Order related properties of Lp spaces

Results

• Lp E p μ is an OrderedAddCommGroup when E is a NormedLatticeAddCommGroup.

TODO

• move definitions of Lp.posPart and Lp.negPart to this file, and define them as PosPart.pos and NegPart.neg given by the lattice structure.

theorem MeasureTheory.Lp.coeFn_le {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ENNReal} [NormedLatticeAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ)) (g : ↥(MeasureTheory.Lp E p μ)) : ↑↑f ≤ᶠ[MeasureTheory.Measure.ae μ] ↑↑g ↔ f ≤ g

theorem MeasureTheory.Lp.coeFn_nonneg {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ENNReal} [NormedLatticeAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ)) : 0 ≤ᶠ[MeasureTheory.Measure.ae μ] ↑↑f ↔ 0 ≤ f

instance MeasureTheory.Lp.instCovariantClassLE {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ENNReal} [NormedLatticeAddCommGroup E] : CovariantClass (↥(MeasureTheory.Lp E p μ)) (↥(MeasureTheory.Lp E p μ)) (fun (x x_1 : ↥(MeasureTheory.Lp E p μ)) => x + x_1) fun (x x_1 : ↥(MeasureTheory.Lp E p μ)) => x ≤ x_1

Equations

• ⋯ = ⋯

instance MeasureTheory.Lp.instOrderedAddCommGroup {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ENNReal} [NormedLatticeAddCommGroup E] : OrderedAddCommGroup ↥(MeasureTheory.Lp E p μ)

Equations

• One or more equations did not get rendered due to their size.

theorem MeasureTheory.Memℒp.sup {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ENNReal} [NormedLatticeAddCommGroup E] {f : α → E} {g : α → E} (hf : MeasureTheory.Memℒp f p μ) (hg : MeasureTheory.Memℒp g p μ) : MeasureTheory.Memℒp (f ⊔ g) p μ

theorem MeasureTheory.Memℒp.inf {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ENNReal} [NormedLatticeAddCommGroup E] {f : α → E} {g : α → E} (hf : MeasureTheory.Memℒp f p μ) (hg : MeasureTheory.Memℒp g p μ) : MeasureTheory.Memℒp (f ⊓ g) p μ

theorem MeasureTheory.Memℒp.abs {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ENNReal} [NormedLatticeAddCommGroup E] {f : α → E} (hf : MeasureTheory.Memℒp f p μ) : MeasureTheory.Memℒp |f| p μ

instance MeasureTheory.Lp.instLattice {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ENNReal} [NormedLatticeAddCommGroup E] : Lattice ↥(MeasureTheory.Lp E p μ)

Equations

• MeasureTheory.Lp.instLattice = Subtype.lattice ⋯ ⋯

theorem MeasureTheory.Lp.coeFn_sup {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ENNReal} [NormedLatticeAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ)) (g : ↥(MeasureTheory.Lp E p μ)) : ↑↑(f ⊔ g) =ᶠ[MeasureTheory.Measure.ae μ] ↑↑f ⊔ ↑↑g

theorem MeasureTheory.Lp.coeFn_inf {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ENNReal} [NormedLatticeAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ)) (g : ↥(MeasureTheory.Lp E p μ)) : ↑↑(f ⊓ g) =ᶠ[MeasureTheory.Measure.ae μ] ↑↑f ⊓ ↑↑g

theorem MeasureTheory.Lp.coeFn_abs {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ENNReal} [NormedLatticeAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ)) : ↑↑|f| =ᶠ[MeasureTheory.Measure.ae μ] fun (x : α) => |↑↑f x|

noncomputable instance MeasureTheory.Lp.instNormedLatticeAddCommGroup {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ENNReal} [NormedLatticeAddCommGroup E] [Fact (1 ≤ p)] : NormedLatticeAddCommGroup ↥(MeasureTheory.Lp E p μ)

Equations

• MeasureTheory.Lp.instNormedLatticeAddCommGroup = let __src := MeasureTheory.Lp.instLattice; let __src_1 := MeasureTheory.Lp.instNormedAddCommGroup; NormedLatticeAddCommGroup.mk ⋯