158 lines
4.5 KiB
Plaintext
158 lines
4.5 KiB
Plaintext
/-
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Chapter 1.6
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The concept of area as a set function
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-/
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import Bookshelf.Real.Function.Step
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import Bookshelf.Real.Geometry.Rectangle
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namespace Real
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/--
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All *measurable sets*, i.e. sets in the plane to which an area can be assigned.
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The existence of such a class is assumed in the axiomatic definition of area
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introduced by Apostol. He denotes this set of sets as `𝓜`.
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-/
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axiom 𝓜 : Set (Set ℝ²)
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/--
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A set function mapping every *measurable set* to a value denoting its area.
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The existence of such a function is assumed in the axiomatic definition of area
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introduced by Apostol. He denotes this function as `a`.
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-/
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axiom area : ∀ ⦃x⦄, x ∈ 𝓜 → ℝ
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/--
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The nonnegative property.
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For each set `S` in `𝓜`, we have `a(S) ≥ 0`.
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-/
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axiom area_ge_zero {S : Set ℝ²} (h : S ∈ 𝓜): area h ≥ 0
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/--
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The additive property (i).
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If `S` and `T` are in `𝓜`, then `S ∪ T` in `𝓜`.
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-/
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axiom measureable_imp_union_measurable {S T : Set ℝ²} (hS : S ∈ 𝓜) (hT : T ∈ 𝓜)
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: S ∪ T ∈ 𝓜
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/--
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The additive property (ii).
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If `S` and `T` are in `𝓜`, then `S ∩ T` in `𝓜`.
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-/
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axiom measurable_imp_inter_measurable {S T : Set ℝ²} (hS : S ∈ 𝓜) (hT : T ∈ 𝓜)
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: S ∩ T ∈ 𝓜
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/--
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The additive property (iii).
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If `S` and `T` are in `𝓜`, then `a(S ∪ T) = a(S) + a(T) - a(S ∩ T)`.
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-/
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axiom union_area_eq_area_add_area_sub_inter_area
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{S T : Set ℝ²} (hS : S ∈ 𝓜) (hT : T ∈ 𝓜)
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: area (measureable_imp_union_measurable hS hT) =
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area hS + area hT - area (measurable_imp_inter_measurable hS hT)
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/--
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The difference property (i).
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If `S` and `T` are in `𝓜` with `S ⊆ T`, then `T - S` is in `𝓜`.
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-/
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axiom measureable_imp_diff_measurable
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{S T : Set ℝ²} (hS : S ∈ 𝓜) (hT : T ∈ 𝓜) (h : S ⊆ T)
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: T \ S ∈ 𝓜
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/--
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The difference property (ii).
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If `S` and `T` are in `𝓜` with `S ⊆ T`, then `a(T - S) = a(T) - a(S)`.
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-/
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axiom diff_area_eq_area_sub_area
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{S T : Set ℝ²} (hS : S ∈ 𝓜) (hT : T ∈ 𝓜) (h : S ⊆ T)
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: area (measureable_imp_diff_measurable hS hT h) = area hT - area hS
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/--
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Invariance under congruence (i).
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If a set `S` is in `𝓜` and if `T` is congruent to `S`, then `T` is also in `𝓜`..
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-/
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axiom measurable_congruent_imp_measurable
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{S T : Set ℝ²} (hS : S ∈ 𝓜) (h : congruent S T)
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: T ∈ 𝓜
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/--
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Invariance under congruence (ii).
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If a set `S` is in `𝓜` and if `T` is congruent to `S`, then `a(S) = a(T)`.
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-/
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axiom congruent_imp_area_eq_area
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{S T : Set ℝ²} (hS : S ∈ 𝓜) (h : congruent S T)
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: area hS = area (measurable_congruent_imp_measurable hS h)
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/--
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Choice of scale (i).
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Every rectangle `R` is in `𝓜`.
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-/
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axiom rectangle_measurable (R : Rectangle)
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: R.set_def ∈ 𝓜
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/--
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Choice of scale (ii).
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If the edges of rectangle `R` have lengths `h` and `k`, then `a(R) = hk`.
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-/
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axiom rectangle_area_eq_mul_edge_lengths (R : Rectangle)
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: area (rectangle_measurable R) = R.width * R.height
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/--
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Every step region is measurable. This follows from the choice of scale axiom,
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and the fact all step regions are equivalent to the union of a collection of
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rectangles.
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-/
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theorem step_function_measurable (S : Function.Step) : S.set_def ∈ 𝓜 := by
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sorry
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/--
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Exhaustion property.
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Let `Q` be a set that can be enclosed between two step regions `S` and `T`, so
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that (1.1) `S ⊆ Q ⊆ T`. If there is one and only one number `k` which satisfies
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the inequalities `a(S) ≤ k ≤ a(T)` for all step regions `S` and `T` satisfying
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(1.1), then `Q` is measurable and `a(Q) = k`.
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-/
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def forall_subset_between_step_imp_le_between_area (k : ℝ) (Q : Set ℝ²) :=
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∀ S T : Function.Step,
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(hS : S.set_def ⊆ Q) →
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(hT : Q ⊆ T.set_def) →
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area (step_function_measurable S) ≤ k ∧ k ≤ area (step_function_measurable T)
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/--
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Exhaustion property (i).
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If there exists some `k` satisfying the description in the above `def`, then `Q`
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is measurable.
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-/
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axiom exhaustion_exists_unique_imp_measurable (Q : Set ℝ²)
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: (∃ k : ℝ, forall_subset_between_step_imp_le_between_area k Q ∧
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(∀ x : ℝ, forall_subset_between_step_imp_le_between_area x Q → x = k))
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→ Q ∈ 𝓜
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/--
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Exhaustion property (ii).
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If there exists some `k` satisfying the description in the above `def`, then `Q`
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satisfies `a(Q) = k`.
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-/
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axiom exhaustion_exists_unique_imp_area_eq (Q : Set ℝ²)
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: ∃ k : ℝ,
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(h : forall_subset_between_step_imp_le_between_area k Q ∧
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(∀ x : ℝ, forall_subset_between_step_imp_le_between_area x Q → x = k))
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→ area (exhaustion_exists_unique_imp_measurable Q ⟨k, h⟩) = k
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end Real
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