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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 | import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
namespace OAI
noncomputable section
open Set MeasureTheory
namespace ProjectionCounterexample
/-- Every level set of a nonzero real linear functional has zero ambient volume. -/
theorem volume_linear_level_eq_zero
{V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
[FiniteDimensional ℝ V] [MeasureSpace V] [BorelSpace V]
[Measure.IsAddHaarMeasure (volume : Measure V)]
(f : V →ₗ[ℝ] ℝ) (hf : f ≠ 0) (c : ℝ) :
volume {x : V | f x = c} = 0 := by
let H : AffineSubspace ℝ V := (affineSpan ℝ ({c} : Set ℝ)).comap f.toAffineMap
have hmem (x : V) : x ∈ H ↔ f x = c := by
simp [H]
have hproper : H ≠ ⊤ := by
intro htop
have heq (x : V) : f x = c := (hmem x).mp (by rw [htop]; trivial)
have hc : c = 0 := by simpa using (heq 0).symm
apply hf
ext x
simpa [hc] using heq x
have hset : (H : Set V) = {x : V | f x = c} := Set.ext hmem
rw [← hset]
exact Measure.addHaar_affineSubspace volume H hproper
end ProjectionCounterexample
end
end OAI
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