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2fca478 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 | import OAI.Geometry.ProjectionBody.FacetCover
import Mathlib.MeasureTheory.Measure.Basic
namespace OAI
noncomputable section
open Set MeasureTheory
namespace ProjectionCounterexample
variable {ι V W : Type*} [Fintype ι]
[AddCommGroup V] [Module ℝ V] [AddCommGroup W] [Module ℝ W]
[MeasurableSpace W]
/-- Finite additivity for shadows of visible faces. The geometric hypotheses
are explicit: the later applications must prove measurability and nullity of
the projected intersections. No facet-volume formula is assumed here. -/
theorem measure_image_halfspaceBody_eq_sum_visible
(a : ι → V →ₗ[ℝ] ℝ) (b : ι → ℝ) (q : V →ₗ[ℝ] W) (μ : Measure W)
{u : V} (hqu : q u = 0) (hu : ∃ i, 0 < a i u)
(hker : LinearMap.ker q = Submodule.span ℝ {u})
(hmeas : ∀ i, MeasurableSet (q '' boundingFace a b i))
(hnull : ∀ i j, i ≠ j →
μ (q '' (boundingFace a b i ∩ boundingFace a b j)) = 0) :
μ (q '' halfspaceBody a b) =
∑ i : {i : ι // 0 < a i u}, μ (q '' boundingFace a b i.1) := by
classical
rw [image_halfspaceBody_eq_visible_faces a b q hqu hu]
have hd : Pairwise (fun i j : {i : ι // 0 < a i u} =>
AEDisjoint μ (q '' boundingFace a b i.1) (q '' boundingFace a b j.1)) := by
intro i j hij
change μ ((q '' boundingFace a b i.1) ∩ (q '' boundingFace a b j.1)) = 0
rw [image_visible_faces_inter a b q hker i.2 j.2]
exact hnull i j (fun h => hij (Subtype.ext h))
rw [measure_iUnion₀ hd (fun i => (hmeas i.1).nullMeasurableSet), tsum_fintype]
end ProjectionCounterexample
end
end OAI
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