l3lab/ntp-mathlib huggingface dataset:
/-- Riemann-Lebesgue Lemma for continuous and compactly-supported functions: the integral
`∫ v, exp (-2 * π * ⟪w, v⟫ * I) • f v` tends to 0 wrt `cocompact V`. Note that this is primarily
of interest as a preparatory step for the more general result
`tendsto_integral_exp_inner_smul_cocompact` in which `f` can be arbitrary. -/
theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support (hf1 : Continuous f)
    (hf2 : HasCompactSupport f) :
    Tendsto (fun w : V => ∫ v : V, e[-⟪v, w⟫] • f v) (cocompact V) (𝓝 0) := by
  refine' NormedAddCommGroup.tendsto_nhds_zero.mpr fun ε hε => _
  suffices ∃ T : ℝ, ∀ w : V, T ≤ ‖w‖ → ‖∫ v : V, e[-⟪v, w⟫] • f v‖ < ε by
    simp_rw [← comap_dist_left_atTop_eq_cocompact (0 : V), eventually_comap, eventually_atTop,
      dist_eq_norm', sub_zero]
    exact
      let ⟨T, hT⟩ := this
      ⟨T, fun b hb v hv => hT v (hv.symm ▸ hb)⟩
  obtain ⟨R, -, hR_bd⟩ : ∃ R : ℝ, 0 < R ∧ ∀ x : V, R ≤ ‖x‖ → f x = 0
  exact hf2.exists_pos_le_norm
  let A := {v : V | ‖v‖ ≤ R + 1}
  

Proof state:
case intro.intro
E : Type u_1
V : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
f : V → E
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : FiniteDimensional ℝ V
inst✝ : CompleteSpace E
hf1 : Continuous f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
⊢ ∃ T, ∀ (w : V), T ≤ ‖w‖ → ‖∫ (v : V), ↑(e (Multiplicative.ofAdd (-inner v w))) • f v‖ < ε

Next tactic:
have mA : MeasurableSet A := by
    suffices A = Metric.closedBall (0 : V) (R + 1) by
      rw [this]
      exact Metric.isClosed_ball.measurableSet
    simp_rw [Metric.closedBall, dist_eq_norm, sub_zero]
/-- Riemann-Lebesgue Lemma for continuous and compactly-supported functions: the integral
`∫ v, exp (-2 * π * ⟪w, v⟫ * I) • f v` tends to 0 wrt `cocompact V`. Note that this is primarily
of interest as a preparatory step for the more general result
`tendsto_integral_exp_inner_smul_cocompact` in which `f` can be arbitrary. -/
theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support (hf1 : Continuous f)
    (hf2 : HasCompactSupport f) :
    Tendsto (fun w : V => ∫ v : V, e[-⟪v, w⟫] • f v) (cocompact V) (𝓝 0) := by
  refine' NormedAddCommGroup.tendsto_nhds_zero.mpr fun ε hε => _
  suffices ∃ T : ℝ, ∀ w : V, T ≤ ‖w‖ → ‖∫ v : V, e[-⟪v, w⟫] • f v‖ < ε by
    simp_rw [← comap_dist_left_atTop_eq_cocompact (0 : V), eventually_comap, eventually_atTop,
      dist_eq_norm', sub_zero]
    exact
      let ⟨T, hT⟩ := this
      ⟨T, fun b hb v hv => hT v (hv.symm ▸ hb)⟩
  obtain ⟨R, -, hR_bd⟩ : ∃ R : ℝ, 0 < R ∧ ∀ x : V, R ≤ ‖x‖ → f x = 0
  exact hf2.exists_pos_le_norm
  let A := {v : V | ‖v‖ ≤ R + 1}
  have mA : MeasurableSet A := by
    

Proof state:
E : Type u_1
V : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
f : V → E
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : FiniteDimensional ℝ V
inst✝ : CompleteSpace E
hf1 : Continuous f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
⊢ MeasurableSet A

Next tactic:
suffices A = Metric.closedBall (0 : V) (R + 1) by
      rw [this]
      exact Metric.isClosed_ball.measurableSet
/-- Riemann-Lebesgue Lemma for continuous and compactly-supported functions: the integral
`∫ v, exp (-2 * π * ⟪w, v⟫ * I) • f v` tends to 0 wrt `cocompact V`. Note that this is primarily
of interest as a preparatory step for the more general result
`tendsto_integral_exp_inner_smul_cocompact` in which `f` can be arbitrary. -/
theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support (hf1 : Continuous f)
    (hf2 : HasCompactSupport f) :
    Tendsto (fun w : V => ∫ v : V, e[-⟪v, w⟫] • f v) (cocompact V) (𝓝 0) := by
  refine' NormedAddCommGroup.tendsto_nhds_zero.mpr fun ε hε => _
  suffices ∃ T : ℝ, ∀ w : V, T ≤ ‖w‖ → ‖∫ v : V, e[-⟪v, w⟫] • f v‖ < ε by
    simp_rw [← comap_dist_left_atTop_eq_cocompact (0 : V), eventually_comap, eventually_atTop,
      dist_eq_norm', sub_zero]
    exact
      let ⟨T, hT⟩ := this
      ⟨T, fun b hb v hv => hT v (hv.symm ▸ hb)⟩
  obtain ⟨R, -, hR_bd⟩ : ∃ R : ℝ, 0 < R ∧ ∀ x : V, R ≤ ‖x‖ → f x = 0
  exact hf2.exists_pos_le_norm
  let A := {v : V | ‖v‖ ≤ R + 1}
  have mA : MeasurableSet A := by
    suffices A = Metric.closedBall (0 : V) (R + 1) by
      

Proof state:
E : Type u_1
V : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
f : V → E
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : FiniteDimensional ℝ V
inst✝ : CompleteSpace E
hf1 : Continuous f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
this : A = Metric.closedBall 0 (R + 1)
⊢ MeasurableSet A

Next tactic:
rw [this]
/-- Riemann-Lebesgue Lemma for continuous and compactly-supported functions: the integral
`∫ v, exp (-2 * π * ⟪w, v⟫ * I) • f v` tends to 0 wrt `cocompact V`. Note that this is primarily
of interest as a preparatory step for the more general result
`tendsto_integral_exp_inner_smul_cocompact` in which `f` can be arbitrary. -/
theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support (hf1 : Continuous f)
    (hf2 : HasCompactSupport f) :
    Tendsto (fun w : V => ∫ v : V, e[-⟪v, w⟫] • f v) (cocompact V) (𝓝 0) := by
  refine' NormedAddCommGroup.tendsto_nhds_zero.mpr fun ε hε => _
  suffices ∃ T : ℝ, ∀ w : V, T ≤ ‖w‖ → ‖∫ v : V, e[-⟪v, w⟫] • f v‖ < ε by
    simp_rw [← comap_dist_left_atTop_eq_cocompact (0 : V), eventually_comap, eventually_atTop,
      dist_eq_norm', sub_zero]
    exact
      let ⟨T, hT⟩ := this
      ⟨T, fun b hb v hv => hT v (hv.symm ▸ hb)⟩
  obtain ⟨R, -, hR_bd⟩ : ∃ R : ℝ, 0 < R ∧ ∀ x : V, R ≤ ‖x‖ → f x = 0
  exact hf2.exists_pos_le_norm
  let A := {v : V | ‖v‖ ≤ R + 1}
  have mA : MeasurableSet A := by
    suffices A = Metric.closedBall (0 : V) (R + 1) by
      rw [this]
      

Proof state:
E : Type u_1
V : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
f : V → E
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : FiniteDimensional ℝ V
inst✝ : CompleteSpace E
hf1 : Continuous f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
this : A = Metric.closedBall 0 (R + 1)
⊢ MeasurableSet (Metric.closedBall 0 (R + 1))

Next tactic:
exact Metric.isClosed_ball.measurableSet
/-- Riemann-Lebesgue Lemma for continuous and compactly-supported functions: the integral
`∫ v, exp (-2 * π * ⟪w, v⟫ * I) • f v` tends to 0 wrt `cocompact V`. Note that this is primarily
of interest as a preparatory step for the more general result
`tendsto_integral_exp_inner_smul_cocompact` in which `f` can be arbitrary. -/
theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support (hf1 : Continuous f)
    (hf2 : HasCompactSupport f) :
    Tendsto (fun w : V => ∫ v : V, e[-⟪v, w⟫] • f v) (cocompact V) (𝓝 0) := by
  refine' NormedAddCommGroup.tendsto_nhds_zero.mpr fun ε hε => _
  suffices ∃ T : ℝ, ∀ w : V, T ≤ ‖w‖ → ‖∫ v : V, e[-⟪v, w⟫] • f v‖ < ε by
    simp_rw [← comap_dist_left_atTop_eq_cocompact (0 : V), eventually_comap, eventually_atTop,
      dist_eq_norm', sub_zero]
    exact
      let ⟨T, hT⟩ := this
      ⟨T, fun b hb v hv => hT v (hv.symm ▸ hb)⟩
  obtain ⟨R, -, hR_bd⟩ : ∃ R : ℝ, 0 < R ∧ ∀ x : V, R ≤ ‖x‖ → f x = 0
  exact hf2.exists_pos_le_norm
  let A := {v : V | ‖v‖ ≤ R + 1}
  have mA : MeasurableSet A := by
    suffices A = Metric.closedBall (0 : V) (R + 1) by
      rw [this]
      exact Metric.isClosed_ball.measurableSet
    

Proof state:
E : Type u_1
V : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
f : V → E
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : FiniteDimensional ℝ V
inst✝ : CompleteSpace E
hf1 : Continuous f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
⊢ A = Metric.closedBall 0 (R + 1)

Next tactic:
simp_rw [Metric.closedBall, dist_eq_norm, sub_zero]
/-- Riemann-Lebesgue Lemma for continuous and compactly-supported functions: the integral
`∫ v, exp (-2 * π * ⟪w, v⟫ * I) • f v` tends to 0 wrt `cocompact V`. Note that this is primarily
of interest as a preparatory step for the more general result
`tendsto_integral_exp_inner_smul_cocompact` in which `f` can be arbitrary. -/
theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support (hf1 : Continuous f)
    (hf2 : HasCompactSupport f) :
    Tendsto (fun w : V => ∫ v : V, e[-⟪v, w⟫] • f v) (cocompact V) (𝓝 0) := by
  refine' NormedAddCommGroup.tendsto_nhds_zero.mpr fun ε hε => _
  suffices ∃ T : ℝ, ∀ w : V, T ≤ ‖w‖ → ‖∫ v : V, e[-⟪v, w⟫] • f v‖ < ε by
    simp_rw [← comap_dist_left_atTop_eq_cocompact (0 : V), eventually_comap, eventually_atTop,
      dist_eq_norm', sub_zero]
    exact
      let ⟨T, hT⟩ := this
      ⟨T, fun b hb v hv => hT v (hv.symm ▸ hb)⟩
  obtain ⟨R, -, hR_bd⟩ : ∃ R : ℝ, 0 < R ∧ ∀ x : V, R ≤ ‖x‖ → f x = 0
  exact hf2.exists_pos_le_norm
  let A := {v : V | ‖v‖ ≤ R + 1}
  have mA : MeasurableSet A := by
    suffices A = Metric.closedBall (0 : V) (R + 1) by
      rw [this]
      exact Metric.isClosed_ball.measurableSet
    simp_rw [Metric.closedBall, dist_eq_norm, sub_zero]
  

Proof state:
case intro.intro
E : Type u_1
V : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
f : V → E
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : FiniteDimensional ℝ V
inst✝ : CompleteSpace E
hf1 : Continuous f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
mA : MeasurableSet A
⊢ ∃ T, ∀ (w : V), T ≤ ‖w‖ → ‖∫ (v : V), ↑(e (Multiplicative.ofAdd (-inner v w))) • f v‖ < ε

Next tactic:
obtain ⟨B, hB_pos, hB_vol⟩ : ∃ B : ℝ≥0, 0 < B ∧ volume A ≤ B := by
    have hc : IsCompact A := by
      simpa only [Metric.closedBall, dist_eq_norm, sub_zero] using isCompact_closedBall (0 : V) _
    let B₀ := volume A
    replace hc : B₀ < ⊤ := hc.measure_lt_top
    refine' ⟨B₀.toNNReal + 1, add_pos_of_nonneg_of_pos B₀.toNNReal.coe_nonneg one_pos, _⟩
    rw [ENNReal.coe_add, ENNReal.coe_one, ENNReal.coe_toNNReal hc.ne]
    exact le_self_add
/-- Riemann-Lebesgue Lemma for continuous and compactly-supported functions: the integral
`∫ v, exp (-2 * π * ⟪w, v⟫ * I) • f v` tends to 0 wrt `cocompact V`. Note that this is primarily
of interest as a preparatory step for the more general result
`tendsto_integral_exp_inner_smul_cocompact` in which `f` can be arbitrary. -/
theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support (hf1 : Continuous f)
    (hf2 : HasCompactSupport f) :
    Tendsto (fun w : V => ∫ v : V, e[-⟪v, w⟫] • f v) (cocompact V) (𝓝 0) := by
  refine' NormedAddCommGroup.tendsto_nhds_zero.mpr fun ε hε => _
  suffices ∃ T : ℝ, ∀ w : V, T ≤ ‖w‖ → ‖∫ v : V, e[-⟪v, w⟫] • f v‖ < ε by
    simp_rw [← comap_dist_left_atTop_eq_cocompact (0 : V), eventually_comap, eventually_atTop,
      dist_eq_norm', sub_zero]
    exact
      let ⟨T, hT⟩ := this
      ⟨T, fun b hb v hv => hT v (hv.symm ▸ hb)⟩
  obtain ⟨R, -, hR_bd⟩ : ∃ R : ℝ, 0 < R ∧ ∀ x : V, R ≤ ‖x‖ → f x = 0
  exact hf2.exists_pos_le_norm
  let A := {v : V | ‖v‖ ≤ R + 1}
  have mA : MeasurableSet A := by
    suffices A = Metric.closedBall (0 : V) (R + 1) by
      rw [this]
      exact Metric.isClosed_ball.measurableSet
    simp_rw [Metric.closedBall, dist_eq_norm, sub_zero]
  obtain ⟨B, hB_pos, hB_vol⟩ : ∃ B : ℝ≥0, 0 < B ∧ volume A ≤ B := by
    

Proof state:
E : Type u_1
V : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
f : V → E
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : FiniteDimensional ℝ V
inst✝ : CompleteSpace E
hf1 : Continuous f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
mA : MeasurableSet A
⊢ ∃ B, 0 < B ∧ ↑↑volume A ≤ ↑B

Next tactic:
have hc : IsCompact A := by
      simpa only [Metric.closedBall, dist_eq_norm, sub_zero] using isCompact_closedBall (0 : V) _
/-- Riemann-Lebesgue Lemma for continuous and compactly-supported functions: the integral
`∫ v, exp (-2 * π * ⟪w, v⟫ * I) • f v` tends to 0 wrt `cocompact V`. Note that this is primarily
of interest as a preparatory step for the more general result
`tendsto_integral_exp_inner_smul_cocompact` in which `f` can be arbitrary. -/
theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support (hf1 : Continuous f)
    (hf2 : HasCompactSupport f) :
    Tendsto (fun w : V => ∫ v : V, e[-⟪v, w⟫] • f v) (cocompact V) (𝓝 0) := by
  refine' NormedAddCommGroup.tendsto_nhds_zero.mpr fun ε hε => _
  suffices ∃ T : ℝ, ∀ w : V, T ≤ ‖w‖ → ‖∫ v : V, e[-⟪v, w⟫] • f v‖ < ε by
    simp_rw [← comap_dist_left_atTop_eq_cocompact (0 : V), eventually_comap, eventually_atTop,
      dist_eq_norm', sub_zero]
    exact
      let ⟨T, hT⟩ := this
      ⟨T, fun b hb v hv => hT v (hv.symm ▸ hb)⟩
  obtain ⟨R, -, hR_bd⟩ : ∃ R : ℝ, 0 < R ∧ ∀ x : V, R ≤ ‖x‖ → f x = 0
  exact hf2.exists_pos_le_norm
  let A := {v : V | ‖v‖ ≤ R + 1}
  have mA : MeasurableSet A := by
    suffices A = Metric.closedBall (0 : V) (R + 1) by
      rw [this]
      exact Metric.isClosed_ball.measurableSet
    simp_rw [Metric.closedBall, dist_eq_norm, sub_zero]
  obtain ⟨B, hB_pos, hB_vol⟩ : ∃ B : ℝ≥0, 0 < B ∧ volume A ≤ B := by
    have hc : IsCompact A := by
      

Proof state:
E : Type u_1
V : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
f : V → E
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : FiniteDimensional ℝ V
inst✝ : CompleteSpace E
hf1 : Continuous f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
mA : MeasurableSet A
⊢ IsCompact A

Next tactic:
simpa only [Metric.closedBall, dist_eq_norm, sub_zero] using isCompact_closedBall (0 : V) _
/-- Riemann-Lebesgue Lemma for continuous and compactly-supported functions: the integral
`∫ v, exp (-2 * π * ⟪w, v⟫ * I) • f v` tends to 0 wrt `cocompact V`. Note that this is primarily
of interest as a preparatory step for the more general result
`tendsto_integral_exp_inner_smul_cocompact` in which `f` can be arbitrary. -/
theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support (hf1 : Continuous f)
    (hf2 : HasCompactSupport f) :
    Tendsto (fun w : V => ∫ v : V, e[-⟪v, w⟫] • f v) (cocompact V) (𝓝 0) := by
  refine' NormedAddCommGroup.tendsto_nhds_zero.mpr fun ε hε => _
  suffices ∃ T : ℝ, ∀ w : V, T ≤ ‖w‖ → ‖∫ v : V, e[-⟪v, w⟫] • f v‖ < ε by
    simp_rw [← comap_dist_left_atTop_eq_cocompact (0 : V), eventually_comap, eventually_atTop,
      dist_eq_norm', sub_zero]
    exact
      let ⟨T, hT⟩ := this
      ⟨T, fun b hb v hv => hT v (hv.symm ▸ hb)⟩
  obtain ⟨R, -, hR_bd⟩ : ∃ R : ℝ, 0 < R ∧ ∀ x : V, R ≤ ‖x‖ → f x = 0
  exact hf2.exists_pos_le_norm
  let A := {v : V | ‖v‖ ≤ R + 1}
  have mA : MeasurableSet A := by
    suffices A = Metric.closedBall (0 : V) (R + 1) by
      rw [this]
      exact Metric.isClosed_ball.measurableSet
    simp_rw [Metric.closedBall, dist_eq_norm, sub_zero]
  obtain ⟨B, hB_pos, hB_vol⟩ : ∃ B : ℝ≥0, 0 < B ∧ volume A ≤ B := by
    have hc : IsCompact A := by
      simpa only [Metric.closedBall, dist_eq_norm, sub_zero] using isCompact_closedBall (0 : V) _
    

Proof state:
E : Type u_1
V : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
f : V → E
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : FiniteDimensional ℝ V
inst✝ : CompleteSpace E
hf1 : Continuous f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
mA : MeasurableSet A
hc : IsCompact A
⊢ ∃ B, 0 < B ∧ ↑↑volume A ≤ ↑B

Next tactic:
let B₀ := volume A
/-- Riemann-Lebesgue Lemma for continuous and compactly-supported functions: the integral
`∫ v, exp (-2 * π * ⟪w, v⟫ * I) • f v` tends to 0 wrt `cocompact V`. Note that this is primarily
of interest as a preparatory step for the more general result
`tendsto_integral_exp_inner_smul_cocompact` in which `f` can be arbitrary. -/
theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support (hf1 : Continuous f)
    (hf2 : HasCompactSupport f) :
    Tendsto (fun w : V => ∫ v : V, e[-⟪v, w⟫] • f v) (cocompact V) (𝓝 0) := by
  refine' NormedAddCommGroup.tendsto_nhds_zero.mpr fun ε hε => _
  suffices ∃ T : ℝ, ∀ w : V, T ≤ ‖w‖ → ‖∫ v : V, e[-⟪v, w⟫] • f v‖ < ε by
    simp_rw [← comap_dist_left_atTop_eq_cocompact (0 : V), eventually_comap, eventually_atTop,
      dist_eq_norm', sub_zero]
    exact
      let ⟨T, hT⟩ := this
      ⟨T, fun b hb v hv => hT v (hv.symm ▸ hb)⟩
  obtain ⟨R, -, hR_bd⟩ : ∃ R : ℝ, 0 < R ∧ ∀ x : V, R ≤ ‖x‖ → f x = 0
  exact hf2.exists_pos_le_norm
  let A := {v : V | ‖v‖ ≤ R + 1}
  have mA : MeasurableSet A := by
    suffices A = Metric.closedBall (0 : V) (R + 1) by
      rw [this]
      exact Metric.isClosed_ball.measurableSet
    simp_rw [Metric.closedBall, dist_eq_norm, sub_zero]
  obtain ⟨B, hB_pos, hB_vol⟩ : ∃ B : ℝ≥0, 0 < B ∧ volume A ≤ B := by
    have hc : IsCompact A := by
      simpa only [Metric.closedBall, dist_eq_norm, sub_zero] using isCompact_closedBall (0 : V) _
    let B₀ := volume A
    

Proof state:
E : Type u_1
V : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
f : V → E
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : FiniteDimensional ℝ V
inst✝ : CompleteSpace E
hf1 : Continuous f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
mA : MeasurableSet A
hc : IsCompact A
B₀ : ℝ≥0∞ := ↑↑volume A
⊢ ∃ B, 0 < B ∧ ↑↑volume A ≤ ↑B

Next tactic:
replace hc : B₀ < ⊤ := hc.measure_lt_top
/-- Riemann-Lebesgue Lemma for continuous and compactly-supported functions: the integral
`∫ v, exp (-2 * π * ⟪w, v⟫ * I) • f v` tends to 0 wrt `cocompact V`. Note that this is primarily
of interest as a preparatory step for the more general result
`tendsto_integral_exp_inner_smul_cocompact` in which `f` can be arbitrary. -/
theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support (hf1 : Continuous f)
    (hf2 : HasCompactSupport f) :
    Tendsto (fun w : V => ∫ v : V, e[-⟪v, w⟫] • f v) (cocompact V) (𝓝 0) := by
  refine' NormedAddCommGroup.tendsto_nhds_zero.mpr fun ε hε => _
  suffices ∃ T : ℝ, ∀ w : V, T ≤ ‖w‖ → ‖∫ v : V, e[-⟪v, w⟫] • f v‖ < ε by
    simp_rw [← comap_dist_left_atTop_eq_cocompact (0 : V), eventually_comap, eventually_atTop,
      dist_eq_norm', sub_zero]
    exact
      let ⟨T, hT⟩ := this
      ⟨T, fun b hb v hv => hT v (hv.symm ▸ hb)⟩
  obtain ⟨R, -, hR_bd⟩ : ∃ R : ℝ, 0 < R ∧ ∀ x : V, R ≤ ‖x‖ → f x = 0
  exact hf2.exists_pos_le_norm
  let A := {v : V | ‖v‖ ≤ R + 1}
  have mA : MeasurableSet A := by
    suffices A = Metric.closedBall (0 : V) (R + 1) by
      rw [this]
      exact Metric.isClosed_ball.measurableSet
    simp_rw [Metric.closedBall, dist_eq_norm, sub_zero]
  obtain ⟨B, hB_pos, hB_vol⟩ : ∃ B : ℝ≥0, 0 < B ∧ volume A ≤ B := by
    have hc : IsCompact A := by
      simpa only [Metric.closedBall, dist_eq_norm, sub_zero] using isCompact_closedBall (0 : V) _
    let B₀ := volume A
    replace hc : B₀ < ⊤ := hc.measure_lt_top
    

Proof state:
E : Type u_1
V : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
f : V → E
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : FiniteDimensional ℝ V
inst✝ : CompleteSpace E
hf1 : Continuous f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
mA : MeasurableSet A
B₀ : ℝ≥0∞ := ↑↑volume A
hc : B₀ < ⊤
⊢ ∃ B, 0 < B ∧ ↑↑volume A ≤ ↑B

Next tactic:
refine' ⟨B₀.toNNReal + 1, add_pos_of_nonneg_of_pos B₀.toNNReal.coe_nonneg one_pos, _⟩
/-- Riemann-Lebesgue Lemma for continuous and compactly-supported functions: the integral
`∫ v, exp (-2 * π * ⟪w, v⟫ * I) • f v` tends to 0 wrt `cocompact V`. Note that this is primarily
of interest as a preparatory step for the more general result
`tendsto_integral_exp_inner_smul_cocompact` in which `f` can be arbitrary. -/
theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support (hf1 : Continuous f)
    (hf2 : HasCompactSupport f) :
    Tendsto (fun w : V => ∫ v : V, e[-⟪v, w⟫] • f v) (cocompact V) (𝓝 0) := by
  refine' NormedAddCommGroup.tendsto_nhds_zero.mpr fun ε hε => _
  suffices ∃ T : ℝ, ∀ w : V, T ≤ ‖w‖ → ‖∫ v : V, e[-⟪v, w⟫] • f v‖ < ε by
    simp_rw [← comap_dist_left_atTop_eq_cocompact (0 : V), eventually_comap, eventually_atTop,
      dist_eq_norm', sub_zero]
    exact
      let ⟨T, hT⟩ := this
      ⟨T, fun b hb v hv => hT v (hv.symm ▸ hb)⟩
  obtain ⟨R, -, hR_bd⟩ : ∃ R : ℝ, 0 < R ∧ ∀ x : V, R ≤ ‖x‖ → f x = 0
  exact hf2.exists_pos_le_norm
  let A := {v : V | ‖v‖ ≤ R + 1}
  have mA : MeasurableSet A := by
    suffices A = Metric.closedBall (0 : V) (R + 1) by
      rw [this]
      exact Metric.isClosed_ball.measurableSet
    simp_rw [Metric.closedBall, dist_eq_norm, sub_zero]
  obtain ⟨B, hB_pos, hB_vol⟩ : ∃ B : ℝ≥0, 0 < B ∧ volume A ≤ B := by
    have hc : IsCompact A := by
      simpa only [Metric.closedBall, dist_eq_norm, sub_zero] using isCompact_closedBall (0 : V) _
    let B₀ := volume A
    replace hc : B₀ < ⊤ := hc.measure_lt_top
    refine' ⟨B₀.toNNReal + 1, add_pos_of_nonneg_of_pos B₀.toNNReal.coe_nonneg one_pos, _⟩
    

Proof state:
E : Type u_1
V : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℂ E
f : V → E
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : MeasurableSpace V
inst✝³ : BorelSpace V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : FiniteDimensional ℝ V
inst✝ : CompleteSpace E
hf1 : Continuous f
hf2 : HasCompactSupport f
ε : ℝ
hε : ε > 0
R : ℝ
hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0
A : Set V := {v | ‖v‖ ≤ R + 1}
mA : MeasurableSet A
B₀ : ℝ≥0∞ := ↑↑volume A
hc : B₀ < ⊤
⊢ ↑↑volume A ≤ ↑(ENNReal.toNNReal B₀ + 1)

Next tactic:
rw [ENNReal.coe_add, ENNReal.coe_one, ENNReal.coe_toNNReal hc.ne]