File: Mathlib/Analysis/BoxIntegral/Partition/Measure.lean

full_name: BoxIntegral.Box.measure_Icc_lt_top
code: theorem measure_Icc_lt_top (μ : Measure (ι → ℝ)) [IsLocallyFiniteMeasure μ] : μ (Box.Icc I) < ∞ 
goal: μ (BoxIntegral.Box.Icc I) < ⊤

full_name: BoxIntegral.Box.measure_coe_lt_top
code: theorem measure_coe_lt_top (μ : Measure (ι → ℝ)) [IsLocallyFiniteMeasure μ] : μ I < ∞ 
goal: μ ↑I < ⊤

full_name: BoxIntegral.Box.measurableSet_coe
code: theorem measurableSet_coe : MeasurableSet (I : Set (ι → ℝ)) 
goal: MeasurableSet ↑I

full_name: BoxIntegral.Box.measurableSet_Icc
code: theorem measurableSet_Icc : MeasurableSet (Box.Icc I) 
goal: MeasurableSet (BoxIntegral.Box.Icc I)

full_name: BoxIntegral.Box.measurableSet_Ioo
code: theorem measurableSet_Ioo : MeasurableSet (Box.Ioo I) 
goal: MeasurableSet (BoxIntegral.Box.Ioo I)

full_name: BoxIntegral.Box.coe_ae_eq_Icc
code: theorem coe_ae_eq_Icc : (I : Set (ι → ℝ)) =ᵐ[volume] Box.Icc I 
goal: ↑I =ᵐ[MeasureTheory.volume] BoxIntegral.Box.Icc I

full_name: BoxIntegral.Box.Ioo_ae_eq_Icc
code: theorem Ioo_ae_eq_Icc : Box.Ioo I =ᵐ[volume] Box.Icc I 
goal: BoxIntegral.Box.Ioo I =ᵐ[MeasureTheory.volume] BoxIntegral.Box.Icc I

full_name: BoxIntegral.Prepartition.measure_iUnion_toReal
code: theorem Prepartition.measure_iUnion_toReal [Finite ι] {I : Box ι} (π : Prepartition I)
    (μ : Measure (ι → ℝ)) [IsLocallyFiniteMeasure μ] :
    (μ π.iUnion).toReal = ∑ J ∈ π.boxes, (μ J).toReal 
goal: (μ π.iUnion).toReal = ∑ J ∈ π.boxes, (μ ↑J).toReal

full_name: MeasureTheory.Measure.toBoxAdditive
code: /-- If `μ` is a locally finite measure on `ℝⁿ`, then `fun J ↦ (μ J).toReal` is a box-additive
function. -/
@[simps]
def toBoxAdditive [Finite ι] (μ : Measure (ι → ℝ)) [IsLocallyFiniteMeasure μ] : ι →ᵇᵃ[⊤] ℝ 
goal: ι →ᵇᵃ[⊤] ℝ

full_name: MeasureTheory.Measure.toBoxAdditive_apply
code: /-- If `μ` is a locally finite measure on `ℝⁿ`, then `fun J ↦ (μ J).toReal` is a box-additive
function. -/
@[simps]
def toBoxAdditive [Finite ι] (μ : Measure (ι → ℝ)) [IsLocallyFiniteMeasure μ] : ι →ᵇᵃ[⊤] ℝ 
goal: μ.toBoxAdditive J = (μ ↑J).toReal

full_name: BoxIntegral.Box.volume_apply
code: theorem volume_apply (I : Box ι) :
    (volume : Measure (ι → ℝ)).toBoxAdditive I = ∏ i, (I.upper i - I.lower i) 
goal: MeasureTheory.volume.toBoxAdditive I = ∏ i : ι, (I.upper i - I.lower i)

full_name: BoxIntegral.Box.volume_apply'
code: @[simp]
theorem volume_apply' (I : Box ι) :
    ((volume : Measure (ι → ℝ)) I).toReal = ∏ i, (I.upper i - I.lower i) 
goal: (MeasureTheory.volume ↑I).toReal = ∏ i : ι, (I.upper i - I.lower i)

full_name: BoxIntegral.Box.volume_face_mul
code: theorem volume_face_mul {n} (i : Fin (n + 1)) (I : Box (Fin (n + 1))) :
    (∏ j, ((I.face i).upper j - (I.face i).lower j)) * (I.upper i - I.lower i) =
      ∏ j, (I.upper j - I.lower j) 
goal: (∏ j : Fin n, ((I.face i).upper j - (I.face i).lower j)) * (I.upper i - I.lower i) =
  ∏ j : Fin (n + 1), (I.upper j - I.lower j)

full_name: BoxIntegral.BoxAdditiveMap.volume
code: /-- Box-additive map sending each box `I` to the continuous linear endomorphism
`x ↦ (volume I).toReal • x`. -/
protected def volume {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] : ι →ᵇᵃ E →L[ℝ] E 
goal: ι →ᵇᵃ[⊤] E →L[ℝ] E

full_name: BoxIntegral.BoxAdditiveMap.volume_apply
code: theorem volume_apply {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] (I : Box ι) (x : E) :
    BoxAdditiveMap.volume I x = (∏ j, (I.upper j - I.lower j)) • x 
goal: (BoxIntegral.BoxAdditiveMap.volume I) x = (∏ j : ι, (I.upper j - I.lower j)) • x