₊ ∂μ ≤ ENNReal.ofReal ε

full_name: MeasureTheory.Memℒp.integral_indicator_norm_ge_nonneg_le_of_meas
code: /-- This lemma is superceded by `MeasureTheory.Memℒp.integral_indicator_norm_ge_nonneg_le`
which does not require measurability. -/
theorem Memℒp.integral_indicator_norm_ge_nonneg_le_of_meas (hf : Memℒp f 1 μ)
    (hmeas : StronglyMeasurable f) {ε : ℝ} (hε : 0 < ε) :
    ∃ M : ℝ, 0 ≤ M ∧ (∫⁻ x, ‖{ x | M ≤ ‖f x‖₊ }.indicator f x‖₊ ∂μ) ≤ ENNReal.ofReal ε 
goal: ∃ M, 0 ≤ M ∧ ∫⁻ (x : α), ↑‖{x | M ≤ ↑‖f x‖₊}.indicator f x‖₊ ∂μ ≤ ENNReal.ofReal ε

full_name: MeasureTheory.Memℒp.integral_indicator_norm_ge_nonneg_le
code: theorem Memℒp.integral_indicator_norm_ge_nonneg_le (hf : Memℒp f 1 μ) {ε : ℝ} (hε : 0 < ε) :
    ∃ M : ℝ, 0 ≤ M ∧ (∫⁻ x, ‖{ x | M ≤ ‖f x‖₊ }.indicator f x‖₊ ∂μ) ≤ ENNReal.ofReal ε 
goal: ∃ M, 0 ≤ M ∧ ∫⁻ (x : α), ↑‖{x | M ≤ ↑‖f x‖₊}.indicator f x‖₊ ∂μ ≤ ENNReal.ofReal ε

full_name: MeasureTheory.Memℒp.eLpNormEssSup_indicator_norm_ge_eq_zero
code: theorem Memℒp.eLpNormEssSup_indicator_norm_ge_eq_zero (hf : Memℒp f ∞ μ)
    (hmeas : StronglyMeasurable f) :
    ∃ M : ℝ, eLpNormEssSup ({ x | M ≤ ‖f x‖₊ }.indicator f) μ = 0 
goal: ∃ M, MeasureTheory.eLpNormEssSup ({x | M ≤ ↑‖f x‖₊}.indicator f) μ = 0

full_name: MeasureTheory.Memℒp.eLpNorm_indicator_norm_ge_le
code: theorem Memℒp.eLpNorm_indicator_norm_ge_le (hf : Memℒp f p μ) (hmeas : StronglyMeasurable f) {ε : ℝ}
    (hε : 0 < ε) : ∃ M : ℝ, eLpNorm ({ x | M ≤ ‖f x‖₊ }.indicator f) p μ ≤ ENNReal.ofReal ε 
goal: ∃ M, MeasureTheory.eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal ε

full_name: MeasureTheory.Memℒp.eLpNorm_indicator_norm_ge_pos_le
code: /-- This lemma implies that a single function is uniformly integrable (in the probability sense). -/
theorem Memℒp.eLpNorm_indicator_norm_ge_pos_le (hf : Memℒp f p μ) (hmeas : StronglyMeasurable f)
    {ε : ℝ} (hε : 0 < ε) :
    ∃ M : ℝ, 0 < M ∧ eLpNorm ({ x | M ≤ ‖f x‖₊ }.indicator f) p μ ≤ ENNReal.ofReal ε 
goal: ∃ M, 0 < M ∧ MeasureTheory.eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal ε

full_name: MeasureTheory.eLpNorm_indicator_le_of_bound
code: theorem eLpNorm_indicator_le_of_bound {f : α → β} (hp_top : p ≠ ∞) {ε : ℝ} (hε : 0 < ε) {M : ℝ}
    (hf : ∀ x, ‖f x‖ < M) :
    ∃ (δ : ℝ) (hδ : 0 < δ), ∀ s, MeasurableSet s →
      μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε 
goal: ∃ δ,
  ∃ (_ : 0 < δ),
    ∀ (s : Set α),
      MeasurableSet s → μ s ≤ ENNReal.ofReal δ → MeasureTheory.eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε

full_name: MeasureTheory.Memℒp.eLpNorm_indicator_le'
code: /-- Auxiliary lemma for `MeasureTheory.Memℒp.eLpNorm_indicator_le`. -/
theorem Memℒp.eLpNorm_indicator_le' (hp_one : 1 ≤ p) (hp_top : p ≠ ∞) (hf : Memℒp f p μ)
    (hmeas : StronglyMeasurable f) {ε : ℝ} (hε : 0 < ε) :
    ∃ (δ : ℝ) (hδ : 0 < δ), ∀ s, MeasurableSet s → μ s ≤ ENNReal.ofReal δ →
      eLpNorm (s.indicator f) p μ ≤ 2 * ENNReal.ofReal ε 
goal: ∃ δ,
  ∃ (_ : 0 < δ),
    ∀ (s : Set α),
      MeasurableSet s → μ s ≤ ENNReal.ofReal δ → MeasureTheory.eLpNorm (s.indicator f) p μ ≤ 2 * ENNReal.ofReal ε

full_name: MeasureTheory.Memℒp.eLpNorm_indicator_le_of_meas
code: /-- This lemma is superceded by `MeasureTheory.Memℒp.eLpNorm_indicator_le` which does not require
measurability on `f`. -/
theorem Memℒp.eLpNorm_indicator_le_of_meas (hp_one : 1 ≤ p) (hp_top : p ≠ ∞) (hf : Memℒp f p μ)
    (hmeas : StronglyMeasurable f) {ε : ℝ} (hε : 0 < ε) :
    ∃ (δ : ℝ) (hδ : 0 < δ), ∀ s, MeasurableSet s → μ s ≤ ENNReal.ofReal δ →
      eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε 
goal: ∃ δ,
  ∃ (_ : 0 < δ),
    ∀ (s : Set α),
      MeasurableSet s → μ s ≤ ENNReal.ofReal δ → MeasureTheory.eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε

full_name: MeasureTheory.Memℒp.eLpNorm_indicator_le
code: theorem Memℒp.eLpNorm_indicator_le (hp_one : 1 ≤ p) (hp_top : p ≠ ∞) (hf : Memℒp f p μ) {ε : ℝ}
    (hε : 0 < ε) :
    ∃ (δ : ℝ) (hδ : 0 < δ), ∀ s, MeasurableSet s → μ s ≤ ENNReal.ofReal δ →
      eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε 
goal: ∃ δ,
  ∃ (_ : 0 < δ),
    ∀ (s : Set α),
      MeasurableSet s → μ s ≤ ENNReal.ofReal δ → MeasureTheory.eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε

full_name: MeasureTheory.unifIntegrable_const
code: /-- A constant function is uniformly integrable. -/
theorem unifIntegrable_const {g : α → β} (hp : 1 ≤ p) (hp_ne_top : p ≠ ∞) (hg : Memℒp g p μ) :
    UnifIntegrable (fun _ : ι => g) p μ 
goal: MeasureTheory.UnifIntegrable (fun x => g) p μ

full_name: MeasureTheory.unifIntegrable_subsingleton
code: /-- A single function is uniformly integrable. -/
theorem unifIntegrable_subsingleton [Subsingleton ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ∞)
    {f : ι → α → β} (hf : ∀ i, Memℒp (f i) p μ) : UnifIntegrable f p μ 
goal: MeasureTheory.UnifIntegrable f p μ

full_name: MeasureTheory.unifIntegrable_fin
code: /-- This lemma is less general than `MeasureTheory.unifIntegrable_finite` which applies to
all sequences indexed by a finite type. -/
theorem unifIntegrable_fin (hp_one : 1 ≤ p) (hp_top : p ≠ ∞) {n : ℕ} {f : Fin n → α → β}
    (hf : ∀ i, Memℒp (f i) p μ) : UnifIntegrable f p μ 
goal: MeasureTheory.UnifIntegrable f p μ

full_name: MeasureTheory.unifIntegrable_finite
code: /-- A finite sequence of Lp functions is uniformly integrable. -/
theorem unifIntegrable_finite [Finite ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ∞) {f : ι → α → β}
    (hf : ∀ i, Memℒp (f i) p μ) : UnifIntegrable f p μ 
goal: MeasureTheory.UnifIntegrable f p μ

full_name: MeasureTheory.eLpNorm_sub_le_of_dist_bdd
code: theorem eLpNorm_sub_le_of_dist_bdd (μ : Measure α)
    {p : ℝ≥0∞} (hp' : p ≠ ∞) {s : Set α} (hs : MeasurableSet[m] s)
    {f g : α → β} {c : ℝ} (hc : 0 ≤ c) (hf : ∀ x ∈ s, dist (f x) (g x) ≤ c) :
    eLpNorm (s.indicator (f - g)) p μ ≤ ENNReal.ofReal c * μ s ^ (1 / p.toReal) 
goal: MeasureTheory.eLpNorm (s.indicator (f - g)) p μ ≤ ENNReal.ofReal c * μ s ^ (1 / p.toReal)

full_name: MeasureTheory.tendsto_Lp_finite_of_tendsto_ae_of_meas
code: /-- A sequence of uniformly integrable functions which converges μ-a.e. converges in Lp. -/
theorem tendsto_Lp_finite_of_tendsto_ae_of_meas [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
    {f : ℕ → α → β} {g : α → β} (hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g)
    (hg' : Memℒp g p μ) (hui : UnifIntegrable f p μ)
    (hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
    Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0) 
goal: Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) Filter.atTop (𝓝 0)

full_name: MeasureTheory.tendsto_Lp_finite_of_tendsto_ae
code: /-- A sequence of uniformly integrable functions which converges μ-a.e. converges in Lp. -/
theorem tendsto_Lp_finite_of_tendsto_ae [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
    {f : ℕ → α → β} {g : α → β} (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hg : Memℒp g p μ)
    (hui : UnifIntegrable f p μ) (hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
    Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0) 
goal: Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) Filter.atTop (𝓝 0)

full_name: MeasureTheory.unifIntegrable_of_tendsto_Lp_zero
code: theorem unifIntegrable_of_tendsto_Lp_zero (hp : 1 ≤ p) (hp' : p ≠ ∞) (hf : ∀ n, Memℒp (f n) p μ)
    (hf_tendsto : Tendsto (fun n => eLpNorm (f n) p μ) atTop (𝓝 0)) : UnifIntegrable f p μ 
goal: MeasureTheory.UnifIntegrable f p μ

full_name: MeasureTheory.unifIntegrable_of_tendsto_Lp
code: /-- Convergence in Lp implies uniform integrability. -/
theorem unifIntegrable_of_tendsto_Lp (hp : 1 ≤ p) (hp' : p ≠ ∞) (hf : ∀ n, Memℒp (f n) p μ)
    (hg : Memℒp g p μ) (hfg : Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)) :
    UnifIntegrable f p μ 
goal: MeasureTheory.UnifIntegrable f p μ

full_name: MeasureTheory.tendsto_Lp_finite_of_tendstoInMeasure
code: /-- Forward direction of Vitali's convergence theorem: if `f` is a sequence of uniformly integrable
functions that converge in measure to some function `g` in a finite measure space, then `f`
converge in Lp to `g`. -/
theorem tendsto_Lp_finite_of_tendstoInMeasure [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hg : Memℒp g p μ) (hui : UnifIntegrable f p μ)
    (hfg : TendstoInMeasure μ f atTop g) : Tendsto (fun n ↦ eLpNorm (f n - g) p μ) atTop (𝓝 0) 
goal: Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) Filter.atTop (𝓝 0)

full_name: MeasureTheory.tendstoInMeasure_iff_tendsto_Lp_finite
code: /-- **Vitali's convergence theorem**: A sequence of functions `f` converges to `g` in Lp if and
only if it is uniformly integrable and converges to `g` in measure. -/
theorem tendstoInMeasure_iff_tendsto_Lp_finite [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
    (hf : ∀ n, Memℒp (f n) p μ) (hg : Memℒp g p μ) :
    TendstoInMeasure μ f atTop g ∧ UnifIntegrable f p μ ↔
      Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0) 
goal: MeasureTheory.TendstoInMeasure μ f Filter.atTop g ∧ MeasureTheory.UnifIntegrable f p μ ↔
  Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) Filter.atTop (𝓝 0)

full_name: MeasureTheory.unifIntegrable_of'
code: /-- This lemma is superceded by `unifIntegrable_of` which do not require `C` to be positive. -/
theorem unifIntegrable_of' (hp : 1 ≤ p) (hp' : p ≠ ∞) {f : ι → α → β}
    (hf : ∀ i, StronglyMeasurable (f i))
    (h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0, 0 < C ∧
      ∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε) :
    UnifIntegrable f p μ 
goal: MeasureTheory.UnifIntegrable f p μ

full_name: MeasureTheory.unifIntegrable_of
code: theorem unifIntegrable_of (hp : 1 ≤ p) (hp' : p ≠ ∞) {f : ι → α → β}
    (hf : ∀ i, AEStronglyMeasurable (f i) μ)
    (h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0,
      ∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε) :
    UnifIntegrable f p μ 
goal: MeasureTheory.UnifIntegrable f p μ

full_name: MeasureTheory.uniformIntegrable_zero_meas
code: theorem uniformIntegrable_zero_meas [MeasurableSpace α] : UniformIntegrable f p (0 : Measure α) 
goal: MeasureTheory.UniformIntegrable f p 0

full_name: MeasureTheory.UniformIntegrable.ae_eq
code: theorem UniformIntegrable.ae_eq {g : ι → α → β} (hf : UniformIntegrable f p μ)
    (hfg : ∀ n, f n =ᵐ[μ] g n) : UniformIntegrable g p μ 
goal: MeasureTheory.UniformIntegrable g p μ

full_name: MeasureTheory.uniformIntegrable_congr_ae
code: theorem uniformIntegrable_congr_ae {g : ι → α → β} (hfg : ∀ n, f n =ᵐ[μ] g n) :
    UniformIntegrable f p μ ↔ UniformIntegrable g p μ 
goal: MeasureTheory.UniformIntegrable f p μ ↔ MeasureTheory.UniformIntegrable g p μ

full_name: MeasureTheory.uniformIntegrable_finite
code: /-- A finite sequence of Lp functions is uniformly integrable in the probability sense. -/
theorem uniformIntegrable_finite [Finite ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ∞)
    (hf : ∀ i, Memℒp (f i) p μ) : UniformIntegrable f p μ 
goal: MeasureTheory.UniformIntegrable f p μ

full_name: MeasureTheory.uniformIntegrable_subsingleton
code: /-- A single function is uniformly integrable in the probability sense. -/
theorem uniformIntegrable_subsingleton [Subsingleton ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ∞)
    (hf : ∀ i, Memℒp (f i) p μ) : UniformIntegrable f p μ 
goal: MeasureTheory.UniformIntegrable f p μ

full_name: MeasureTheory.uniformIntegrable_const
code: /-- A constant sequence of functions is uniformly integrable in the probability sense. -/
theorem uniformIntegrable_const {g : α → β} (hp : 1 ≤ p) (hp_ne_top : p ≠ ∞) (hg : Memℒp g p μ) :
    UniformIntegrable (fun _ : ι => g) p μ 
goal: MeasureTheory.UniformIntegrable (fun x => g) p μ

full_name: MeasureTheory.uniformIntegrable_of'
code: /-- This lemma is superceded by `uniformIntegrable_of` which only requires
`AEStronglyMeasurable`. -/
theorem uniformIntegrable_of' [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
    (hf : ∀ i, StronglyMeasurable (f i))
    (h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0,
      ∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε) :
    UniformIntegrable f p μ 
goal: MeasureTheory.UniformIntegrable f p μ

full_name: MeasureTheory.uniformIntegrable_of
code: /-- A sequence of functions `(fₙ)` is uniformly integrable in the probability sense if for all
`ε > 0`, there exists some `C` such that `∫ x in {|fₙ| ≥ C}, fₙ x ∂μ ≤ ε` for all `n`. -/
theorem uniformIntegrable_of [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
    (hf : ∀ i, AEStronglyMeasurable (f i) μ)
    (h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0,
      ∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε) :
    UniformIntegrable f p μ 
goal: MeasureTheory.UniformIntegrable f p μ

full_name: MeasureTheory.UniformIntegrable.spec'
code: /-- This lemma is superceded by `UniformIntegrable.spec` which does not require measurability. -/
theorem UniformIntegrable.spec' (hp : p ≠ 0) (hp' : p ≠ ∞) (hf : ∀ i, StronglyMeasurable (f i))
    (hfu : UniformIntegrable f p μ) {ε : ℝ} (hε : 0 < ε) :
    ∃ C : ℝ≥0, ∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε 
goal: ∃ C, ∀ (i : ι), MeasureTheory.eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε

full_name: MeasureTheory.UniformIntegrable.spec
code: theorem UniformIntegrable.spec (hp : p ≠ 0) (hp' : p ≠ ∞) (hfu : UniformIntegrable f p μ) {ε : ℝ}
    (hε : 0 < ε) :
    ∃ C : ℝ≥0, ∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε 
goal: ∃ C, ∀ (i : ι), MeasureTheory.eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε

full_name: MeasureTheory.uniformIntegrable_iff
code: /-- The definition of uniform integrable in mathlib is equivalent to the definition commonly
found in literature. -/
theorem uniformIntegrable_iff [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞) :
    UniformIntegrable f p μ ↔
      (∀ i, AEStronglyMeasurable (f i) μ) ∧
        ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0,
          ∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε 
goal: MeasureTheory.UniformIntegrable f p μ ↔
  (∀ (i : ι), MeasureTheory.AEStronglyMeasurable (f i) μ) ∧
    ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), MeasureTheory.eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε

full_name: MeasureTheory.uniformIntegrable_average
code: /-- The averaging of a uniformly integrable sequence is also uniformly integrable. -/
theorem uniformIntegrable_average
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    (hp : 1 ≤ p) {f : ℕ → α → E} (hf : UniformIntegrable f p μ) :
    UniformIntegrable (fun (n : ℕ) => (n : ℝ)⁻¹ • (∑ i ∈ Finset.range n, f i)) p μ 
goal: MeasureTheory.UniformIntegrable (fun n => (↑n)⁻¹ • ∑ i ∈ Finset.range n, f i) p μ

full_name: MeasureTheory.uniformIntegrable_average_real
code: /-- The averaging of a uniformly integrable real-valued sequence is also uniformly integrable. -/
theorem uniformIntegrable_average_real (hp : 1 ≤ p) {f : ℕ → α → ℝ} (hf : UniformIntegrable f p μ) :
    UniformIntegrable (fun n => (∑ i ∈ Finset.range n, f i) / (n : α → ℝ)) p μ 
goal: MeasureTheory.UniformIntegrable (fun n => (∑ i ∈ Finset.range n, f i) / ↑n) p μ