Box additive functions

We say that a function f : Box ι → M from boxes in ℝⁿ to a commutative additive monoid M is box additive on subboxes of I₀ : WithTop (Box ι) if for any box J, ↑J ≤ I₀, and a partition π of J, f J = ∑ J' ∈ π.boxes, f J'. We use I₀ : WithTop (Box ι) instead of I₀ : Box ι to use the same definition for functions box additive on subboxes of a box and for functions box additive on all boxes.

Examples of box-additive functions include the measure of a box and the integral of a fixed integrable function over a box.

In this file we define box-additive functions and prove that a function such that f J = f (J ∩ {x | x i < y}) + f (J ∩ {x | y ≤ x i}) is box-additive.

Tags

rectangular box, additive function

structure BoxIntegral.BoxAdditiveMap (ι : Type u_3) (M : Type u_4) [AddCommMonoid M] (I : WithTop (Box ι)) : Type (max u_3 u_4)

A function on Box ι is called box additive if for every box J and a partition π of J we have f J = ∑ Ji ∈ π.boxes, f Ji. A function is called box additive on subboxes of I : Box ι if the same property holds for J ≤ I. We formalize these two notions in the same definition using I : WithBot (Box ι): the value I = ⊤ corresponds to functions box additive on the whole space.

• toFun : Box ι → M

  The function underlying this additive map.

• sum_partition_boxes' (J : Box ι) : ↑J ≤ I → ∀ (π : Prepartition J), π.IsPartition → ∑ Ji ∈ π.boxes, self.toFun Ji = self.toFun J

def BoxIntegral.«term_→ᵇᵃ_» : Lean.TrailingParserDescr

A function on Box ι is called box additive if for every box J and a partition π of J we have f J = ∑ Ji ∈ π.boxes, f Ji.

def BoxIntegral.«term_→ᵇᵃ[_]_» : Lean.TrailingParserDescr

A function on Box ι is called box additive if for every box J and a partition π of J we have f J = ∑ Ji ∈ π.boxes, f Ji. A function is called box additive on subboxes of I : Box ι if the same property holds for J ≤ I. We formalize these two notions in the same definition using I : WithBot (Box ι): the value I = ⊤ corresponds to functions box additive on the whole space.

instance BoxIntegral.BoxAdditiveMap.instFunLikeBox {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (Box ι)} : FunLike (BoxAdditiveMap ι M I₀) (Box ι) M

@[simp]

theorem BoxIntegral.BoxAdditiveMap.coe_mk {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (Box ι)} (f : Box ι → M) (h : ∀ (J : Box ι), ↑J ≤ I₀ → ∀ (π : Prepartition J), π.IsPartition → ∑ Ji ∈ π.boxes, f Ji = f J) : ⇑{ toFun := f, sum_partition_boxes' := h } = f

theorem BoxIntegral.BoxAdditiveMap.coe_injective {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (Box ι)} : Function.Injective fun (f : BoxAdditiveMap ι M I₀) (x : Box ι) => f x

theorem BoxIntegral.BoxAdditiveMap.coe_inj {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (Box ι)} {f g : BoxAdditiveMap ι M I₀} : ⇑f = ⇑g ↔ f = g

theorem BoxIntegral.BoxAdditiveMap.sum_partition_boxes {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (Box ι)} {I : Box ι} (f : BoxAdditiveMap ι M I₀) (hI : ↑I ≤ I₀) {π : Prepartition I} (h : π.IsPartition) : ∑ J ∈ π.boxes, f J = f I

instance BoxIntegral.BoxAdditiveMap.instZero {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (Box ι)} : Zero (BoxAdditiveMap ι M I₀)

@[simp]

theorem BoxIntegral.BoxAdditiveMap.zero_apply {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (Box ι)} : ⇑0 = 0

instance BoxIntegral.BoxAdditiveMap.instInhabited {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (Box ι)} : Inhabited (BoxAdditiveMap ι M I₀)

instance BoxIntegral.BoxAdditiveMap.instAdd {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (Box ι)} : Add (BoxAdditiveMap ι M I₀)

instance BoxIntegral.BoxAdditiveMap.instSMulOfDistribMulAction {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (Box ι)} {R : Type u_4} [Monoid R] [DistribMulAction R M] : SMul R (BoxAdditiveMap ι M I₀)

instance BoxIntegral.BoxAdditiveMap.instAddCommMonoid {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (Box ι)} : AddCommMonoid (BoxAdditiveMap ι M I₀)

@[simp]

theorem BoxIntegral.BoxAdditiveMap.map_split_add {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (Box ι)} {I : Box ι} (f : BoxAdditiveMap ι M I₀) (hI : ↑I ≤ I₀) (i : ι) (x : ℝ) : Option.elim' 0 (⇑f) (I.splitLower i x) + Option.elim' 0 (⇑f) (I.splitUpper i x) = f I

def BoxIntegral.BoxAdditiveMap.restrict {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (Box ι)} (f : BoxAdditiveMap ι M I₀) (I : WithTop (Box ι)) (hI : I ≤ I₀) : BoxAdditiveMap ι M I

If f is box-additive on subboxes of I₀, then it is box-additive on subboxes of any I ≤ I₀.

@[simp]

theorem BoxIntegral.BoxAdditiveMap.restrict_apply {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (Box ι)} (f : BoxAdditiveMap ι M I₀) (I : WithTop (Box ι)) (hI : I ≤ I₀) (a : Box ι) : (f.restrict I hI) a = f a

def BoxIntegral.BoxAdditiveMap.ofMapSplitAdd {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] [Finite ι] (f : Box ι → M) (I₀ : WithTop (Box ι)) (hf : ∀ (I : Box ι), ↑I ≤ I₀ → ∀ {i : ι} {x : ℝ}, x ∈ Set.Ioo (I.lower i) (I.upper i) → Option.elim' 0 f (I.splitLower i x) + Option.elim' 0 f (I.splitUpper i x) = f I) : BoxAdditiveMap ι M I₀

If f : Box ι → M is box additive on partitions of the form split I i x, then it is box additive.

def BoxIntegral.BoxAdditiveMap.map {ι : Type u_1} {M : Type u_2} {N : Type u_3} [AddCommMonoid M] [AddCommMonoid N] {I₀ : WithTop (Box ι)} (f : BoxAdditiveMap ι M I₀) (g : M →+ N) : BoxAdditiveMap ι N I₀

If g : M → N is an additive map and f is a box additive map, then g ∘ f is a box additive map.

@[simp]

theorem BoxIntegral.BoxAdditiveMap.map_apply {ι : Type u_1} {M : Type u_2} {N : Type u_3} [AddCommMonoid M] [AddCommMonoid N] {I₀ : WithTop (Box ι)} (f : BoxAdditiveMap ι M I₀) (g : M →+ N) : ⇑(f.map g) = ⇑g ∘ ⇑f

theorem BoxIntegral.BoxAdditiveMap.sum_boxes_congr {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (Box ι)} {I : Box ι} [Finite ι] (f : BoxAdditiveMap ι M I₀) (hI : ↑I ≤ I₀) {π₁ π₂ : Prepartition I} (h : π₁.iUnion = π₂.iUnion) : ∑ J ∈ π₁.boxes, f J = ∑ J ∈ π₂.boxes, f J

If f is a box additive function on subboxes of I and π₁, π₂ are two prepartitions of I that cover the same part of I, then ∑ J ∈ π₁.boxes, f J = ∑ J ∈ π₂.boxes, f J.

def BoxIntegral.BoxAdditiveMap.toSMul {ι : Type u_1} {I₀ : WithTop (Box ι)} {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : BoxAdditiveMap ι ℝ I₀) : BoxAdditiveMap ι (E →L[ℝ] E) I₀

If f is a box-additive map, then so is the map sending I to the scalar multiplication by f I as a continuous linear map from E to itself.

@[simp]

theorem BoxIntegral.BoxAdditiveMap.toSMul_apply {ι : Type u_1} {I₀ : WithTop (Box ι)} {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : BoxAdditiveMap ι ℝ I₀) (I : Box ι) (x : E) : (f.toSMul I) x = f I • x

def BoxIntegral.BoxAdditiveMap.upperSubLower {n : ℕ} {G : Type u} [AddCommGroup G] (I₀ : Box (Fin (n + 1))) (i : Fin (n + 1)) (f : ℝ → Box (Fin n) → G) (fb : ↑(Set.Icc (I₀.lower i) (I₀.upper i)) → BoxAdditiveMap (Fin n) G ↑(I₀.face i)) (hf : ∀ (x : ℝ) (hx : x ∈ Set.Icc (I₀.lower i) (I₀.upper i)) (J : Box (Fin n)), f x J = (fb ⟨x, hx⟩) J) : BoxAdditiveMap (Fin (n + 1)) G ↑I₀

Given a box I₀ in ℝⁿ⁺¹, f x : Box (Fin n) → G is a family of functions indexed by a real x and for x ∈ [I₀.lower i, I₀.upper i], f x is box-additive on subboxes of the i-th face of I₀, then fun J ↦ f (J.upper i) (J.face i) - f (J.lower i) (J.face i) is box-additive on subboxes of I₀.

@[simp]

theorem BoxIntegral.BoxAdditiveMap.upperSubLower_apply {n : ℕ} {G : Type u} [AddCommGroup G] (I₀ : Box (Fin (n + 1))) (i : Fin (n + 1)) (f : ℝ → Box (Fin n) → G) (fb : ↑(Set.Icc (I₀.lower i) (I₀.upper i)) → BoxAdditiveMap (Fin n) G ↑(I₀.face i)) (hf : ∀ (x : ℝ) (hx : x ∈ Set.Icc (I₀.lower i) (I₀.upper i)) (J : Box (Fin n)), f x J = (fb ⟨x, hx⟩) J) (J : Box (Fin (n + 1))) : (upperSubLower I₀ i f fb hf) J = f (J.upper i) (J.face i) - f (J.lower i) (J.face i)