File: Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean

full_name: MeasureTheory.measure_unitBall_eq_integral_div_gamma
code: theorem MeasureTheory.measure_unitBall_eq_integral_div_gamma {E : Type*} {p : ℝ}
    [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E]
    [BorelSpace E] (μ : Measure E) [IsAddHaarMeasure μ] (hp : 0 < p) :
    μ (Metric.ball 0 1) =
      .ofReal ((∫ (x : E), Real.exp (- ‖x‖ ^ p) ∂μ) / Real.Gamma (finrank ℝ E / p + 1)) 
goal: μ (Metric.ball 0 1) =
  ENNReal.ofReal ((∫ (x : E), Real.exp (-‖x‖ ^ p) ∂μ) / Real.Gamma (↑(FiniteDimensional.finrank ℝ E) / p + 1))

full_name: MeasureTheory.measure_lt_one_eq_integral_div_gamma
code: theorem MeasureTheory.measure_lt_one_eq_integral_div_gamma {p : ℝ} (hp : 0 < p) :
    μ {x : E | g x < 1} =
      .ofReal ((∫ (x : E), Real.exp (- (g x) ^ p) ∂μ) / Real.Gamma (finrank ℝ E / p + 1)) 
goal: μ {x | g x < 1} =
  ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(FiniteDimensional.finrank ℝ E) / p + 1))

full_name: MeasureTheory.measure_le_eq_lt
code: theorem MeasureTheory.measure_le_eq_lt [Nontrivial E] (r : ℝ) :
    μ {x : E | g x ≤ r} = μ {x : E | g x < r} 
goal: μ {x | g x ≤ r} = μ {x | g x < r}

full_name: MeasureTheory.volume_sum_rpow_lt_one
code: theorem MeasureTheory.volume_sum_rpow_lt_one :
    volume {x : ι → ℝ | ∑ i, |x i| ^ p < 1} =
      .ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (card ι / p + 1)) 
goal: MeasureTheory.volume {x | ∑ i : ι, |x i| ^ p < 1} =
  ENNReal.ofReal ((2 * Real.Gamma (1 / p + 1)) ^ Fintype.card ι / Real.Gamma (↑(Fintype.card ι) / p + 1))

full_name: MeasureTheory.volume_sum_rpow_lt
code: theorem MeasureTheory.volume_sum_rpow_lt [Nonempty ι] {p : ℝ} (hp : 1 ≤ p) (r : ℝ) :
    volume {x : ι → ℝ | (∑ i, |x i| ^ p) ^ (1 / p) < r} = (.ofReal r) ^ card ι *
      .ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (card ι / p + 1)) 
goal: MeasureTheory.volume {x | (∑ i : ι, |x i| ^ p) ^ (1 / p) < r} =
  ENNReal.ofReal r ^ Fintype.card ι *
    ENNReal.ofReal ((2 * Real.Gamma (1 / p + 1)) ^ Fintype.card ι / Real.Gamma (↑(Fintype.card ι) / p + 1))

full_name: MeasureTheory.volume_sum_rpow_le
code: theorem MeasureTheory.volume_sum_rpow_le [Nonempty ι] {p : ℝ} (hp : 1 ≤ p) (r : ℝ) :
    volume {x : ι → ℝ | (∑ i, |x i| ^ p) ^ (1 / p) ≤ r} = (.ofReal r) ^ card ι *
      .ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (card ι / p + 1)) 
goal: MeasureTheory.volume {x | (∑ i : ι, |x i| ^ p) ^ (1 / p) ≤ r} =
  ENNReal.ofReal r ^ Fintype.card ι *
    ENNReal.ofReal ((2 * Real.Gamma (1 / p + 1)) ^ Fintype.card ι / Real.Gamma (↑(Fintype.card ι) / p + 1))

full_name: Complex.volume_sum_rpow_lt_one
code: theorem Complex.volume_sum_rpow_lt_one {p : ℝ} (hp : 1 ≤ p) :
    volume {x : ι → ℂ | ∑ i, ‖x i‖ ^ p < 1} =
      .ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * card ι / p + 1)) 
goal: MeasureTheory.volume {x | ∑ i : ι, ‖x i‖ ^ p < 1} =
  ENNReal.ofReal ((Real.pi * Real.Gamma (2 / p + 1)) ^ Fintype.card ι / Real.Gamma (2 * ↑(Fintype.card ι) / p + 1))

full_name: Complex.volume_sum_rpow_lt
code: theorem Complex.volume_sum_rpow_lt [Nonempty ι] {p : ℝ} (hp : 1 ≤ p) (r : ℝ) :
    volume {x : ι → ℂ | (∑ i, ‖x i‖ ^ p) ^ (1 / p) < r} = (.ofReal r) ^ (2 * card ι) *
      .ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * card ι / p + 1)) 
goal: MeasureTheory.volume {x | (∑ i : ι, ‖x i‖ ^ p) ^ (1 / p) < r} =
  ENNReal.ofReal r ^ (2 * Fintype.card ι) *
    ENNReal.ofReal ((Real.pi * Real.Gamma (2 / p + 1)) ^ Fintype.card ι / Real.Gamma (2 * ↑(Fintype.card ι) / p + 1))

full_name: Complex.volume_sum_rpow_le
code: theorem Complex.volume_sum_rpow_le [Nonempty ι] {p : ℝ} (hp : 1 ≤ p) (r : ℝ) :
    volume {x : ι → ℂ | (∑ i, ‖x i‖ ^ p) ^ (1 / p) ≤ r} = (.ofReal r) ^ (2 * card ι) *
      .ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * card ι / p + 1)) 
goal: MeasureTheory.volume {x | (∑ i : ι, ‖x i‖ ^ p) ^ (1 / p) ≤ r} =
  ENNReal.ofReal r ^ (2 * Fintype.card ι) *
    ENNReal.ofReal ((Real.pi * Real.Gamma (2 / p + 1)) ^ Fintype.card ι / Real.Gamma (2 * ↑(Fintype.card ι) / p + 1))

full_name: EuclideanSpace.volume_ball
code: theorem EuclideanSpace.volume_ball (x : EuclideanSpace ℝ ι) (r : ℝ) :
    volume (Metric.ball x r) = (.ofReal r) ^ card ι *
      .ofReal (Real.sqrt π ^ card ι / Gamma (card ι / 2 + 1)) 
goal: MeasureTheory.volume (Metric.ball x r) =
  ENNReal.ofReal r ^ Fintype.card ι *
    ENNReal.ofReal (√Real.pi ^ Fintype.card ι / Real.Gamma (↑(Fintype.card ι) / 2 + 1))

full_name: EuclideanSpace.volume_closedBall
code: theorem EuclideanSpace.volume_closedBall (x : EuclideanSpace ℝ ι) (r : ℝ) :
    volume (Metric.closedBall x r) = (.ofReal r) ^ card ι *
      .ofReal (sqrt π ^ card ι / Gamma (card ι / 2 + 1)) 
goal: MeasureTheory.volume (Metric.closedBall x r) =
  ENNReal.ofReal r ^ Fintype.card ι *
    ENNReal.ofReal (√Real.pi ^ Fintype.card ι / Real.Gamma (↑(Fintype.card ι) / 2 + 1))

full_name: InnerProductSpace.volume_ball
code: theorem InnerProductSpace.volume_ball (x : E) (r : ℝ) :
    volume (Metric.ball x r) = (.ofReal r) ^ finrank ℝ E *
      .ofReal (sqrt π ^ finrank ℝ E / Gamma (finrank ℝ E / 2 + 1)) 
goal: MeasureTheory.volume (Metric.ball x r) =
  ENNReal.ofReal r ^ FiniteDimensional.finrank ℝ E *
    ENNReal.ofReal (√Real.pi ^ FiniteDimensional.finrank ℝ E / Real.Gamma (↑(FiniteDimensional.finrank ℝ E) / 2 + 1))

full_name: InnerProductSpace.volume_closedBall
code: theorem InnerProductSpace.volume_closedBall (x : E) (r : ℝ) :
    volume (Metric.closedBall x r) = (.ofReal r) ^ finrank ℝ E *
      .ofReal (sqrt π ^ finrank ℝ E / Gamma (finrank ℝ E / 2 + 1)) 
goal: MeasureTheory.volume (Metric.closedBall x r) =
  ENNReal.ofReal r ^ FiniteDimensional.finrank ℝ E *
    ENNReal.ofReal (√Real.pi ^ FiniteDimensional.finrank ℝ E / Real.Gamma (↑(FiniteDimensional.finrank ℝ E) / 2 + 1))

full_name: Complex.volume_ball
code: @[simp]
theorem Complex.volume_ball (a : ℂ) (r : ℝ) :
    volume (Metric.ball a r) = .ofReal r ^ 2 * NNReal.pi 
goal: MeasureTheory.volume (Metric.ball a r) = ENNReal.ofReal r ^ 2 * ↑NNReal.pi

full_name: Complex.volume_closedBall
code: @[simp]
theorem Complex.volume_closedBall (a : ℂ) (r : ℝ) :
    volume (Metric.closedBall a r) = .ofReal r ^ 2 * NNReal.pi 
goal: MeasureTheory.volume (Metric.closedBall a r) = ENNReal.ofReal r ^ 2 * ↑NNReal.pi