File: Mathlib/MeasureTheory/Constructions/HaarToSphere.lean

full_name: _aux_Mathlib_MeasureTheory_Constructions_HaarToSphere___unexpand_FiniteDimensional_finrank_1
code: local notation "dim" => FiniteDimensional.finrank ℝ
goal: Lean.PrettyPrinter.Unexpander

full_name: _aux_Mathlib_MeasureTheory_Constructions_HaarToSphere___macroRules_termDim_1
code: local notation "dim" => FiniteDimensional.finrank ℝ
goal: Lean.Macro

full_name: MeasureTheory.Measure.toSphere
code: /-- If `μ` is an additive Haar measure on a normed space `E`,
then `μ.toSphere` is the measure on the unit sphere in `E`
such that `μ.toSphere s = FiniteDimensional.finrank ℝ E • μ (Set.Ioo (0 : ℝ) 1 • s)`. -/
def toSphere (μ : Measure E) : Measure (sphere (0 : E) 1) 
goal: MeasureTheory.Measure ↑(Metric.sphere 0 1)

full_name: MeasureTheory.Measure.toSphere_apply_aux
code: theorem toSphere_apply_aux (s : Set (sphere (0 : E) 1)) (r : Ioi (0 : ℝ)) :
    μ ((↑) '' (homeomorphUnitSphereProd E ⁻¹' s ×ˢ Iio r)) = μ (Ioo (0 : ℝ) r • ((↑) '' s)) 
goal: μ (Subtype.val '' (⇑(homeomorphUnitSphereProd E) ⁻¹' s ×ˢ Set.Iio r)) = μ (Set.Ioo 0 ↑r • Subtype.val '' s)

full_name: MeasureTheory.Measure.toSphere_apply'
code: theorem toSphere_apply' {s : Set (sphere (0 : E) 1)} (hs : MeasurableSet s) :
    μ.toSphere s = dim E * μ (Ioo (0 : ℝ) 1 • ((↑) '' s)) 
goal: μ.toSphere s = ↑(dim E) * μ (Set.Ioo 0 1 • Subtype.val '' s)

full_name: MeasureTheory.Measure.toSphere_apply_univ'
code: theorem toSphere_apply_univ' : μ.toSphere univ = dim E * μ (ball 0 1 \ {0}) 
goal: μ.toSphere Set.univ = ↑(dim E) * μ (Metric.ball 0 1 \ {0})

full_name: MeasureTheory.Measure.toSphere_apply_univ
code: @[simp]
theorem toSphere_apply_univ : μ.toSphere univ = dim E * μ (ball 0 1) 
goal: μ.toSphere Set.univ = ↑(dim E) * μ (Metric.ball 0 1)

full_name: MeasureTheory.Measure.instIsFiniteMeasureElemSphereOfNatRealToSphere
code: instance : IsFiniteMeasure μ.toSphere 
goal: MeasureTheory.IsFiniteMeasure μ.toSphere

full_name: MeasureTheory.Measure.volumeIoiPow
code: /-- The measure on `(0, +∞)` that has density `(· ^ n)` with respect to the Lebesgue measure. -/
def volumeIoiPow (n : ℕ) : Measure (Ioi (0 : ℝ)) 
goal: MeasureTheory.Measure ↑(Set.Ioi 0)

full_name: MeasureTheory.Measure.volumeIoiPow_apply_Iio
code: lemma volumeIoiPow_apply_Iio (n : ℕ) (x : Ioi (0 : ℝ)) :
    volumeIoiPow n (Iio x) = ENNReal.ofReal (x.1 ^ (n + 1) / (n + 1)) 
goal: (MeasureTheory.Measure.volumeIoiPow n) (Set.Iio x) = ENNReal.ofReal (↑x ^ (n + 1) / (↑n + 1))

full_name: MeasureTheory.Measure.finiteSpanningSetsIn_volumeIoiPow_range_Iio
code: /-- The intervals `(0, k + 1)` have finite measure `MeasureTheory.Measure.volumeIoiPow _`
and cover the whole open ray `(0, +∞)`. -/
def finiteSpanningSetsIn_volumeIoiPow_range_Iio (n : ℕ) :
    FiniteSpanningSetsIn (volumeIoiPow n) (range Iio) 
goal: (MeasureTheory.Measure.volumeIoiPow n).FiniteSpanningSetsIn (Set.range Set.Iio)

full_name: MeasureTheory.Measure.instSigmaFiniteElemRealIoiOfNatVolumeIoiPow
code: instance (n : ℕ) : SigmaFinite (volumeIoiPow n) 
goal: MeasureTheory.SigmaFinite (MeasureTheory.Measure.volumeIoiPow n)

full_name: MeasureTheory.Measure.measurePreserving_homeomorphUnitSphereProd
code: /-- The homeomorphism `homeomorphUnitSphereProd E` sends an additive Haar measure `μ`
to the product of `μ.toSphere` and `MeasureTheory.Measure.volumeIoiPow (dim E - 1)`,
where `dim E = FiniteDimensional.finrank ℝ E` is the dimension of `E`. -/
theorem measurePreserving_homeomorphUnitSphereProd :
    MeasurePreserving (homeomorphUnitSphereProd E) (μ.comap (↑))
      (μ.toSphere.prod (volumeIoiPow (dim E - 1))) 
goal: MeasureTheory.MeasurePreserving (⇑(homeomorphUnitSphereProd E)) (MeasureTheory.Measure.comap Subtype.val μ)
  (μ.toSphere.prod (MeasureTheory.Measure.volumeIoiPow (dim E - 1)))

full_name: MeasureTheory.integral_fun_norm_addHaar
code: lemma integral_fun_norm_addHaar (f : ℝ → F) :
    ∫ x, f (‖x‖) ∂μ = dim E • (μ (ball 0 1)).toReal • ∫ y in Ioi (0 : ℝ), y ^ (dim E - 1) • f y 
goal: ∫ (x : E), f ‖x‖ ∂μ = dim E • (μ (Metric.ball 0 1)).toReal • ∫ (y : ℝ) in Set.Ioi 0, y ^ (dim E - 1) • f y