l3lab/ntp-mathlib huggingface dataset:
theorem set_integral_dirac' {mα : MeasurableSpace α} {f : α → E} (hf : StronglyMeasurable f) (a : α)
    {s : Set α} (hs : MeasurableSet s) [Decidable (a ∈ s)] :
    ∫ x in s, f x ∂Measure.dirac a = if a ∈ s then f a else 0 := by
  rw [restrict_dirac' hs]
  split_ifs
  · 

Proof state:
case pos
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : SMulCommClass ℝ 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : CompleteSpace F
G : Type u_5
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
ν : Measure α
mα : MeasurableSpace α
f : α → E
hf : StronglyMeasurable f
a : α
s : Set α
hs : MeasurableSet s
inst✝ : Decidable (a ∈ s)
h✝ : a ∈ s
⊢ ∫ (x : α), f x ∂Measure.dirac a = f a

Next tactic:
exact integral_dirac' _ _ hf
theorem set_integral_dirac' {mα : MeasurableSpace α} {f : α → E} (hf : StronglyMeasurable f) (a : α)
    {s : Set α} (hs : MeasurableSet s) [Decidable (a ∈ s)] :
    ∫ x in s, f x ∂Measure.dirac a = if a ∈ s then f a else 0 := by
  rw [restrict_dirac' hs]
  split_ifs
  · exact integral_dirac' _ _ hf
  · 

Proof state:
case neg
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : SMulCommClass ℝ 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : CompleteSpace F
G : Type u_5
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
ν : Measure α
mα : MeasurableSpace α
f : α → E
hf : StronglyMeasurable f
a : α
s : Set α
hs : MeasurableSet s
inst✝ : Decidable (a ∈ s)
h✝ : a ∉ s
⊢ ∫ (x : α), f x ∂0 = 0

Next tactic:
exact integral_zero_measure _
theorem set_integral_dirac [MeasurableSpace α] [MeasurableSingletonClass α] (f : α → E) (a : α)
    (s : Set α) [Decidable (a ∈ s)] :
    ∫ x in s, f x ∂Measure.dirac a = if a ∈ s then f a else 0 := by
  

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : NormedSpace 𝕜 E
inst✝¹⁰ : SMulCommClass ℝ 𝕜 E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : CompleteSpace F
G : Type u_5
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝⁴ : TopologicalSpace X
inst✝³ : FirstCountableTopology X
ν : Measure α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSingletonClass α
f : α → E
a : α
s : Set α
inst✝ : Decidable (a ∈ s)
⊢ ∫ (x : α) in s, f x ∂Measure.dirac a = if a ∈ s then f a else 0

Next tactic:
rw [restrict_dirac]
theorem set_integral_dirac [MeasurableSpace α] [MeasurableSingletonClass α] (f : α → E) (a : α)
    (s : Set α) [Decidable (a ∈ s)] :
    ∫ x in s, f x ∂Measure.dirac a = if a ∈ s then f a else 0 := by
  rw [restrict_dirac]
  

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : NormedSpace 𝕜 E
inst✝¹⁰ : SMulCommClass ℝ 𝕜 E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : CompleteSpace F
G : Type u_5
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝⁴ : TopologicalSpace X
inst✝³ : FirstCountableTopology X
ν : Measure α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSingletonClass α
f : α → E
a : α
s : Set α
inst✝ : Decidable (a ∈ s)
⊢ (∫ (x : α), f x ∂if a ∈ s then Measure.dirac a else 0) = if a ∈ s then f a else 0

Next tactic:
split_ifs
theorem set_integral_dirac [MeasurableSpace α] [MeasurableSingletonClass α] (f : α → E) (a : α)
    (s : Set α) [Decidable (a ∈ s)] :
    ∫ x in s, f x ∂Measure.dirac a = if a ∈ s then f a else 0 := by
  rw [restrict_dirac]
  split_ifs
  · 

Proof state:
case pos
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : NormedSpace 𝕜 E
inst✝¹⁰ : SMulCommClass ℝ 𝕜 E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : CompleteSpace F
G : Type u_5
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝⁴ : TopologicalSpace X
inst✝³ : FirstCountableTopology X
ν : Measure α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSingletonClass α
f : α → E
a : α
s : Set α
inst✝ : Decidable (a ∈ s)
h✝ : a ∈ s
⊢ ∫ (x : α), f x ∂Measure.dirac a = f a

Next tactic:
exact integral_dirac _ _
theorem set_integral_dirac [MeasurableSpace α] [MeasurableSingletonClass α] (f : α → E) (a : α)
    (s : Set α) [Decidable (a ∈ s)] :
    ∫ x in s, f x ∂Measure.dirac a = if a ∈ s then f a else 0 := by
  rw [restrict_dirac]
  split_ifs
  · exact integral_dirac _ _
  · 

Proof state:
case neg
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : NormedSpace 𝕜 E
inst✝¹⁰ : SMulCommClass ℝ 𝕜 E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : CompleteSpace F
G : Type u_5
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝⁴ : TopologicalSpace X
inst✝³ : FirstCountableTopology X
ν : Measure α
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSingletonClass α
f : α → E
a : α
s : Set α
inst✝ : Decidable (a ∈ s)
h✝ : a ∉ s
⊢ ∫ (x : α), f x ∂0 = 0

Next tactic:
exact integral_zero_measure _
/-- **Markov's inequality** also known as **Chebyshev's first inequality**. -/
theorem mul_meas_ge_le_integral_of_nonneg {f : α → ℝ} (hf_nonneg : 0 ≤ᵐ[μ] f)
    (hf_int : Integrable f μ) (ε : ℝ) : ε * (μ { x | ε ≤ f x }).toReal ≤ ∫ x, f x ∂μ := by
  

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : CompleteSpace F
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
f : α → ℝ
hf_nonneg : 0 ≤ᵐ[μ] f
hf_int : Integrable f
ε : ℝ
⊢ ε * ENNReal.toReal (↑↑μ {x | ε ≤ f x}) ≤ ∫ (x : α), f x ∂μ

Next tactic:
cases' eq_top_or_lt_top (μ {x | ε ≤ f x}) with hμ hμ
/-- **Markov's inequality** also known as **Chebyshev's first inequality**. -/
theorem mul_meas_ge_le_integral_of_nonneg {f : α → ℝ} (hf_nonneg : 0 ≤ᵐ[μ] f)
    (hf_int : Integrable f μ) (ε : ℝ) : ε * (μ { x | ε ≤ f x }).toReal ≤ ∫ x, f x ∂μ := by
  cases' eq_top_or_lt_top (μ {x | ε ≤ f x}) with hμ hμ
  · 

Proof state:
case inl
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : CompleteSpace F
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
f : α → ℝ
hf_nonneg : 0 ≤ᵐ[μ] f
hf_int : Integrable f
ε : ℝ
hμ : ↑↑μ {x | ε ≤ f x} = ⊤
⊢ ε * ENNReal.toReal (↑↑μ {x | ε ≤ f x}) ≤ ∫ (x : α), f x ∂μ

Next tactic:
simpa [hμ] using integral_nonneg_of_ae hf_nonneg
/-- **Markov's inequality** also known as **Chebyshev's first inequality**. -/
theorem mul_meas_ge_le_integral_of_nonneg {f : α → ℝ} (hf_nonneg : 0 ≤ᵐ[μ] f)
    (hf_int : Integrable f μ) (ε : ℝ) : ε * (μ { x | ε ≤ f x }).toReal ≤ ∫ x, f x ∂μ := by
  cases' eq_top_or_lt_top (μ {x | ε ≤ f x}) with hμ hμ
  · simpa [hμ] using integral_nonneg_of_ae hf_nonneg
  · 

Proof state:
case inr
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : CompleteSpace F
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
f : α → ℝ
hf_nonneg : 0 ≤ᵐ[μ] f
hf_int : Integrable f
ε : ℝ
hμ : ↑↑μ {x | ε ≤ f x} < ⊤
⊢ ε * ENNReal.toReal (↑↑μ {x | ε ≤ f x}) ≤ ∫ (x : α), f x ∂μ

Next tactic:
have := Fact.mk hμ
/-- **Markov's inequality** also known as **Chebyshev's first inequality**. -/
theorem mul_meas_ge_le_integral_of_nonneg {f : α → ℝ} (hf_nonneg : 0 ≤ᵐ[μ] f)
    (hf_int : Integrable f μ) (ε : ℝ) : ε * (μ { x | ε ≤ f x }).toReal ≤ ∫ x, f x ∂μ := by
  cases' eq_top_or_lt_top (μ {x | ε ≤ f x}) with hμ hμ
  · simpa [hμ] using integral_nonneg_of_ae hf_nonneg
  · have := Fact.mk hμ
    

Proof state:
case inr
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : CompleteSpace F
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
f : α → ℝ
hf_nonneg : 0 ≤ᵐ[μ] f
hf_int : Integrable f
ε : ℝ
hμ : ↑↑μ {x | ε ≤ f x} < ⊤
this : Fact (↑↑μ {x | ε ≤ f x} < ⊤)
⊢ ε * ENNReal.toReal (↑↑μ {x | ε ≤ f x}) ≤ ∫ (x : α), f x ∂μ

Next tactic:
calc
      ε * (μ { x | ε ≤ f x }).toReal = ∫ _ in {x | ε ≤ f x}, ε ∂μ := by simp [mul_comm]
      _ ≤ ∫ x in {x | ε ≤ f x}, f x ∂μ :=
        integral_mono_ae (integrable_const _) (hf_int.mono_measure μ.restrict_le_self) <|
          ae_restrict_mem₀ <| hf_int.aemeasurable.nullMeasurable measurableSet_Ici
      _ ≤ _ := integral_mono_measure μ.restrict_le_self hf_nonneg hf_int
/-- **Markov's inequality** also known as **Chebyshev's first inequality**. -/
theorem mul_meas_ge_le_integral_of_nonneg {f : α → ℝ} (hf_nonneg : 0 ≤ᵐ[μ] f)
    (hf_int : Integrable f μ) (ε : ℝ) : ε * (μ { x | ε ≤ f x }).toReal ≤ ∫ x, f x ∂μ := by
  cases' eq_top_or_lt_top (μ {x | ε ≤ f x}) with hμ hμ
  · simpa [hμ] using integral_nonneg_of_ae hf_nonneg
  · have := Fact.mk hμ
    calc
      ε * (μ { x | ε ≤ f x }).toReal = ∫ _ in {x | ε ≤ f x}, ε ∂μ := by 

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : CompleteSpace F
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
f : α → ℝ
hf_nonneg : 0 ≤ᵐ[μ] f
hf_int : Integrable f
ε : ℝ
hμ : ↑↑μ {x | ε ≤ f x} < ⊤
this : Fact (↑↑μ {x | ε ≤ f x} < ⊤)
⊢ ε * ENNReal.toReal (↑↑μ {x | ε ≤ f x}) = ∫ (x : α) in {x | ε ≤ f x}, ε ∂μ

Next tactic:
simp [mul_comm]
/-- Hölder's inequality for the integral of a product of norms. The integral of the product of two
norms of functions is bounded by the product of their `ℒp` and `ℒq` seminorms when `p` and `q` are
conjugate exponents. -/
theorem integral_mul_norm_le_Lp_mul_Lq {E} [NormedAddCommGroup E] {f g : α → E} {p q : ℝ}
    (hpq : p.IsConjugateExponent q) (hf : Memℒp f (ENNReal.ofReal p) μ)
    (hg : Memℒp g (ENNReal.ofReal q) μ) :
    ∫ a, ‖f a‖ * ‖g a‖ ∂μ ≤ (∫ a, ‖f a‖ ^ p ∂μ) ^ (1 / p) * (∫ a, ‖g a‖ ^ q ∂μ) ^ (1 / q) := by
  -- translate the Bochner integrals into Lebesgue integrals.
  

Proof state:
α : Type u_1
E✝ : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹² : NormedAddCommGroup E✝
inst✝¹¹ : NormedSpace ℝ E✝
hE : CompleteSpace E✝
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E✝
inst✝⁸ : SMulCommClass ℝ 𝕜 E✝
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : CompleteSpace F
G : Type u_5
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f✝ g✝ : α → E✝
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
ν : Measure α
E : Type u_7
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : Real.IsConjugateExponent p q
hf : Memℒp f (ENNReal.ofReal p)
hg : Memℒp g (ENNReal.ofReal q)
⊢ ∫ (a : α), ‖f a‖ * ‖g a‖ ∂μ ≤ (∫ (a : α), ‖f a‖ ^ p ∂μ) ^ (1 / p) * (∫ (a : α), ‖g a‖ ^ q ∂μ) ^ (1 / q)

Next tactic:
rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae,
    integral_eq_lintegral_of_nonneg_ae]
/-- Hölder's inequality for the integral of a product of norms. The integral of the product of two
norms of functions is bounded by the product of their `ℒp` and `ℒq` seminorms when `p` and `q` are
conjugate exponents. -/
theorem integral_mul_norm_le_Lp_mul_Lq {E} [NormedAddCommGroup E] {f g : α → E} {p q : ℝ}
    (hpq : p.IsConjugateExponent q) (hf : Memℒp f (ENNReal.ofReal p) μ)
    (hg : Memℒp g (ENNReal.ofReal q) μ) :
    ∫ a, ‖f a‖ * ‖g a‖ ∂μ ≤ (∫ a, ‖f a‖ ^ p ∂μ) ^ (1 / p) * (∫ a, ‖g a‖ ^ q ∂μ) ^ (1 / q) := by
  -- translate the Bochner integrals into Lebesgue integrals.
  rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae,
    integral_eq_lintegral_of_nonneg_ae]
  rotate_left
  · 

Proof state:
case hf
α : Type u_1
E✝ : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹² : NormedAddCommGroup E✝
inst✝¹¹ : NormedSpace ℝ E✝
hE : CompleteSpace E✝
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E✝
inst✝⁸ : SMulCommClass ℝ 𝕜 E✝
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : CompleteSpace F
G : Type u_5
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f✝ g✝ : α → E✝
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
ν : Measure α
E : Type u_7
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : Real.IsConjugateExponent p q
hf : Memℒp f (ENNReal.ofReal p)
hg : Memℒp g (ENNReal.ofReal q)
⊢ 0 ≤ᵐ[μ] fun a => ‖g a‖ ^ q

Next tactic:
exact eventually_of_forall fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
/-- Hölder's inequality for the integral of a product of norms. The integral of the product of two
norms of functions is bounded by the product of their `ℒp` and `ℒq` seminorms when `p` and `q` are
conjugate exponents. -/
theorem integral_mul_norm_le_Lp_mul_Lq {E} [NormedAddCommGroup E] {f g : α → E} {p q : ℝ}
    (hpq : p.IsConjugateExponent q) (hf : Memℒp f (ENNReal.ofReal p) μ)
    (hg : Memℒp g (ENNReal.ofReal q) μ) :
    ∫ a, ‖f a‖ * ‖g a‖ ∂μ ≤ (∫ a, ‖f a‖ ^ p ∂μ) ^ (1 / p) * (∫ a, ‖g a‖ ^ q ∂μ) ^ (1 / q) := by
  -- translate the Bochner integrals into Lebesgue integrals.
  rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae,
    integral_eq_lintegral_of_nonneg_ae]
  rotate_left
  · exact eventually_of_forall fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
  · 

Proof state:
case hfm
α : Type u_1
E✝ : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹² : NormedAddCommGroup E✝
inst✝¹¹ : NormedSpace ℝ E✝
hE : CompleteSpace E✝
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E✝
inst✝⁸ : SMulCommClass ℝ 𝕜 E✝
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : CompleteSpace F
G : Type u_5
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f✝ g✝ : α → E✝
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
ν : Measure α
E : Type u_7
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : Real.IsConjugateExponent p q
hf : Memℒp f (ENNReal.ofReal p)
hg : Memℒp g (ENNReal.ofReal q)
⊢ AEStronglyMeasurable (fun a => ‖g a‖ ^ q) μ

Next tactic:
exact (hg.1.norm.aemeasurable.pow aemeasurable_const).aestronglyMeasurable
/-- Hölder's inequality for the integral of a product of norms. The integral of the product of two
norms of functions is bounded by the product of their `ℒp` and `ℒq` seminorms when `p` and `q` are
conjugate exponents. -/
theorem integral_mul_norm_le_Lp_mul_Lq {E} [NormedAddCommGroup E] {f g : α → E} {p q : ℝ}
    (hpq : p.IsConjugateExponent q) (hf : Memℒp f (ENNReal.ofReal p) μ)
    (hg : Memℒp g (ENNReal.ofReal q) μ) :
    ∫ a, ‖f a‖ * ‖g a‖ ∂μ ≤ (∫ a, ‖f a‖ ^ p ∂μ) ^ (1 / p) * (∫ a, ‖g a‖ ^ q ∂μ) ^ (1 / q) := by
  -- translate the Bochner integrals into Lebesgue integrals.
  rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae,
    integral_eq_lintegral_of_nonneg_ae]
  rotate_left
  · exact eventually_of_forall fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
  · exact (hg.1.norm.aemeasurable.pow aemeasurable_const).aestronglyMeasurable
  · 

Proof state:
case hf
α : Type u_1
E✝ : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹² : NormedAddCommGroup E✝
inst✝¹¹ : NormedSpace ℝ E✝
hE : CompleteSpace E✝
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E✝
inst✝⁸ : SMulCommClass ℝ 𝕜 E✝
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : CompleteSpace F
G : Type u_5
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f✝ g✝ : α → E✝
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
ν : Measure α
E : Type u_7
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : Real.IsConjugateExponent p q
hf : Memℒp f (ENNReal.ofReal p)
hg : Memℒp g (ENNReal.ofReal q)
⊢ 0 ≤ᵐ[μ] fun a => ‖f a‖ ^ p

Next tactic:
exact eventually_of_forall fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
/-- Hölder's inequality for the integral of a product of norms. The integral of the product of two
norms of functions is bounded by the product of their `ℒp` and `ℒq` seminorms when `p` and `q` are
conjugate exponents. -/
theorem integral_mul_norm_le_Lp_mul_Lq {E} [NormedAddCommGroup E] {f g : α → E} {p q : ℝ}
    (hpq : p.IsConjugateExponent q) (hf : Memℒp f (ENNReal.ofReal p) μ)
    (hg : Memℒp g (ENNReal.ofReal q) μ) :
    ∫ a, ‖f a‖ * ‖g a‖ ∂μ ≤ (∫ a, ‖f a‖ ^ p ∂μ) ^ (1 / p) * (∫ a, ‖g a‖ ^ q ∂μ) ^ (1 / q) := by
  -- translate the Bochner integrals into Lebesgue integrals.
  rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae,
    integral_eq_lintegral_of_nonneg_ae]
  rotate_left
  · exact eventually_of_forall fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
  · exact (hg.1.norm.aemeasurable.pow aemeasurable_const).aestronglyMeasurable
  · exact eventually_of_forall fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
  · 

Proof state:
case hfm
α : Type u_1
E✝ : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹² : NormedAddCommGroup E✝
inst✝¹¹ : NormedSpace ℝ E✝
hE : CompleteSpace E✝
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E✝
inst✝⁸ : SMulCommClass ℝ 𝕜 E✝
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : CompleteSpace F
G : Type u_5
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f✝ g✝ : α → E✝
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
ν : Measure α
E : Type u_7
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : Real.IsConjugateExponent p q
hf : Memℒp f (ENNReal.ofReal p)
hg : Memℒp g (ENNReal.ofReal q)
⊢ AEStronglyMeasurable (fun a => ‖f a‖ ^ p) μ

Next tactic:
exact (hf.1.norm.aemeasurable.pow aemeasurable_const).aestronglyMeasurable
/-- Hölder's inequality for the integral of a product of norms. The integral of the product of two
norms of functions is bounded by the product of their `ℒp` and `ℒq` seminorms when `p` and `q` are
conjugate exponents. -/
theorem integral_mul_norm_le_Lp_mul_Lq {E} [NormedAddCommGroup E] {f g : α → E} {p q : ℝ}
    (hpq : p.IsConjugateExponent q) (hf : Memℒp f (ENNReal.ofReal p) μ)
    (hg : Memℒp g (ENNReal.ofReal q) μ) :
    ∫ a, ‖f a‖ * ‖g a‖ ∂μ ≤ (∫ a, ‖f a‖ ^ p ∂μ) ^ (1 / p) * (∫ a, ‖g a‖ ^ q ∂μ) ^ (1 / q) := by
  -- translate the Bochner integrals into Lebesgue integrals.
  rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae,
    integral_eq_lintegral_of_nonneg_ae]
  rotate_left
  · exact eventually_of_forall fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
  · exact (hg.1.norm.aemeasurable.pow aemeasurable_const).aestronglyMeasurable
  · exact eventually_of_forall fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
  · exact (hf.1.norm.aemeasurable.pow aemeasurable_const).aestronglyMeasurable
  · 

Proof state:
case hf
α : Type u_1
E✝ : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹² : NormedAddCommGroup E✝
inst✝¹¹ : NormedSpace ℝ E✝
hE : CompleteSpace E✝
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E✝
inst✝⁸ : SMulCommClass ℝ 𝕜 E✝
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : CompleteSpace F
G : Type u_5
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f✝ g✝ : α → E✝
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
ν : Measure α
E : Type u_7
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : Real.IsConjugateExponent p q
hf : Memℒp f (ENNReal.ofReal p)
hg : Memℒp g (ENNReal.ofReal q)
⊢ 0 ≤ᵐ[μ] fun a => ‖f a‖ * ‖g a‖

Next tactic:
exact eventually_of_forall fun x => mul_nonneg (norm_nonneg _) (norm_nonneg _)
/-- Hölder's inequality for the integral of a product of norms. The integral of the product of two
norms of functions is bounded by the product of their `ℒp` and `ℒq` seminorms when `p` and `q` are
conjugate exponents. -/
theorem integral_mul_norm_le_Lp_mul_Lq {E} [NormedAddCommGroup E] {f g : α → E} {p q : ℝ}
    (hpq : p.IsConjugateExponent q) (hf : Memℒp f (ENNReal.ofReal p) μ)
    (hg : Memℒp g (ENNReal.ofReal q) μ) :
    ∫ a, ‖f a‖ * ‖g a‖ ∂μ ≤ (∫ a, ‖f a‖ ^ p ∂μ) ^ (1 / p) * (∫ a, ‖g a‖ ^ q ∂μ) ^ (1 / q) := by
  -- translate the Bochner integrals into Lebesgue integrals.
  rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae,
    integral_eq_lintegral_of_nonneg_ae]
  rotate_left
  · exact eventually_of_forall fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
  · exact (hg.1.norm.aemeasurable.pow aemeasurable_const).aestronglyMeasurable
  · exact eventually_of_forall fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _
  · exact (hf.1.norm.aemeasurable.pow aemeasurable_const).aestronglyMeasurable
  · exact eventually_of_forall fun x => mul_nonneg (norm_nonneg _) (norm_nonneg _)
  · 

Proof state:
case hfm
α : Type u_1
E✝ : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹² : NormedAddCommGroup E✝
inst✝¹¹ : NormedSpace ℝ E✝
hE : CompleteSpace E✝
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E✝
inst✝⁸ : SMulCommClass ℝ 𝕜 E✝
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : CompleteSpace F
G : Type u_5
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
f✝ g✝ : α → E✝
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
ν : Measure α
E : Type u_7
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : Real.IsConjugateExponent p q
hf : Memℒp f (ENNReal.ofReal p)
hg : Memℒp g (ENNReal.ofReal q)
⊢ AEStronglyMeasurable (fun a => ‖f a‖ * ‖g a‖) μ

Next tactic:
exact hf.1.norm.mul hg.1.norm