| 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 | |