File: Mathlib/MeasureTheory/Group/LIntegral.lean

full_name: MeasureTheory.lintegral_add_left_eq_self
code: /-- Translating a function by left-multiplication does not change its Lebesgue integral
with respect to a left-invariant measure. -/
@[to_additive
      "Translating a function by left-addition does not change its Lebesgue integral with
      respect to a left-invariant measure."]
theorem lintegral_mul_left_eq_self [IsMulLeftInvariant μ] (f : G → ℝ≥0∞) (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.lintegral_mul_left_eq_self
code: /-- Translating a function by left-multiplication does not change its Lebesgue integral
with respect to a left-invariant measure. -/
@[to_additive
      "Translating a function by left-addition does not change its Lebesgue integral with
      respect to a left-invariant measure."]
theorem lintegral_mul_left_eq_self [IsMulLeftInvariant μ] (f : G → ℝ≥0∞) (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.lintegral_mul_right_eq_self
code: /-- Translating a function by right-multiplication does not change its Lebesgue integral
with respect to a right-invariant measure. -/
@[to_additive
      "Translating a function by right-addition does not change its Lebesgue integral with
      respect to a right-invariant measure."]
theorem lintegral_mul_right_eq_self [IsMulRightInvariant μ] (f : G → ℝ≥0∞) (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.lintegral_add_right_eq_self
code: /-- Translating a function by right-multiplication does not change its Lebesgue integral
with respect to a right-invariant measure. -/
@[to_additive
      "Translating a function by right-addition does not change its Lebesgue integral with
      respect to a right-invariant measure."]
theorem lintegral_mul_right_eq_self [IsMulRightInvariant μ] (f : G → ℝ≥0∞) (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.lintegral_div_right_eq_self
code: @[to_additive] -- Porting note: was `@[simp]`
theorem lintegral_div_right_eq_self [IsMulRightInvariant μ] (f : G → ℝ≥0∞) (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.lintegral_sub_right_eq_self
code: @[to_additive] -- Porting note: was `@[simp]`
theorem lintegral_div_right_eq_self [IsMulRightInvariant μ] (f : G → ℝ≥0∞) (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.lintegral_eq_zero_of_isMulLeftInvariant
code: /-- For nonzero regular left invariant measures, the integral of a continuous nonnegative function
  `f` is 0 iff `f` is 0. -/
@[to_additive
      "For nonzero regular left invariant measures, the integral of a continuous nonnegative
      function `f` is 0 iff `f` is 0."]
theorem lintegral_eq_zero_of_isMulLeftInvariant [Regular μ] [NeZero μ] {f : G → ℝ≥0∞}
    (hf : Continuous f) : ∫⁻ x, f x ∂μ = 0 ↔ f = 0 
goal: ∫⁻ (x : G), f x ∂μ = 0 ↔ f = 0

full_name: MeasureTheory.lintegral_eq_zero_of_isAddLeftInvariant
code: /-- For nonzero regular left invariant measures, the integral of a continuous nonnegative function
  `f` is 0 iff `f` is 0. -/
@[to_additive
      "For nonzero regular left invariant measures, the integral of a continuous nonnegative
      function `f` is 0 iff `f` is 0."]
theorem lintegral_eq_zero_of_isMulLeftInvariant [Regular μ] [NeZero μ] {f : G → ℝ≥0∞}
    (hf : Continuous f) : ∫⁻ x, f x ∂μ = 0 ↔ f = 0 
goal: ∫⁻ (x : G), f x ∂μ = 0 ↔ f = 0