Box-additive functions defined by measures

In this file we prove a few simple facts about rectangular boxes, partitions, and measures:

• given a box I : Box ι, its coercion to Set (ι → ℝ) and I.Icc are measurable sets;
• if μ is a locally finite measure, then (I : Set (ι → ℝ)) and I.Icc have finite measure;
• if μ is a locally finite measure, then fun J ↦ (μ J).toReal is a box additive function.

For the last statement, we both prove it as a proposition and define a bundled BoxIntegral.BoxAdditiveMap function.

Tags

rectangular box, measure

theorem BoxIntegral.Box.measure_Icc_lt_top {ι : Type u_1} (I : BoxIntegral.Box ι) (μ : MeasureTheory.Measure (ι → ℝ)) [MeasureTheory.IsLocallyFiniteMeasure μ] : ↑↑μ (BoxIntegral.Box.Icc I) < ⊤

theorem BoxIntegral.Box.measure_coe_lt_top {ι : Type u_1} (I : BoxIntegral.Box ι) (μ : MeasureTheory.Measure (ι → ℝ)) [MeasureTheory.IsLocallyFiniteMeasure μ] : ↑↑μ ↑I < ⊤

theorem BoxIntegral.Box.measurableSet_coe {ι : Type u_1} (I : BoxIntegral.Box ι) [Countable ι] : MeasurableSet ↑I

theorem BoxIntegral.Box.measurableSet_Icc {ι : Type u_1} (I : BoxIntegral.Box ι) [Countable ι] : MeasurableSet (BoxIntegral.Box.Icc I)

theorem BoxIntegral.Box.measurableSet_Ioo {ι : Type u_1} (I : BoxIntegral.Box ι) [Countable ι] : MeasurableSet (BoxIntegral.Box.Ioo I)

theorem BoxIntegral.Box.coe_ae_eq_Icc {ι : Type u_1} (I : BoxIntegral.Box ι) [Fintype ι] : ↑I =ᶠ[MeasureTheory.Measure.ae MeasureTheory.volume] BoxIntegral.Box.Icc I

theorem BoxIntegral.Box.Ioo_ae_eq_Icc {ι : Type u_1} (I : BoxIntegral.Box ι) [Fintype ι] : BoxIntegral.Box.Ioo I =ᶠ[MeasureTheory.Measure.ae MeasureTheory.volume] BoxIntegral.Box.Icc I

theorem BoxIntegral.Prepartition.measure_iUnion_toReal {ι : Type u_1} [Finite ι] {I : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (μ : MeasureTheory.Measure (ι → ℝ)) [MeasureTheory.IsLocallyFiniteMeasure μ] : (↑↑μ (BoxIntegral.Prepartition.iUnion π)).toReal = Finset.sum π.boxes fun (J : BoxIntegral.Box ι) => (↑↑μ ↑J).toReal

@[simp]

theorem MeasureTheory.Measure.toBoxAdditive_apply {ι : Type u_1} [Finite ι] (μ : MeasureTheory.Measure (ι → ℝ)) [MeasureTheory.IsLocallyFiniteMeasure μ] (J : BoxIntegral.Box ι) : (MeasureTheory.Measure.toBoxAdditive μ) J = (↑↑μ ↑J).toReal

def MeasureTheory.Measure.toBoxAdditive {ι : Type u_1} [Finite ι] (μ : MeasureTheory.Measure (ι → ℝ)) [MeasureTheory.IsLocallyFiniteMeasure μ] : BoxIntegral.BoxAdditiveMap ι ℝ ⊤

If μ is a locally finite measure on ℝⁿ, then fun J ↦ (μ J).toReal is a box-additive function.

Equations

• MeasureTheory.Measure.toBoxAdditive μ = { toFun := fun (J : BoxIntegral.Box ι) => (↑↑μ ↑J).toReal, sum_partition_boxes' := ⋯ }

theorem BoxIntegral.Box.volume_apply {ι : Type u_1} [Fintype ι] (I : BoxIntegral.Box ι) : (MeasureTheory.Measure.toBoxAdditive MeasureTheory.volume) I = Finset.prod Finset.univ fun (i : ι) => I.upper i - I.lower i

@[simp]

theorem BoxIntegral.Box.volume_apply' {ι : Type u_1} [Fintype ι] (I : BoxIntegral.Box ι) : (↑↑MeasureTheory.volume ↑I).toReal = Finset.prod Finset.univ fun (i : ι) => I.upper i - I.lower i

theorem BoxIntegral.Box.volume_face_mul {n : ℕ} (i : Fin (n + 1)) (I : BoxIntegral.Box (Fin (n + 1))) : (Finset.prod Finset.univ fun (j : Fin n) => (BoxIntegral.Box.face I i).upper j - (BoxIntegral.Box.face I i).lower j) * (I.upper i - I.lower i) = Finset.prod Finset.univ fun (j : Fin (n + 1)) => I.upper j - I.lower j

def BoxIntegral.BoxAdditiveMap.volume {ι : Type u_1} [Fintype ι] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] : BoxIntegral.BoxAdditiveMap ι (E →L[ℝ] E) ⊤

Box-additive map sending each box I to the continuous linear endomorphism x ↦ (volume I).toReal • x.

Equations

• BoxIntegral.BoxAdditiveMap.volume = BoxIntegral.BoxAdditiveMap.toSMul (MeasureTheory.Measure.toBoxAdditive MeasureTheory.volume)

theorem BoxIntegral.BoxAdditiveMap.volume_apply {ι : Type u_1} [Fintype ι] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (I : BoxIntegral.Box ι) (x : E) : (BoxIntegral.BoxAdditiveMap.volume I) x = (Finset.prod Finset.univ fun (j : ι) => I.upper j - I.lower j) • x