Theorem: integrable_one_add_norm
File: Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean

tactic: constructor
state_before: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
⊢ Integrable (fun x => (1 + ‖x‖) ^ (-r)) μ
state_after: case left
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
⊢ AEStronglyMeasurable (fun x => (1 + ‖x‖) ^ (-r)) μ

case right
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
⊢ HasFiniteIntegral (fun x => (1 + ‖x‖) ^ (-r)) μ

tactic: have : (∫⁻ a : E, ‖(1 + ‖a‖) ^ (-r)‖₊ ∂μ) = ∫⁻ a : E, ENNReal.ofReal ((1 + ‖a‖) ^ (-r)) ∂μ :=
  lintegral_nnnorm_eq_of_nonneg fun _ => rpow_nonneg (by positivity) _
state_before: case right
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
⊢ HasFiniteIntegral (fun x => (1 + ‖x‖) ^ (-r)) μ
state_after: case right
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
this : ∫⁻ (a : E), ↑‖(1 + ‖a‖) ^ (-r)‖₊ ∂μ = ∫⁻ (a : E), ENNReal.ofReal ((1 + ‖a‖) ^ (-r)) ∂μ
⊢ HasFiniteIntegral (fun x => (1 + ‖x‖) ^ (-r)) μ

tactic: rw [HasFiniteIntegral, this]
state_before: case right
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
this : ∫⁻ (a : E), ↑‖(1 + ‖a‖) ^ (-r)‖₊ ∂μ = ∫⁻ (a : E), ENNReal.ofReal ((1 + ‖a‖) ^ (-r)) ∂μ
⊢ HasFiniteIntegral (fun x => (1 + ‖x‖) ^ (-r)) μ
state_after: case right
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
this : ∫⁻ (a : E), ↑‖(1 + ‖a‖) ^ (-r)‖₊ ∂μ = ∫⁻ (a : E), ENNReal.ofReal ((1 + ‖a‖) ^ (-r)) ∂μ
⊢ ∫⁻ (a : E), ENNReal.ofReal ((1 + ‖a‖) ^ (-r)) ∂μ < ⊤

tactic: exact finite_integral_one_add_norm hnr
state_before: case right
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
this : ∫⁻ (a : E), ↑‖(1 + ‖a‖) ^ (-r)‖₊ ∂μ = ∫⁻ (a : E), ENNReal.ofReal ((1 + ‖a‖) ^ (-r)) ∂μ
⊢ ∫⁻ (a : E), ENNReal.ofReal ((1 + ‖a‖) ^ (-r)) ∂μ < ⊤
state_after: no goals

tactic: measurability
state_before: case left
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
⊢ AEStronglyMeasurable (fun x => (1 + ‖x‖) ^ (-r)) μ
state_after: no goals

tactic: positivity
state_before: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
x✝ : E
⊢ 0 ≤ 1 + ‖x✝‖
state_after: no goals