# mathlibdocumentation

analysis.box_integral.partition.measure

# 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 λ J, (μ J).to_real is a box additive function.

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

### Tags #

rectangular box, measure

theorem box_integral.box.measure_Icc_lt_top {ι : Type u_1} (I : box_integral.box ι) (μ : measure_theory.measure (ι → ))  :
theorem box_integral.box.measure_coe_lt_top {ι : Type u_1} (I : box_integral.box ι) (μ : measure_theory.measure (ι → ))  :
theorem box_integral.box.measurable_set_coe {ι : Type u_1} [fintype ι] (I : box_integral.box ι) :
theorem box_integral.box.measurable_set_Icc {ι : Type u_1} [fintype ι] (I : box_integral.box ι) :
theorem box_integral.box.measurable_set_Ioo {ι : Type u_1} [fintype ι] (I : box_integral.box ι) :
theorem box_integral.box.coe_ae_eq_Icc {ι : Type u_1} [fintype ι] (I : box_integral.box ι) :
theorem box_integral.box.Ioo_ae_eq_Icc {ι : Type u_1} [fintype ι] (I : box_integral.box ι) :
theorem box_integral.prepartition.measure_Union_to_real {ι : Type u_1} [fintype ι] {I : box_integral.box ι} (μ : measure_theory.measure (ι → ))  :
(μ π.Union).to_real = π.boxes.sum (λ (J : , (μ J).to_real)
@[simp]
noncomputable def measure_theory.measure.to_box_additive {ι : Type u_1} [fintype ι] (μ : measure_theory.measure (ι → ))  :

If μ is a locally finite measure on ℝⁿ, then λ J, (μ J).to_real is a box-additive function.

Equations
@[simp]
theorem box_integral.box.volume_apply {ι : Type u_1} [fintype ι] (I : box_integral.box ι) :
= finset.univ.prod (λ (i : ι), I.upper i - I.lower i)
theorem box_integral.box.volume_face_mul {n : } (i : fin (n + 1)) (I : box_integral.box (fin (n + 1))) :
finset.univ.prod (λ (j : fin n), (I.face i).upper j - (I.face i).lower j) * (I.upper i - I.lower i) = finset.univ.prod (λ (j : fin (n + 1)), I.upper j - I.lower j)
@[protected]
noncomputable def box_integral.box_additive_map.volume {ι : Type u_1} [fintype ι] {E : Type u_2} [normed_group E] [ E] :

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

Equations
theorem box_integral.box_additive_map.volume_apply {ι : Type u_1} [fintype ι] {E : Type u_2} [normed_group E] [ E] (I : box_integral.box ι) (x : E) :
= finset.univ.prod (λ (j : ι), I.upper j - I.lower j) x