|
| 1 | +/- |
| 2 | +Copyright (c) 2021 Yury G. Kudryashov. All rights reserved. |
| 3 | +Released under Apache 2.0 license as described in the file LICENSE. |
| 4 | +Authors: Yury G. Kudryashov |
| 5 | +-/ |
| 6 | +import measure_theory.group.arithmetic |
| 7 | + |
| 8 | +/-! |
| 9 | +# (Scalar) multiplication and (vector) addition as measurable equivalences |
| 10 | +
|
| 11 | +In this file we define the following measurable equivalences: |
| 12 | +
|
| 13 | +* `measurable_equiv.smul`: if a group `G` acts on `α` by measurable maps, then each element `c : G` |
| 14 | + defines a measurable automorphism of `α`; |
| 15 | +* `measurable_equiv.vadd`: additive version of `measurable_equiv.smul`; |
| 16 | +* `measurable_equiv.smul₀`: if a group with zero `G` acts on `α` by measurable maps, then each |
| 17 | + nonzero element `c : G` defines a measurable automorphism of `α`; |
| 18 | +* `measurable_equiv.mul_left`: if `G` is a group with measurable multiplication, then left |
| 19 | + multiplication by `g : G` is a measurable automorphism of `G`; |
| 20 | +* `measurable_equiv.add_left`: additive version of `measurable_equiv.mul_left`; |
| 21 | +* `measurable_equiv.mul_right`: if `G` is a group with measurable multiplication, then right |
| 22 | + multiplication by `g : G` is a measurable automorphism of `G`; |
| 23 | +* `measurable_equiv.add_right`: additive version of `measurable_equiv.mul_right`; |
| 24 | +* `measurable_equiv.mul_left₀`, `measurable_equiv.mul_right₀`: versions of |
| 25 | + `measurable_equiv.mul_left` and `measurable_equiv.mul_right` for groups with zero; |
| 26 | +* `measurable_equiv.inv`, `measurable_equiv.inv₀`: `has_inv.inv` as a measurable automorphism |
| 27 | + of a group (or a group with zero); |
| 28 | +* `measurable_equiv.neg`: negation as a measurable automorphism of an additive group. |
| 29 | +
|
| 30 | +We also deduce that the corresponding maps are measurable embeddings. |
| 31 | +
|
| 32 | +## Tags |
| 33 | +
|
| 34 | +measurable, equivalence, group action |
| 35 | +-/ |
| 36 | + |
| 37 | +namespace measurable_equiv |
| 38 | + |
| 39 | +variables {G G₀ α : Type*} [measurable_space G] [measurable_space G₀] [measurable_space α] |
| 40 | + [group G] [group_with_zero G₀] [mul_action G α] [mul_action G₀ α] |
| 41 | + [has_measurable_smul G α] [has_measurable_smul G₀ α] |
| 42 | + |
| 43 | +/-- If a group `G` acts on `α` by measurable maps, then each element `c : G` defines a measurable |
| 44 | +automorphism of `α`. -/ |
| 45 | +@[to_additive "If an additive group `G` acts on `α` by measurable maps, then each element `c : G` |
| 46 | +defines a measurable automorphism of `α`.", simps to_equiv apply { fully_applied := ff }] |
| 47 | +def smul (c : G) : α ≃ᵐ α := |
| 48 | +{ to_equiv := mul_action.to_perm c, |
| 49 | + measurable_to_fun := measurable_const_smul c, |
| 50 | + measurable_inv_fun := measurable_const_smul c⁻¹ } |
| 51 | + |
| 52 | +@[to_additive] |
| 53 | +lemma _root_.measurable_embedding_const_smul (c : G) : measurable_embedding ((•) c : α → α) := |
| 54 | +(smul c).measurable_embedding |
| 55 | + |
| 56 | +@[simp, to_additive] |
| 57 | +lemma symm_smul (c : G) : (smul c : α ≃ᵐ α).symm = smul c⁻¹ := ext rfl |
| 58 | + |
| 59 | +/-- If a group with zero `G₀` acts on `α` by measurable maps, then each nonzero element `c : G₀` |
| 60 | +defines a measurable automorphism of `α` -/ |
| 61 | +def smul₀ (c : G₀) (hc : c ≠ 0) : α ≃ᵐ α := |
| 62 | +measurable_equiv.smul (units.mk0 c hc) |
| 63 | + |
| 64 | +@[simp] lemma coe_smul₀ {c : G₀} (hc : c ≠ 0) : ⇑(smul₀ c hc : α ≃ᵐ α) = (•) c := rfl |
| 65 | + |
| 66 | +@[simp] lemma symm_smul₀ {c : G₀} (hc : c ≠ 0) : |
| 67 | + (smul₀ c hc : α ≃ᵐ α).symm = smul₀ c⁻¹ (inv_ne_zero hc) := |
| 68 | +ext rfl |
| 69 | + |
| 70 | +lemma _root_.measurable_embedding_const_smul₀ {c : G₀} (hc : c ≠ 0) : |
| 71 | + measurable_embedding ((•) c : α → α) := |
| 72 | +(smul₀ c hc).measurable_embedding |
| 73 | + |
| 74 | +section mul |
| 75 | + |
| 76 | +variables [has_measurable_mul G] [has_measurable_mul G₀] |
| 77 | + |
| 78 | +/-- If `G` is a group with measurable multiplication, then left multiplication by `g : G` is a |
| 79 | +measurable automorphism of `G`. -/ |
| 80 | +@[to_additive "If `G` is an additive group with measurable addition, then addition of `g : G` |
| 81 | +on the left is a measurable automorphism of `G`."] |
| 82 | +def mul_left (g : G) : G ≃ᵐ G := smul g |
| 83 | + |
| 84 | +@[simp, to_additive] lemma coe_mul_left (g : G) : ⇑(mul_left g) = (*) g := rfl |
| 85 | + |
| 86 | +@[simp, to_additive] lemma symm_mul_left (g : G) : (mul_left g).symm = mul_left g⁻¹ := ext rfl |
| 87 | + |
| 88 | +@[simp, to_additive] lemma to_equiv_mul_left (g : G) : |
| 89 | + (mul_left g).to_equiv = equiv.mul_left g := rfl |
| 90 | + |
| 91 | +@[to_additive] |
| 92 | +lemma _root_.measurable_embedding_mul_left (g : G) : measurable_embedding ((*) g) := |
| 93 | +(mul_left g).measurable_embedding |
| 94 | + |
| 95 | +/-- If `G` is a group with measurable multiplication, then right multiplication by `g : G` is a |
| 96 | +measurable automorphism of `G`. -/ |
| 97 | +@[to_additive "If `G` is an additive group with measurable addition, then addition of `g : G` |
| 98 | +on the right is a measurable automorphism of `G`."] |
| 99 | +def mul_right (g : G) : G ≃ᵐ G := |
| 100 | +{ to_equiv := equiv.mul_right g, |
| 101 | + measurable_to_fun := measurable_mul_const g, |
| 102 | + measurable_inv_fun := measurable_mul_const g⁻¹ } |
| 103 | + |
| 104 | +@[to_additive] |
| 105 | +lemma _root_.measurable_embedding_mul_right (g : G) : measurable_embedding (λ x, x * g) := |
| 106 | +(mul_right g).measurable_embedding |
| 107 | + |
| 108 | +@[simp, to_additive] lemma coe_mul_right (g : G) : ⇑(mul_right g) = (λ x, x * g) := rfl |
| 109 | + |
| 110 | +@[simp, to_additive] lemma symm_mul_right (g : G) : (mul_right g).symm = mul_right g⁻¹ := ext rfl |
| 111 | + |
| 112 | +@[simp, to_additive] lemma to_equiv_mul_right (g : G) : |
| 113 | + (mul_right g).to_equiv = equiv.mul_right g := rfl |
| 114 | + |
| 115 | +/-- If `G₀` is a group with zero with measurable multiplication, then left multiplication by a |
| 116 | +nonzero element `g : G₀` is a measurable automorphism of `G₀`. -/ |
| 117 | +def mul_left₀ (g : G₀) (hg : g ≠ 0) : G₀ ≃ᵐ G₀ := smul₀ g hg |
| 118 | + |
| 119 | +lemma _root_.measurable_embedding_mul_left₀ {g : G₀} (hg : g ≠ 0) : measurable_embedding ((*) g) := |
| 120 | +(mul_left₀ g hg).measurable_embedding |
| 121 | + |
| 122 | +@[simp] lemma coe_mul_left₀ {g : G₀} (hg : g ≠ 0) : ⇑(mul_left₀ g hg) = (*) g := rfl |
| 123 | + |
| 124 | +@[simp] lemma symm_mul_left₀ {g : G₀} (hg : g ≠ 0) : |
| 125 | + (mul_left₀ g hg).symm = mul_left₀ g⁻¹ (inv_ne_zero hg) := ext rfl |
| 126 | + |
| 127 | +@[simp] lemma to_equiv_mul_left₀ {g : G₀} (hg : g ≠ 0) : |
| 128 | + (mul_left₀ g hg).to_equiv = equiv.mul_left₀ g hg := rfl |
| 129 | + |
| 130 | +/-- If `G₀` is a group with zero with measurable multiplication, then right multiplication by a |
| 131 | +nonzero element `g : G₀` is a measurable automorphism of `G₀`. -/ |
| 132 | +def mul_right₀ (g : G₀) (hg : g ≠ 0) : G₀ ≃ᵐ G₀ := |
| 133 | +{ to_equiv := equiv.mul_right₀ g hg, |
| 134 | + measurable_to_fun := measurable_mul_const g, |
| 135 | + measurable_inv_fun := measurable_mul_const g⁻¹ } |
| 136 | + |
| 137 | +lemma _root_.measurable_embedding_mul_right₀ {g : G₀} (hg : g ≠ 0) : |
| 138 | + measurable_embedding (λ x, x * g) := |
| 139 | +(mul_right₀ g hg).measurable_embedding |
| 140 | + |
| 141 | +@[simp] lemma coe_mul_right₀ {g : G₀} (hg : g ≠ 0) : ⇑(mul_right₀ g hg) = λ x, x * g := rfl |
| 142 | + |
| 143 | +@[simp] lemma symm_mul_right₀ {g : G₀} (hg : g ≠ 0) : |
| 144 | + (mul_right₀ g hg).symm = mul_right₀ g⁻¹ (inv_ne_zero hg) := ext rfl |
| 145 | + |
| 146 | +@[simp] lemma to_equiv_mul_right₀ {g : G₀} (hg : g ≠ 0) : |
| 147 | + (mul_right₀ g hg).to_equiv = equiv.mul_right₀ g hg := rfl |
| 148 | + |
| 149 | +end mul |
| 150 | + |
| 151 | +variables (G G₀) |
| 152 | + |
| 153 | +/-- Inversion as a measurable automorphism of a group. -/ |
| 154 | +@[to_additive "Negation as a measurable automorphism of an additive group.", |
| 155 | + simps to_equiv apply { fully_applied := ff }] |
| 156 | +def inv [has_measurable_inv G] : G ≃ᵐ G := |
| 157 | +{ to_equiv := equiv.inv G, |
| 158 | + measurable_to_fun := measurable_inv, |
| 159 | + measurable_inv_fun := measurable_inv } |
| 160 | + |
| 161 | +/-- Inversion as a measurable automorphism of a group with zero. -/ |
| 162 | +@[simps to_equiv apply { fully_applied := ff }] |
| 163 | +def inv₀ [has_measurable_inv G₀] : G₀ ≃ᵐ G₀ := |
| 164 | +{ to_equiv := equiv.inv₀ G₀, |
| 165 | + measurable_to_fun := measurable_inv, |
| 166 | + measurable_inv_fun := measurable_inv } |
| 167 | + |
| 168 | +variables {G G₀} |
| 169 | + |
| 170 | +@[simp] lemma symm_inv [has_measurable_inv G] : (inv G).symm = inv G := rfl |
| 171 | +@[simp] lemma symm_inv₀ [has_measurable_inv G₀] : (inv₀ G₀).symm = inv₀ G₀ := rfl |
| 172 | + |
| 173 | +end measurable_equiv |
0 commit comments