| import Mathlib |
|
|
| namespace OAI |
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| namespace BorsukNine |
| open Set MeasureTheory |
| open scoped Topology |
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|
| section NullImage |
| variable {D T : Type*} [NormedAddCommGroup D] [NormedSpace ℝ D] |
| [NormedAddCommGroup T] [NormedSpace ℝ T] [FiniteDimensional ℝ T] |
| [MeasurableSpace T] [BorelSpace T] |
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| lemma differentiable_image_null_of_projection (μ : Measure T) [μ.IsAddHaarMeasure] |
| (p : T →L[ℝ] D) (hp : Function.Surjective p) (hpn : ¬ Function.Injective p) |
| {f : D → T} {S : Set D} (hf : ∀ x ∈ S, DifferentiableAt ℝ f x) : μ (f '' S) = 0 := by |
| let g : T → T := fun x => f (p x) |
| let g' : T → T →L[ℝ] T := fun x => (fderiv ℝ f (p x)).comp p |
| have hd : ∀ x ∈ p ⁻¹' S, HasFDerivWithinAt g (g' x) (p ⁻¹' S) x := by |
| intro x hx |
| exact ((hf (p x) hx).hasFDerivAt.comp x p.hasFDerivAt).hasFDerivWithinAt |
| have hdet : ∀ x ∈ p ⁻¹' S, (g' x).det = 0 := by |
| intro x _ |
| change ((g' x).toLinearMap).det = 0 |
| apply LinearMap.det_eq_zero_iff_ker_ne_bot.mpr |
| apply mt LinearMap.ker_eq_bot.mp |
| intro hinj |
| exact hpn (fun u v huv => hinj (by change fderiv ℝ f (p x) (p u) = fderiv ℝ f (p x) (p v); rw [huv])) |
| have he : g '' (p ⁻¹' S) = f '' S := by |
| ext y |
| constructor |
| · rintro ⟨x,hx,rfl⟩ |
| exact ⟨p x,hx,rfl⟩ |
| · rintro ⟨x,hx,rfl⟩ |
| obtain ⟨z,rfl⟩ := hp x |
| exact ⟨z,hx,rfl⟩ |
| rw [← he] |
| exact addHaar_image_eq_zero_of_det_fderivWithin_eq_zero μ hd hdet |
|
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| end NullImage |
|
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| noncomputable def initialProjection (n m : ℕ) (h : n ≤ m) : |
| EuclideanSpace ℝ (Fin m) →L[ℝ] EuclideanSpace ℝ (Fin n) := |
| LinearMap.toContinuousLinearMap |
| { toFun := fun x => WithLp.toLp 2 (fun i => x (Fin.castLE h i)) |
| map_add' := by intros; rfl |
| map_smul' := by intros; rfl } |
|
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| lemma initialProjection_apply (n m : ℕ) (h : n ≤ m) (x : EuclideanSpace ℝ (Fin m)) (i : Fin n) : |
| initialProjection n m h x i = x (Fin.castLE h i) := rfl |
|
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| lemma initialProjection_surjective (n m : ℕ) (h : n ≤ m) : |
| Function.Surjective (initialProjection n m h) := by |
| intro x |
| refine ⟨WithLp.toLp 2 (fun j : Fin m => if h : j.val < n then x ⟨j.val,h⟩ else 0),?_⟩ |
| apply PiLp.ext |
| intro i |
| change (if hj : i.val < n then x ⟨i.val,hj⟩ else 0) = x i |
| simp only [dite_eq_left i.isLt] |
|
|
| lemma initialProjection_not_injective (n m : ℕ) (h : n < m) : |
| ¬ Function.Injective (initialProjection n m h.le) := by |
| intro hh |
| have hl := LinearMap.finrank_le_finrank_of_injective |
| (f := (initialProjection n m h.le).toLinearMap) hh |
| simp only [finrank_euclideanSpace_fin] at hl |
| omega |
|
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| lemma differentiable_image_null_of_dim_lt (n m : ℕ) (h : n < m) |
| {f : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin m)} |
| {S : Set (EuclideanSpace ℝ (Fin n))} (hf : ∀ x ∈ S, DifferentiableAt ℝ f x) : |
| volume (f '' S) = 0 := |
| differentiable_image_null_of_projection volume (initialProjection n m h.le) |
| (initialProjection_surjective n m h.le) (initialProjection_not_injective n m h) hf |
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| lemma open_exists_not_mem_differentiable_image (n m : ℕ) (h : n < m) |
| {f : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin m)} |
| {S : Set (EuclideanSpace ℝ (Fin n))} (hf : ∀ x ∈ S, DifferentiableAt ℝ f x) |
| {U : Set (EuclideanSpace ℝ (Fin m))} (hU : IsOpen U) (hne : U.Nonempty) : |
| ∃ y ∈ U, y ∉ f '' S := by |
| by_contra hn |
| push Not at hn |
| have hl : volume U ≤ volume (f '' S) := measure_mono hn |
| rw [differentiable_image_null_of_dim_lt n m h hf] at hl |
| exact (hU.measure_ne_zero volume hne) (le_antisymm hl (zero_le)) |
|
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| end BorsukNine |
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| end OAI |
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