File: Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean

full_name: MeasureTheory.Measure.MapContinuousLinearEquiv.isAddHaarMeasure
code: instance MapContinuousLinearEquiv.isAddHaarMeasure (e : G ≃L[𝕜] H) : IsAddHaarMeasure (μ.map e) 
goal: (MeasureTheory.Measure.map (⇑e) μ).IsAddHaarMeasure

full_name: MeasureTheory.Measure.MapLinearEquiv.isAddHaarMeasure
code: instance MapLinearEquiv.isAddHaarMeasure (e : G ≃ₗ[𝕜] H) : IsAddHaarMeasure (μ.map e) 
goal: (MeasureTheory.Measure.map (⇑e) μ).IsAddHaarMeasure

full_name: MeasureTheory.Measure.integral_comp_smul
code: /-- The integral of `f (R • x)` with respect to an additive Haar measure is a multiple of the
integral of `f`. The formula we give works even when `f` is not integrable or `R = 0`
thanks to the convention that a non-integrable function has integral zero. -/
theorem integral_comp_smul (f : E → F) (R : ℝ) :
    ∫ x, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ x, f x ∂μ 
goal: ∫ (x : E), f (R • x) ∂μ = |(R ^ FiniteDimensional.finrank ℝ E)⁻¹| • ∫ (x : E), f x ∂μ

full_name: MeasureTheory.Measure.integral_comp_smul_of_nonneg
code: /-- The integral of `f (R • x)` with respect to an additive Haar measure is a multiple of the
integral of `f`. The formula we give works even when `f` is not integrable or `R = 0`
thanks to the convention that a non-integrable function has integral zero. -/
theorem integral_comp_smul_of_nonneg (f : E → F) (R : ℝ) {hR : 0 ≤ R} :
    ∫ x, f (R • x) ∂μ = (R ^ finrank ℝ E)⁻¹ • ∫ x, f x ∂μ 
goal: ∫ (x : E), f (R • x) ∂μ = (R ^ FiniteDimensional.finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ

full_name: MeasureTheory.Measure.integral_comp_inv_smul
code: /-- The integral of `f (R⁻¹ • x)` with respect to an additive Haar measure is a multiple of the
integral of `f`. The formula we give works even when `f` is not integrable or `R = 0`
thanks to the convention that a non-integrable function has integral zero. -/
theorem integral_comp_inv_smul (f : E → F) (R : ℝ) :
    ∫ x, f (R⁻¹ • x) ∂μ = |R ^ finrank ℝ E| • ∫ x, f x ∂μ 
goal: ∫ (x : E), f (R⁻¹ • x) ∂μ = |R ^ FiniteDimensional.finrank ℝ E| • ∫ (x : E), f x ∂μ

full_name: MeasureTheory.Measure.integral_comp_inv_smul_of_nonneg
code: /-- The integral of `f (R⁻¹ • x)` with respect to an additive Haar measure is a multiple of the
integral of `f`. The formula we give works even when `f` is not integrable or `R = 0`
thanks to the convention that a non-integrable function has integral zero. -/
theorem integral_comp_inv_smul_of_nonneg (f : E → F) {R : ℝ} (hR : 0 ≤ R) :
    ∫ x, f (R⁻¹ • x) ∂μ = R ^ finrank ℝ E • ∫ x, f x ∂μ 
goal: ∫ (x : E), f (R⁻¹ • x) ∂μ = R ^ FiniteDimensional.finrank ℝ E • ∫ (x : E), f x ∂μ

full_name: MeasureTheory.Measure.setIntegral_comp_smul
code: theorem setIntegral_comp_smul (f : E → F) {R : ℝ} (s : Set E) (hR : R ≠ 0) :
    ∫ x in s, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ x in R • s, f x ∂μ 
goal: ∫ (x : E) in s, f (R • x) ∂μ = |(R ^ FiniteDimensional.finrank ℝ E)⁻¹| • ∫ (x : E) in R • s, f x ∂μ

full_name: MeasureTheory.Measure.setIntegral_comp_smul_of_pos
code: theorem setIntegral_comp_smul_of_pos (f : E → F) {R : ℝ} (s : Set E) (hR : 0 < R) :
    ∫ x in s, f (R • x) ∂μ = (R ^ finrank ℝ E)⁻¹ • ∫ x in R • s, f x ∂μ 
goal: ∫ (x : E) in s, f (R • x) ∂μ = (R ^ FiniteDimensional.finrank ℝ E)⁻¹ • ∫ (x : E) in R • s, f x ∂μ

full_name: MeasureTheory.Measure.integral_comp_mul_left
code: theorem integral_comp_mul_left (g : ℝ → F) (a : ℝ) :
    (∫ x : ℝ, g (a * x)) = |a⁻¹| • ∫ y : ℝ, g y 
goal: ∫ (x : ℝ), g (a * x) = |a⁻¹| • ∫ (y : ℝ), g y

full_name: MeasureTheory.Measure.integral_comp_inv_mul_left
code: theorem integral_comp_inv_mul_left (g : ℝ → F) (a : ℝ) :
    (∫ x : ℝ, g (a⁻¹ * x)) = |a| • ∫ y : ℝ, g y 
goal: ∫ (x : ℝ), g (a⁻¹ * x) = |a| • ∫ (y : ℝ), g y

full_name: MeasureTheory.Measure.integral_comp_mul_right
code: theorem integral_comp_mul_right (g : ℝ → F) (a : ℝ) :
    (∫ x : ℝ, g (x * a)) = |a⁻¹| • ∫ y : ℝ, g y 
goal: ∫ (x : ℝ), g (x * a) = |a⁻¹| • ∫ (y : ℝ), g y

full_name: MeasureTheory.Measure.integral_comp_inv_mul_right
code: theorem integral_comp_inv_mul_right (g : ℝ → F) (a : ℝ) :
    (∫ x : ℝ, g (x * a⁻¹)) = |a| • ∫ y : ℝ, g y 
goal: ∫ (x : ℝ), g (x * a⁻¹) = |a| • ∫ (y : ℝ), g y

full_name: MeasureTheory.Measure.integral_comp_div
code: theorem integral_comp_div (g : ℝ → F) (a : ℝ) : (∫ x : ℝ, g (x / a)) = |a| • ∫ y : ℝ, g y 
goal: ∫ (x : ℝ), g (x / a) = |a| • ∫ (y : ℝ), g y

full_name: MeasureTheory.integrable_comp_smul_iff
code: theorem integrable_comp_smul_iff {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ]
    (f : E → F) {R : ℝ} (hR : R ≠ 0) : Integrable (fun x => f (R • x)) μ ↔ Integrable f μ 
goal: MeasureTheory.Integrable (fun x => f (R • x)) μ ↔ MeasureTheory.Integrable f μ

full_name: MeasureTheory.Integrable.comp_smul
code: theorem Integrable.comp_smul {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] {μ : Measure E} [IsAddHaarMeasure μ]
    {f : E → F} (hf : Integrable f μ) {R : ℝ} (hR : R ≠ 0) : Integrable (fun x => f (R • x)) μ 
goal: MeasureTheory.Integrable (fun x => f (R • x)) μ

full_name: MeasureTheory.integrable_comp_mul_left_iff
code: theorem integrable_comp_mul_left_iff (g : ℝ → F) {R : ℝ} (hR : R ≠ 0) :
    (Integrable fun x => g (R * x)) ↔ Integrable g 
goal: MeasureTheory.Integrable (fun x => g (R * x)) MeasureTheory.volume ↔ MeasureTheory.Integrable g MeasureTheory.volume

full_name: MeasureTheory.Integrable.comp_mul_left'
code: theorem Integrable.comp_mul_left' {g : ℝ → F} (hg : Integrable g) {R : ℝ} (hR : R ≠ 0) :
    Integrable fun x => g (R * x) 
goal: MeasureTheory.Integrable (fun x => g (R * x)) MeasureTheory.volume

full_name: MeasureTheory.integrable_comp_mul_right_iff
code: theorem integrable_comp_mul_right_iff (g : ℝ → F) {R : ℝ} (hR : R ≠ 0) :
    (Integrable fun x => g (x * R)) ↔ Integrable g 
goal: MeasureTheory.Integrable (fun x => g (x * R)) MeasureTheory.volume ↔ MeasureTheory.Integrable g MeasureTheory.volume

full_name: MeasureTheory.Integrable.comp_mul_right'
code: theorem Integrable.comp_mul_right' {g : ℝ → F} (hg : Integrable g) {R : ℝ} (hR : R ≠ 0) :
    Integrable fun x => g (x * R) 
goal: MeasureTheory.Integrable (fun x => g (x * R)) MeasureTheory.volume

full_name: MeasureTheory.integrable_comp_div_iff
code: theorem integrable_comp_div_iff (g : ℝ → F) {R : ℝ} (hR : R ≠ 0) :
    (Integrable fun x => g (x / R)) ↔ Integrable g 
goal: MeasureTheory.Integrable (fun x => g (x / R)) MeasureTheory.volume ↔ MeasureTheory.Integrable g MeasureTheory.volume

full_name: MeasureTheory.Integrable.comp_div
code: theorem Integrable.comp_div {g : ℝ → F} (hg : Integrable g) {R : ℝ} (hR : R ≠ 0) :
    Integrable fun x => g (x / R) 
goal: MeasureTheory.Integrable (fun x => g (x / R)) MeasureTheory.volume

full_name: MeasureTheory.integrable_comp
code: theorem integrable_comp (g : F' → A) : Integrable (g ∘ f) ↔ Integrable g 
goal: MeasureTheory.Integrable (g ∘ ⇑f) MeasureTheory.volume ↔ MeasureTheory.Integrable g MeasureTheory.volume

full_name: MeasureTheory.integral_comp
code: theorem integral_comp (g : F' → A) : ∫ (x : E'), g (f x) = ∫ (y : F'), g y 
goal: ∫ (x : E'), g (f x) = ∫ (y : F'), g y