File size: 2,884 Bytes
ddc84c5 | 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 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped ENNReal NNReal Topology
universe uX u_1
namespace ExactModuli
variable {X : Type uX} [NormedAddCommGroup X] [NormedSpace ℝ X]
/-- Closed linear subspaces of finite codimension. -/
def Cofinite (F : Submodule ℝ X) : Prop :=
IsClosed (F : Set X) ∧ FiniteDimensional ℝ (X ⧸ F)
/-- Averaged increment for directions of norm at least one. -/
def averagedIncrement (N : Seminorm ℝ X) (x : X) (t : ℝ) (y : X) : ℝ :=
(N (x + t • y) + N (x - t • y)) / 2 - 1
def H (N : Seminorm ℝ X) (x : X) (t : ℝ) : ℝ :=
sSup {r | ∃ F : Submodule ℝ X, Cofinite F ∧
r = sInf {a | ∃ y ∈ F, 1 ≤ N y ∧ a = averagedIncrement N x t y}}
def D (N : Seminorm ℝ X) (x : X) (t : ℝ) : ℝ :=
sSup {r | ∃ F : Submodule ℝ X, Cofinite F ∧
r = sInf {a | ∃ y ∈ F, N y = 1 ∧ a = N (x + t • y) - 1}}
def averagedModulus (N : Seminorm ℝ X) (t : ℝ) : ℝ :=
sInf {r | ∃ x, N x = 1 ∧ r = H N x t}
def oneSidedModulus (N : Seminorm ℝ X) (t : ℝ) : ℝ :=
sInf {r | ∃ x, N x = 1 ∧ r = D N x t}
/-- Two-sided comparison makes this seminorm a genuine norm equivalent to the original. -/
def EquivalentNorm (N : Seminorm ℝ X) : Prop :=
∃ a b : ℝ, 0 < a ∧ 0 < b ∧ ∀ x, a * ‖x‖ ≤ N x ∧ N x ≤ b * ‖x‖
def AUC (N : Seminorm ℝ X) : Prop :=
∀ t : ℝ, 0 < t → 0 < oneSidedModulus N t
/-- Every rank-one operator has the form `ell.smulRight v`. -/
def Daugavet (X : Type uX) [NormedAddCommGroup X] [NormedSpace ℝ X] : Prop :=
∀ (ell : X →L[ℝ] ℝ) (v : X),
‖ContinuousLinearMap.id ℝ X + ell.smulRight v‖ = 1 + ‖ell.smulRight v‖
variable {Ω : Type u_1} [MeasurableSpace Ω] (μ : Measure Ω)
/-- Convergence-in-measure distance on L1 equivalence classes. -/
def measureDistance (f g : Lp ℝ 1 μ) : ℝ :=
sInf {a : ℝ | 0 < a ∧ μ.real {ω | a < |f ω - g ω|} < a}
/-- Finite covers of the entire unit ball in convergence-in-measure distance. -/
def MeasurePrecompactBall (E : Submodule ℝ (Lp ℝ 1 μ)) : Prop :=
∀ ε : ℝ, 0 < ε → ∃ s : Finset (Lp ℝ 1 μ),
∀ y : E, ‖y‖ ≤ 1 → ∃ b ∈ s, measureDistance μ (y : Lp ℝ 1 μ) b < ε
/-- Every nonempty unit-ball slice contains an almost antipodal unit vector. -/
def UnitSliceProperty (X : Type uX) [NormedAddCommGroup X] [NormedSpace ℝ X] : Prop :=
∀ (w : X), ‖w‖ = 1 → ∀ (ell : X →L[ℝ] ℝ) (α : ℝ),
(∃ u : X, ‖u‖ ≤ 1 ∧ α < ell u) →
∀ ρ : ℝ, 0 < ρ → ∃ y : X, ‖y‖ = 1 ∧ α < ell y ∧ 2 - ρ ≤ ‖w + y‖
/-- The countable product of unit intervals. -/
abbrev KWOmega := ℕ → unitInterval
def kwMeasure : Measure KWOmega := Measure.infinitePi (fun _ : ℕ => (volume : Measure unitInterval))
end ExactModuli
end
end OAI
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