File: Mathlib/MeasureTheory/Group/Integral.lean

full_name: MeasureTheory.Integrable.comp_neg
code: @[to_additive]
theorem Integrable.comp_inv [IsInvInvariant μ] {f : G → F} (hf : Integrable f μ) :
    Integrable (fun t => f t⁻¹) μ 
goal: MeasureTheory.Integrable (fun t => f (-t)) μ

full_name: MeasureTheory.Integrable.comp_inv
code: @[to_additive]
theorem Integrable.comp_inv [IsInvInvariant μ] {f : G → F} (hf : Integrable f μ) :
    Integrable (fun t => f t⁻¹) μ 
goal: MeasureTheory.Integrable (fun t => f t⁻¹) μ

full_name: MeasureTheory.integral_inv_eq_self
code: @[to_additive]
theorem integral_inv_eq_self (f : G → E) (μ : Measure G) [IsInvInvariant μ] :
    ∫ x, f x⁻¹ ∂μ = ∫ x, f x ∂μ 
goal: ∫ (x : G), f x⁻¹ ∂μ = ∫ (x : G), f x ∂μ

full_name: MeasureTheory.integral_neg_eq_self
code: @[to_additive]
theorem integral_inv_eq_self (f : G → E) (μ : Measure G) [IsInvInvariant μ] :
    ∫ x, f x⁻¹ ∂μ = ∫ x, f x ∂μ 
goal: ∫ (x : G), f (-x) ∂μ = ∫ (x : G), f x ∂μ

full_name: MeasureTheory.integral_add_left_eq_self
code: /-- Translating a function by left-multiplication does not change its integral with respect to a
left-invariant measure. -/
@[to_additive
      "Translating a function by left-addition does not change its integral with respect to a
      left-invariant measure."] -- Porting note: was `@[simp]`
theorem integral_mul_left_eq_self [IsMulLeftInvariant μ] (f : G → E) (g : G) :
    (∫ x, f (g * x) ∂μ) = ∫ x, f x ∂μ 
goal: ∫ (x : G), f (g + x) ∂μ = ∫ (x : G), f x ∂μ

full_name: MeasureTheory.integral_mul_left_eq_self
code: /-- Translating a function by left-multiplication does not change its integral with respect to a
left-invariant measure. -/
@[to_additive
      "Translating a function by left-addition does not change its integral with respect to a
      left-invariant measure."] -- Porting note: was `@[simp]`
theorem integral_mul_left_eq_self [IsMulLeftInvariant μ] (f : G → E) (g : G) :
    (∫ x, f (g * x) ∂μ) = ∫ x, f x ∂μ 
goal: ∫ (x : G), f (g * x) ∂μ = ∫ (x : G), f x ∂μ

full_name: MeasureTheory.integral_add_right_eq_self
code: /-- Translating a function by right-multiplication does not change its integral with respect to a
right-invariant measure. -/
@[to_additive
      "Translating a function by right-addition does not change its integral with respect to a
      right-invariant measure."] -- Porting note: was `@[simp]`
theorem integral_mul_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
    (∫ x, f (x * g) ∂μ) = ∫ x, f x ∂μ 
goal: ∫ (x : G), f (x + g) ∂μ = ∫ (x : G), f x ∂μ

full_name: MeasureTheory.integral_mul_right_eq_self
code: /-- Translating a function by right-multiplication does not change its integral with respect to a
right-invariant measure. -/
@[to_additive
      "Translating a function by right-addition does not change its integral with respect to a
      right-invariant measure."] -- Porting note: was `@[simp]`
theorem integral_mul_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
    (∫ x, f (x * g) ∂μ) = ∫ x, f x ∂μ 
goal: ∫ (x : G), f (x * g) ∂μ = ∫ (x : G), f x ∂μ

full_name: MeasureTheory.integral_sub_right_eq_self
code: @[to_additive] -- Porting note: was `@[simp]`
theorem integral_div_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
    (∫ x, f (x / g) ∂μ) = ∫ x, f x ∂μ 
goal: ∫ (x : G), f (x - g) ∂μ = ∫ (x : G), f x ∂μ

full_name: MeasureTheory.integral_div_right_eq_self
code: @[to_additive] -- Porting note: was `@[simp]`
theorem integral_div_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
    (∫ x, f (x / g) ∂μ) = ∫ x, f x ∂μ 
goal: ∫ (x : G), f (x / g) ∂μ = ∫ (x : G), f x ∂μ

full_name: MeasureTheory.integral_eq_zero_of_mul_left_eq_neg
code: /-- If some left-translate of a function negates it, then the integral of the function with respect
to a left-invariant measure is 0. -/
@[to_additive
      "If some left-translate of a function negates it, then the integral of the function with
      respect to a left-invariant measure is 0."]
theorem integral_eq_zero_of_mul_left_eq_neg [IsMulLeftInvariant μ] (hf' : ∀ x, f (g * x) = -f x) :
    ∫ x, f x ∂μ = 0 
goal: ∫ (x : G), f x ∂μ = 0

full_name: MeasureTheory.integral_eq_zero_of_add_left_eq_neg
code: /-- If some left-translate of a function negates it, then the integral of the function with respect
to a left-invariant measure is 0. -/
@[to_additive
      "If some left-translate of a function negates it, then the integral of the function with
      respect to a left-invariant measure is 0."]
theorem integral_eq_zero_of_mul_left_eq_neg [IsMulLeftInvariant μ] (hf' : ∀ x, f (g * x) = -f x) :
    ∫ x, f x ∂μ = 0 
goal: ∫ (x : G), f x ∂μ = 0

full_name: MeasureTheory.integral_eq_zero_of_mul_right_eq_neg
code: /-- If some right-translate of a function negates it, then the integral of the function with respect
to a right-invariant measure is 0. -/
@[to_additive
      "If some right-translate of a function negates it, then the integral of the function with
      respect to a right-invariant measure is 0."]
theorem integral_eq_zero_of_mul_right_eq_neg [IsMulRightInvariant μ] (hf' : ∀ x, f (x * g) = -f x) :
    ∫ x, f x ∂μ = 0 
goal: ∫ (x : G), f x ∂μ = 0

full_name: MeasureTheory.integral_eq_zero_of_add_right_eq_neg
code: /-- If some right-translate of a function negates it, then the integral of the function with respect
to a right-invariant measure is 0. -/
@[to_additive
      "If some right-translate of a function negates it, then the integral of the function with
      respect to a right-invariant measure is 0."]
theorem integral_eq_zero_of_mul_right_eq_neg [IsMulRightInvariant μ] (hf' : ∀ x, f (x * g) = -f x) :
    ∫ x, f x ∂μ = 0 
goal: ∫ (x : G), f x ∂μ = 0

full_name: MeasureTheory.Integrable.comp_mul_left
code: @[to_additive]
theorem Integrable.comp_mul_left {f : G → F} [IsMulLeftInvariant μ] (hf : Integrable f μ) (g : G) :
    Integrable (fun t => f (g * t)) μ 
goal: MeasureTheory.Integrable (fun t => f (g * t)) μ

full_name: MeasureTheory.Integrable.comp_add_left
code: @[to_additive]
theorem Integrable.comp_mul_left {f : G → F} [IsMulLeftInvariant μ] (hf : Integrable f μ) (g : G) :
    Integrable (fun t => f (g * t)) μ 
goal: MeasureTheory.Integrable (fun t => f (g + t)) μ

full_name: MeasureTheory.Integrable.comp_mul_right
code: @[to_additive]
theorem Integrable.comp_mul_right {f : G → F} [IsMulRightInvariant μ] (hf : Integrable f μ)
    (g : G) : Integrable (fun t => f (t * g)) μ 
goal: MeasureTheory.Integrable (fun t => f (t * g)) μ

full_name: MeasureTheory.Integrable.comp_add_right
code: @[to_additive]
theorem Integrable.comp_mul_right {f : G → F} [IsMulRightInvariant μ] (hf : Integrable f μ)
    (g : G) : Integrable (fun t => f (t * g)) μ 
goal: MeasureTheory.Integrable (fun t => f (t + g)) μ

full_name: MeasureTheory.Integrable.comp_sub_right
code: @[to_additive]
theorem Integrable.comp_div_right {f : G → F} [IsMulRightInvariant μ] (hf : Integrable f μ)
    (g : G) : Integrable (fun t => f (t / g)) μ 
goal: MeasureTheory.Integrable (fun t => f (t - g)) μ

full_name: MeasureTheory.Integrable.comp_div_right
code: @[to_additive]
theorem Integrable.comp_div_right {f : G → F} [IsMulRightInvariant μ] (hf : Integrable f μ)
    (g : G) : Integrable (fun t => f (t / g)) μ 
goal: MeasureTheory.Integrable (fun t => f (t / g)) μ

full_name: MeasureTheory.Integrable.comp_div_left
code: @[to_additive]
theorem Integrable.comp_div_left {f : G → F} [IsInvInvariant μ] [IsMulLeftInvariant μ]
    (hf : Integrable f μ) (g : G) : Integrable (fun t => f (g / t)) μ 
goal: MeasureTheory.Integrable (fun t => f (g / t)) μ

full_name: MeasureTheory.Integrable.comp_sub_left
code: @[to_additive]
theorem Integrable.comp_div_left {f : G → F} [IsInvInvariant μ] [IsMulLeftInvariant μ]
    (hf : Integrable f μ) (g : G) : Integrable (fun t => f (g / t)) μ 
goal: MeasureTheory.Integrable (fun t => f (g - t)) μ

full_name: MeasureTheory.integrable_comp_sub_left
code: @[to_additive] -- Porting note: was `@[simp]`
theorem integrable_comp_div_left (f : G → F) [IsInvInvariant μ] [IsMulLeftInvariant μ] (g : G) :
    Integrable (fun t => f (g / t)) μ ↔ Integrable f μ 
goal: MeasureTheory.Integrable (fun t => f (g - t)) μ ↔ MeasureTheory.Integrable f μ

full_name: MeasureTheory.integrable_comp_div_left
code: @[to_additive] -- Porting note: was `@[simp]`
theorem integrable_comp_div_left (f : G → F) [IsInvInvariant μ] [IsMulLeftInvariant μ] (g : G) :
    Integrable (fun t => f (g / t)) μ ↔ Integrable f μ 
goal: MeasureTheory.Integrable (fun t => f (g / t)) μ ↔ MeasureTheory.Integrable f μ

full_name: MeasureTheory.integral_sub_left_eq_self
code: @[to_additive] -- Porting note: was `@[simp]`
theorem integral_div_left_eq_self (f : G → E) (μ : Measure G) [IsInvInvariant μ]
    [IsMulLeftInvariant μ] (x' : G) : (∫ x, f (x' / x) ∂μ) = ∫ x, f x ∂μ 
goal: ∫ (x : G), f (x' - x) ∂μ = ∫ (x : G), f x ∂μ

full_name: MeasureTheory.integral_div_left_eq_self
code: @[to_additive] -- Porting note: was `@[simp]`
theorem integral_div_left_eq_self (f : G → E) (μ : Measure G) [IsInvInvariant μ]
    [IsMulLeftInvariant μ] (x' : G) : (∫ x, f (x' / x) ∂μ) = ∫ x, f x ∂μ 
goal: ∫ (x : G), f (x' / x) ∂μ = ∫ (x : G), f x ∂μ

full_name: MeasureTheory.integral_vadd_eq_self
code: @[to_additive] -- Porting note: was `@[simp]`
theorem integral_smul_eq_self {μ : Measure α} [SMulInvariantMeasure G α μ] (f : α → E) {g : G} :
    (∫ x, f (g • x) ∂μ) = ∫ x, f x ∂μ 
goal: ∫ (x : α), f (g +ᵥ x) ∂μ = ∫ (x : α), f x ∂μ

full_name: MeasureTheory.integral_smul_eq_self
code: @[to_additive] -- Porting note: was `@[simp]`
theorem integral_smul_eq_self {μ : Measure α} [SMulInvariantMeasure G α μ] (f : α → E) {g : G} :
    (∫ x, f (g • x) ∂μ) = ∫ x, f x ∂μ 
goal: ∫ (x : α), f (g • x) ∂μ = ∫ (x : α), f x ∂μ