Induction on subboxes #

In this file we prove (see BoxIntegral.Box.exists_taggedPartition_isHenstock_isSubordinate_homothetic) that for every box I in ℝⁿ and a function r : ℝⁿ → ℝ positive on I there exists a tagged partition π of I such that

π is a Henstock partition;
π is subordinate to r;
each box in π is homothetic to I with coefficient of the form 1 / 2 ^ n.

Later we will use this lemma to prove that the Henstock filter is nontrivial, hence the Henstock integral is well-defined.

Tags #

partition, tagged partition, Henstock integral

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def BoxIntegral.Prepartition.splitCenter {ι : Type u_1} [Fintype ι] (I : BoxIntegral.Box ι) :

BoxIntegral.Prepartition I

Split a box in ℝⁿ into 2 ^ n boxes by hyperplanes passing through its center.

Equations

BoxIntegral.Prepartition.splitCenter I = { boxes := Finset.map (BoxIntegral.Box.splitCenterBoxEmb I) Finset.univ, le_of_mem' := ⋯, pairwiseDisjoint := ⋯ }

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@[simp]

theorem BoxIntegral.Prepartition.mem_splitCenter {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} {J : BoxIntegral.Box ι} :

J ∈ BoxIntegral.Prepartition.splitCenter I ↔ ∃ (s : Set ι), BoxIntegral.Box.splitCenterBox I s = J

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theorem BoxIntegral.Prepartition.isPartition_splitCenter {ι : Type u_1} [Fintype ι] (I : BoxIntegral.Box ι) :

BoxIntegral.Prepartition.IsPartition (BoxIntegral.Prepartition.splitCenter I)

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theorem BoxIntegral.Prepartition.upper_sub_lower_of_mem_splitCenter {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} {J : BoxIntegral.Box ι} (h : J ∈ BoxIntegral.Prepartition.splitCenter I) (i : ι) :

J.upper i - J.lower i = (I.upper i - I.lower i) / 2

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theorem BoxIntegral.Box.subbox_induction_on {ι : Type u_1} [Fintype ι] {p : BoxIntegral.Box ι → Prop} (I : BoxIntegral.Box ι) (H_ind : ∀ J ≤ I, (∀ J' ∈ BoxIntegral.Prepartition.splitCenter J, p J') → p J) (H_nhds : ∀ z ∈ BoxIntegral.Box.Icc I, ∃ U ∈ nhdsWithin z (BoxIntegral.Box.Icc I), ∀ J ≤ I, ∀ (m : ℕ), z ∈ BoxIntegral.Box.Icc J → BoxIntegral.Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J) :

p I

Let p be a predicate on Box ι, let I be a box. Suppose that the following two properties hold true.

Consider a smaller box J ≤ I. The hyperplanes passing through the center of J split it into 2 ^ n boxes. If p holds true on each of these boxes, then it true on J.
For each z in the closed box I.Icc there exists a neighborhood U of z within I.Icc such that for every box J ≤ I such that z ∈ J.Icc ⊆ U, if J is homothetic to I with a coefficient of the form 1 / 2 ^ m, then p is true on J.

Then p I is true. See also BoxIntegral.Box.subbox_induction_on' for a version using BoxIntegral.Box.splitCenterBox instead of BoxIntegral.Prepartition.splitCenter.

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theorem BoxIntegral.Box.exists_taggedPartition_isHenstock_isSubordinate_homothetic {ι : Type u_1} [Fintype ι] (I : BoxIntegral.Box ι) (r : (ι → ℝ) → ↑(Set.Ioi 0)) :

∃ (π : BoxIntegral.TaggedPrepartition I), BoxIntegral.TaggedPrepartition.IsPartition π ∧ BoxIntegral.TaggedPrepartition.IsHenstock π ∧ BoxIntegral.TaggedPrepartition.IsSubordinate π r ∧ (∀ J ∈ π, ∃ (m : ℕ), ∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) ∧ BoxIntegral.TaggedPrepartition.distortion π = BoxIntegral.Box.distortion I

Given a box I in ℝⁿ and a function r : ℝⁿ → (0, ∞), there exists a tagged partition π of I such that

π is a Henstock partition;
π is subordinate to r;
each box in π is homothetic to I with coefficient of the form 1 / 2 ^ m.

This lemma implies that the Henstock filter is nontrivial, hence the Henstock integral is well-defined.

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theorem BoxIntegral.Prepartition.exists_tagged_le_isHenstock_isSubordinate_iUnion_eq {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} (r : (ι → ℝ) → ↑(Set.Ioi 0)) (π : BoxIntegral.Prepartition I) :

∃ (π' : BoxIntegral.TaggedPrepartition I), π'.toPrepartition ≤ π ∧ BoxIntegral.TaggedPrepartition.IsHenstock π' ∧ BoxIntegral.TaggedPrepartition.IsSubordinate π' r ∧ BoxIntegral.TaggedPrepartition.distortion π' = BoxIntegral.Prepartition.distortion π ∧ BoxIntegral.TaggedPrepartition.iUnion π' = BoxIntegral.Prepartition.iUnion π

Given a box I in ℝⁿ, a function r : ℝⁿ → (0, ∞), and a prepartition π of I, there exists a tagged prepartition π' of I such that

each box of π' is included in some box of π;
π' is a Henstock partition;
π' is subordinate to r;
π' covers exactly the same part of I as π;
the distortion of π' is equal to the distortion of π.

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def BoxIntegral.Prepartition.toSubordinate {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (r : (ι → ℝ) → ↑(Set.Ioi 0)) :

BoxIntegral.TaggedPrepartition I

Given a prepartition π of a box I and a function r : ℝⁿ → (0, ∞), π.toSubordinate r is a tagged partition π' such that

each box of π' is included in some box of π;
π' is a Henstock partition;
π' is subordinate to r;
π' covers exactly the same part of I as π;
the distortion of π' is equal to the distortion of π.

Equations

BoxIntegral.Prepartition.toSubordinate π r = Exists.choose ⋯

Instances For

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theorem BoxIntegral.Prepartition.toSubordinate_toPrepartition_le {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (r : (ι → ℝ) → ↑(Set.Ioi 0)) :

(BoxIntegral.Prepartition.toSubordinate π r).toPrepartition ≤ π

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theorem BoxIntegral.Prepartition.isHenstock_toSubordinate {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (r : (ι → ℝ) → ↑(Set.Ioi 0)) :

BoxIntegral.TaggedPrepartition.IsHenstock (BoxIntegral.Prepartition.toSubordinate π r)

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theorem BoxIntegral.Prepartition.isSubordinate_toSubordinate {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (r : (ι → ℝ) → ↑(Set.Ioi 0)) :

BoxIntegral.TaggedPrepartition.IsSubordinate (BoxIntegral.Prepartition.toSubordinate π r) r

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@[simp]

theorem BoxIntegral.Prepartition.distortion_toSubordinate {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (r : (ι → ℝ) → ↑(Set.Ioi 0)) :

BoxIntegral.TaggedPrepartition.distortion (BoxIntegral.Prepartition.toSubordinate π r) = BoxIntegral.Prepartition.distortion π

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@[simp]

theorem BoxIntegral.Prepartition.iUnion_toSubordinate {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (r : (ι → ℝ) → ↑(Set.Ioi 0)) :

BoxIntegral.TaggedPrepartition.iUnion (BoxIntegral.Prepartition.toSubordinate π r) = BoxIntegral.Prepartition.iUnion π

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def BoxIntegral.TaggedPrepartition.unionComplToSubordinate {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} (π₁ : BoxIntegral.TaggedPrepartition I) (π₂ : BoxIntegral.Prepartition I) (hU : BoxIntegral.Prepartition.iUnion π₂ = ↑I \ BoxIntegral.TaggedPrepartition.iUnion π₁) (r : (ι → ℝ) → ↑(Set.Ioi 0)) :

BoxIntegral.TaggedPrepartition I

Given a tagged prepartition π₁, a prepartition π₂ that covers exactly I \ π₁.iUnion, and a function r : ℝⁿ → (0, ∞), returns the union of π₁ and π₂.toSubordinate r. This partition π has the following properties:

π is a partition, i.e. it covers the whole I;
π₁.boxes ⊆ π.boxes;
π.tag J = π₁.tag J whenever J ∈ π₁;
π is Henstock outside of π₁: π.tag J ∈ J.Icc whenever J ∈ π, J ∉ π₁;
π is subordinate to r outside of π₁;
the distortion of π is equal to the maximum of the distortions of π₁ and π₂.