l3lab/ntp-mathlib huggingface dataset:
theorem tendsto_indicator_ge (f : α → β) (x : α) :
    Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) := by
  

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
x : α
⊢ Tendsto (fun M => indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)

Next tactic:
refine' tendsto_atTop_of_eventually_const (i₀ := Nat.ceil (‖f x‖₊ : ℝ) + 1) fun n hn => _
theorem tendsto_indicator_ge (f : α → β) (x : α) :
    Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) := by
  refine' tendsto_atTop_of_eventually_const (i₀ := Nat.ceil (‖f x‖₊ : ℝ) + 1) fun n hn => _
  

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
x : α
n : ℕ
hn : n ≥ ⌈↑‖f x‖₊⌉₊ + 1
⊢ indicator {x | ↑n ≤ ↑‖f x‖₊} f x = 0

Next tactic:
rw [Set.indicator_of_not_mem]
theorem tendsto_indicator_ge (f : α → β) (x : α) :
    Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) := by
  refine' tendsto_atTop_of_eventually_const (i₀ := Nat.ceil (‖f x‖₊ : ℝ) + 1) fun n hn => _
  rw [Set.indicator_of_not_mem]
  

Proof state:
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
x : α
n : ℕ
hn : n ≥ ⌈↑‖f x‖₊⌉₊ + 1
⊢ x ∉ {x | ↑n ≤ ↑‖f x‖₊}

Next tactic:
simp only [not_le, Set.mem_setOf_eq]
theorem tendsto_indicator_ge (f : α → β) (x : α) :
    Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) := by
  refine' tendsto_atTop_of_eventually_const (i₀ := Nat.ceil (‖f x‖₊ : ℝ) + 1) fun n hn => _
  rw [Set.indicator_of_not_mem]
  simp only [not_le, Set.mem_setOf_eq]
  

Proof state:
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
x : α
n : ℕ
hn : n ≥ ⌈↑‖f x‖₊⌉₊ + 1
⊢ ↑‖f x‖₊ < ↑n

Next tactic:
refine' lt_of_le_of_lt (Nat.le_ceil _) _
theorem tendsto_indicator_ge (f : α → β) (x : α) :
    Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) := by
  refine' tendsto_atTop_of_eventually_const (i₀ := Nat.ceil (‖f x‖₊ : ℝ) + 1) fun n hn => _
  rw [Set.indicator_of_not_mem]
  simp only [not_le, Set.mem_setOf_eq]
  refine' lt_of_le_of_lt (Nat.le_ceil _) _
  

Proof state:
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
x : α
n : ℕ
hn : n ≥ ⌈↑‖f x‖₊⌉₊ + 1
⊢ ↑⌈↑‖f x‖₊⌉₊ < ↑n

Next tactic:
refine' lt_of_lt_of_le (lt_add_one _) _
theorem tendsto_indicator_ge (f : α → β) (x : α) :
    Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) := by
  refine' tendsto_atTop_of_eventually_const (i₀ := Nat.ceil (‖f x‖₊ : ℝ) + 1) fun n hn => _
  rw [Set.indicator_of_not_mem]
  simp only [not_le, Set.mem_setOf_eq]
  refine' lt_of_le_of_lt (Nat.le_ceil _) _
  refine' lt_of_lt_of_le (lt_add_one _) _
  

Proof state:
case h
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
x : α
n : ℕ
hn : n ≥ ⌈↑‖f x‖₊⌉₊ + 1
⊢ ↑⌈↑‖f x‖₊⌉₊ + 1 ≤ ↑n

Next tactic:
norm_cast
/-- 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 := by
  

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
⊢ ∃ M, ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖₊ ∂μ ≤ ENNReal.ofReal ε

Next tactic:
have htendsto :
      ∀ᵐ x ∂μ, Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) :=
    univ_mem' (id fun x => tendsto_indicator_ge f x)
/-- 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 := by
  have htendsto :
      ∀ᵐ x ∂μ, Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) :=
    univ_mem' (id fun x => tendsto_indicator_ge f x)
  

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
⊢ ∃ M, ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖₊ ∂μ ≤ ENNReal.ofReal ε

Next tactic:
have hmeas : ∀ M : ℕ, AEStronglyMeasurable ({ x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f) μ := by
    intro M
    apply hf.1.indicator
    apply StronglyMeasurable.measurableSet_le stronglyMeasurable_const
      hmeas.nnnorm.measurable.coe_nnreal_real.stronglyMeasurable
/-- 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 := by
  have htendsto :
      ∀ᵐ x ∂μ, Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) :=
    univ_mem' (id fun x => tendsto_indicator_ge f x)
  have hmeas : ∀ M : ℕ, AEStronglyMeasurable ({ x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f) μ := by
    

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
⊢ ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ

Next tactic:
intro M
/-- 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 := by
  have htendsto :
      ∀ᵐ x ∂μ, Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) :=
    univ_mem' (id fun x => tendsto_indicator_ge f x)
  have hmeas : ∀ M : ℕ, AEStronglyMeasurable ({ x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f) μ := by
    intro M
    

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
M : ℕ
⊢ AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ

Next tactic:
apply hf.1.indicator
/-- 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 := by
  have htendsto :
      ∀ᵐ x ∂μ, Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) :=
    univ_mem' (id fun x => tendsto_indicator_ge f x)
  have hmeas : ∀ M : ℕ, AEStronglyMeasurable ({ x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f) μ := by
    intro M
    apply hf.1.indicator
    

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
M : ℕ
⊢ MeasurableSet {x | ↑M ≤ ↑‖f x‖₊}

Next tactic:
apply StronglyMeasurable.measurableSet_le stronglyMeasurable_const
      hmeas.nnnorm.measurable.coe_nnreal_real.stronglyMeasurable
/-- 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 := by
  have htendsto :
      ∀ᵐ x ∂μ, Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) :=
    univ_mem' (id fun x => tendsto_indicator_ge f x)
  have hmeas : ∀ M : ℕ, AEStronglyMeasurable ({ x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f) μ := by
    intro M
    apply hf.1.indicator
    apply StronglyMeasurable.measurableSet_le stronglyMeasurable_const
      hmeas.nnnorm.measurable.coe_nnreal_real.stronglyMeasurable
  

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
⊢ ∃ M, ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖₊ ∂μ ≤ ENNReal.ofReal ε

Next tactic:
have hbound : HasFiniteIntegral (fun x => ‖f x‖) μ := by
    rw [memℒp_one_iff_integrable] at hf
    exact hf.norm.2
/-- 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 := by
  have htendsto :
      ∀ᵐ x ∂μ, Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) :=
    univ_mem' (id fun x => tendsto_indicator_ge f x)
  have hmeas : ∀ M : ℕ, AEStronglyMeasurable ({ x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f) μ := by
    intro M
    apply hf.1.indicator
    apply StronglyMeasurable.measurableSet_le stronglyMeasurable_const
      hmeas.nnnorm.measurable.coe_nnreal_real.stronglyMeasurable
  have hbound : HasFiniteIntegral (fun x => ‖f x‖) μ := by
    

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
⊢ HasFiniteIntegral fun x => ‖f x‖

Next tactic:
rw [memℒp_one_iff_integrable] at hf
/-- 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 := by
  have htendsto :
      ∀ᵐ x ∂μ, Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) :=
    univ_mem' (id fun x => tendsto_indicator_ge f x)
  have hmeas : ∀ M : ℕ, AEStronglyMeasurable ({ x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f) μ := by
    intro M
    apply hf.1.indicator
    apply StronglyMeasurable.measurableSet_le stronglyMeasurable_const
      hmeas.nnnorm.measurable.coe_nnreal_real.stronglyMeasurable
  have hbound : HasFiniteIntegral (fun x => ‖f x‖) μ := by
    rw [memℒp_one_iff_integrable] at hf
    

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Integrable f
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
⊢ HasFiniteIntegral fun x => ‖f x‖

Next tactic:
exact hf.norm.2
/-- 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 := by
  have htendsto :
      ∀ᵐ x ∂μ, Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) :=
    univ_mem' (id fun x => tendsto_indicator_ge f x)
  have hmeas : ∀ M : ℕ, AEStronglyMeasurable ({ x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f) μ := by
    intro M
    apply hf.1.indicator
    apply StronglyMeasurable.measurableSet_le stronglyMeasurable_const
      hmeas.nnnorm.measurable.coe_nnreal_real.stronglyMeasurable
  have hbound : HasFiniteIntegral (fun x => ‖f x‖) μ := by
    rw [memℒp_one_iff_integrable] at hf
    exact hf.norm.2
  

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
⊢ ∃ M, ∫⁻ (x : α), ↑‖Set.indicator {x | M ≤ ↑‖f x‖₊} f x‖₊ ∂μ ≤ ENNReal.ofReal ε

Next tactic:
have : Tendsto (fun n : ℕ ↦ ∫⁻ a, ENNReal.ofReal ‖{ x | n ≤ ‖f x‖₊ }.indicator f a - 0‖ ∂μ)
      atTop (𝓝 0) := by
    refine' tendsto_lintegral_norm_of_dominated_convergence hmeas hbound _ htendsto
    refine' fun n => univ_mem' (id fun x => _)
    by_cases hx : (n : ℝ) ≤ ‖f x‖
    · dsimp
      rwa [Set.indicator_of_mem]
    · dsimp
      rw [Set.indicator_of_not_mem, norm_zero]
      · exact norm_nonneg _
      · assumption
/-- 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 := by
  have htendsto :
      ∀ᵐ x ∂μ, Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) :=
    univ_mem' (id fun x => tendsto_indicator_ge f x)
  have hmeas : ∀ M : ℕ, AEStronglyMeasurable ({ x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f) μ := by
    intro M
    apply hf.1.indicator
    apply StronglyMeasurable.measurableSet_le stronglyMeasurable_const
      hmeas.nnnorm.measurable.coe_nnreal_real.stronglyMeasurable
  have hbound : HasFiniteIntegral (fun x => ‖f x‖) μ := by
    rw [memℒp_one_iff_integrable] at hf
    exact hf.norm.2
  have : Tendsto (fun n : ℕ ↦ ∫⁻ a, ENNReal.ofReal ‖{ x | n ≤ ‖f x‖₊ }.indicator f a - 0‖ ∂μ)
      atTop (𝓝 0) := by
    

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
⊢ Tendsto (fun n => ∫⁻ (a : α), ENNReal.ofReal ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a - 0‖ ∂μ) atTop (𝓝 0)

Next tactic:
refine' tendsto_lintegral_norm_of_dominated_convergence hmeas hbound _ htendsto
/-- 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 := by
  have htendsto :
      ∀ᵐ x ∂μ, Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) :=
    univ_mem' (id fun x => tendsto_indicator_ge f x)
  have hmeas : ∀ M : ℕ, AEStronglyMeasurable ({ x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f) μ := by
    intro M
    apply hf.1.indicator
    apply StronglyMeasurable.measurableSet_le stronglyMeasurable_const
      hmeas.nnnorm.measurable.coe_nnreal_real.stronglyMeasurable
  have hbound : HasFiniteIntegral (fun x => ‖f x‖) μ := by
    rw [memℒp_one_iff_integrable] at hf
    exact hf.norm.2
  have : Tendsto (fun n : ℕ ↦ ∫⁻ a, ENNReal.ofReal ‖{ x | n ≤ ‖f x‖₊ }.indicator f a - 0‖ ∂μ)
      atTop (𝓝 0) := by
    refine' tendsto_lintegral_norm_of_dominated_convergence hmeas hbound _ htendsto
    

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
⊢ ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a‖ ≤ ‖f a‖

Next tactic:
refine' fun n => univ_mem' (id fun x => _)
/-- 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 := by
  have htendsto :
      ∀ᵐ x ∂μ, Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) :=
    univ_mem' (id fun x => tendsto_indicator_ge f x)
  have hmeas : ∀ M : ℕ, AEStronglyMeasurable ({ x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f) μ := by
    intro M
    apply hf.1.indicator
    apply StronglyMeasurable.measurableSet_le stronglyMeasurable_const
      hmeas.nnnorm.measurable.coe_nnreal_real.stronglyMeasurable
  have hbound : HasFiniteIntegral (fun x => ‖f x‖) μ := by
    rw [memℒp_one_iff_integrable] at hf
    exact hf.norm.2
  have : Tendsto (fun n : ℕ ↦ ∫⁻ a, ENNReal.ofReal ‖{ x | n ≤ ‖f x‖₊ }.indicator f a - 0‖ ∂μ)
      atTop (𝓝 0) := by
    refine' tendsto_lintegral_norm_of_dominated_convergence hmeas hbound _ htendsto
    refine' fun n => univ_mem' (id fun x => _)
    

Proof state:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
n : ℕ
x : α
⊢ x ∈ {x | (fun a => ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a‖ ≤ ‖f a‖) x}

Next tactic:
by_cases hx : (n : ℝ) ≤ ‖f x‖
/-- 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 := by
  have htendsto :
      ∀ᵐ x ∂μ, Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) :=
    univ_mem' (id fun x => tendsto_indicator_ge f x)
  have hmeas : ∀ M : ℕ, AEStronglyMeasurable ({ x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f) μ := by
    intro M
    apply hf.1.indicator
    apply StronglyMeasurable.measurableSet_le stronglyMeasurable_const
      hmeas.nnnorm.measurable.coe_nnreal_real.stronglyMeasurable
  have hbound : HasFiniteIntegral (fun x => ‖f x‖) μ := by
    rw [memℒp_one_iff_integrable] at hf
    exact hf.norm.2
  have : Tendsto (fun n : ℕ ↦ ∫⁻ a, ENNReal.ofReal ‖{ x | n ≤ ‖f x‖₊ }.indicator f a - 0‖ ∂μ)
      atTop (𝓝 0) := by
    refine' tendsto_lintegral_norm_of_dominated_convergence hmeas hbound _ htendsto
    refine' fun n => univ_mem' (id fun x => _)
    by_cases hx : (n : ℝ) ≤ ‖f x‖
    · 

Proof state:
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
n : ℕ
x : α
hx : ↑n ≤ ‖f x‖
⊢ x ∈ {x | (fun a => ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f a‖ ≤ ‖f a‖) x}

Next tactic:
dsimp
/-- 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 := by
  have htendsto :
      ∀ᵐ x ∂μ, Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) :=
    univ_mem' (id fun x => tendsto_indicator_ge f x)
  have hmeas : ∀ M : ℕ, AEStronglyMeasurable ({ x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f) μ := by
    intro M
    apply hf.1.indicator
    apply StronglyMeasurable.measurableSet_le stronglyMeasurable_const
      hmeas.nnnorm.measurable.coe_nnreal_real.stronglyMeasurable
  have hbound : HasFiniteIntegral (fun x => ‖f x‖) μ := by
    rw [memℒp_one_iff_integrable] at hf
    exact hf.norm.2
  have : Tendsto (fun n : ℕ ↦ ∫⁻ a, ENNReal.ofReal ‖{ x | n ≤ ‖f x‖₊ }.indicator f a - 0‖ ∂μ)
      atTop (𝓝 0) := by
    refine' tendsto_lintegral_norm_of_dominated_convergence hmeas hbound _ htendsto
    refine' fun n => univ_mem' (id fun x => _)
    by_cases hx : (n : ℝ) ≤ ‖f x‖
    · dsimp
      

Proof state:
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f 1
hmeas✝ : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f x) atTop (𝓝 0)
hmeas : ∀ (M : ℕ), AEStronglyMeasurable (Set.indicator {x | ↑M ≤ ↑‖f x‖₊} f) μ
hbound : HasFiniteIntegral fun x => ‖f x‖
n : ℕ
x : α
hx : ↑n ≤ ‖f x‖
⊢ ‖Set.indicator {x | ↑n ≤ ‖f x‖₊} f x‖ ≤ ‖f x‖

Next tactic:
rwa [Set.indicator_of_mem]