Lp space

This file provides the space Lp E p μ as the subtype of elements of α →ₘ[μ] E (see ae_eq_fun) such that eLpNorm f p μ is finite. For 1 ≤ p, eLpNorm defines a norm and Lp is a complete metric space.

Main definitions

• Lp E p μ : elements of α →ₘ[μ] E (see ae_eq_fun) such that eLpNorm f p μ is finite. Defined as an AddSubgroup of α →ₘ[μ] E.

Lipschitz functions vanishing at zero act by composition on Lp. We define this action, and prove that it is continuous. In particular,

• ContinuousLinearMap.compLp defines the action on Lp of a continuous linear map.
• Lp.posPart is the positive part of an Lp function.
• Lp.negPart is the negative part of an Lp function.

Notations

• α →₁[μ] E : the type Lp E 1 μ.
• α →₂[μ] E : the type Lp E 2 μ.

Implementation

Since Lp is defined as an AddSubgroup, dot notation does not work. Use Lp.Measurable f to say that the coercion of f to a genuine function is measurable, instead of the non-working f.Measurable.

To prove that two Lp elements are equal, it suffices to show that their coercions to functions coincide almost everywhere (this is registered as an ext rule). This can often be done using filter_upwards. For instance, a proof from first principles that f + (g + h) = (f + g) + h could read (in the Lp namespace)

example (f g h : Lp E p μ) : (f + g) + h = f + (g + h) := by
  ext1
  filter_upwards [coeFn_add (f + g) h, coeFn_add f g, coeFn_add f (g + h), coeFn_add g h]
    with _ ha1 ha2 ha3 ha4
  simp only [ha1, ha2, ha3, ha4, add_assoc]

The lemma coeFn_add states that the coercion of f + g coincides almost everywhere with the sum of the coercions of f and g. All such lemmas use coeFn in their name, to distinguish the function coercion from the coercion to almost everywhere defined functions.

Lp space

The space of equivalence classes of measurable functions for which eLpNorm f p μ < ∞.

@[simp]

theorem MeasureTheory.eLpNorm_aeeqFun {α : Type u_7} {E : Type u_8} [MeasurableSpace α] {μ : Measure α} [NormedAddCommGroup E] {p : ENNReal} {f : α → E} (hf : AEStronglyMeasurable f μ) : eLpNorm (↑(AEEqFun.mk f hf)) p μ = eLpNorm f p μ

theorem MeasureTheory.Memℒp.eLpNorm_mk_lt_top {α : Type u_7} {E : Type u_8} [MeasurableSpace α] {μ : Measure α} [NormedAddCommGroup E] {p : ENNReal} {f : α → E} (hfp : Memℒp f p μ) : eLpNorm (↑(AEEqFun.mk f ⋯)) p μ < ⊤

def MeasureTheory.Lp {α : Type u_8} (E : Type u_7) {m : MeasurableSpace α} [NormedAddCommGroup E] (p : ENNReal) (μ : Measure α := by volume_tac) : AddSubgroup (α →ₘ[μ] E)

Lp space

Equations

• MeasureTheory.Lp E p μ = { carrier := {f : α →ₘ[μ] E | MeasureTheory.eLpNorm (↑f) p μ < ⊤}, add_mem' := ⋯, zero_mem' := ⋯, neg_mem' := ⋯ }

def MeasureTheory.«term_→₁[_]_» : Lean.TrailingParserDescr

Equations

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

def MeasureTheory.«term_→₂[_]_» : Lean.TrailingParserDescr

Equations

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

def MeasureTheory.Memℒp.toLp {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : α → E) (h_mem_ℒp : Memℒp f p μ) : ↥(Lp E p μ)

make an element of Lp from a function verifying Memℒp

Equations

• MeasureTheory.Memℒp.toLp f h_mem_ℒp = ⟨MeasureTheory.AEEqFun.mk f ⋯, ⋯⟩

theorem MeasureTheory.Memℒp.toLp_val {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} (h : Memℒp f p μ) : ↑(toLp f h) = AEEqFun.mk f ⋯

theorem MeasureTheory.Memℒp.coeFn_toLp {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} (hf : Memℒp f p μ) : ↑↑(toLp f hf) =ᶠ[ae μ] f

theorem MeasureTheory.Memℒp.toLp_congr {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) (hfg : f =ᶠ[ae μ] g) : toLp f hf = toLp g hg

@[simp]

theorem MeasureTheory.Memℒp.toLp_eq_toLp_iff {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) : toLp f hf = toLp g hg ↔ f =ᶠ[ae μ] g

@[simp]

theorem MeasureTheory.Memℒp.toLp_zero {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (h : Memℒp 0 p μ) : toLp 0 h = 0

theorem MeasureTheory.Memℒp.toLp_add {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) : toLp (f + g) ⋯ = toLp f hf + toLp g hg

theorem MeasureTheory.Memℒp.toLp_neg {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} (hf : Memℒp f p μ) : toLp (-f) ⋯ = -toLp f hf

theorem MeasureTheory.Memℒp.toLp_sub {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) : toLp (f - g) ⋯ = toLp f hf - toLp g hg

instance MeasureTheory.Lp.instCoeFun {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] : CoeFun ↥(Lp E p μ) fun (x : ↥(Lp E p μ)) => α → E

Equations

• MeasureTheory.Lp.instCoeFun = { coe := fun (f : ↥(MeasureTheory.Lp E p μ)) => ↑↑f }

theorem MeasureTheory.Lp.ext {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f g : ↥(Lp E p μ)} (h : ↑↑f =ᶠ[ae μ] ↑↑g) : f = g

theorem MeasureTheory.Lp.mem_Lp_iff_eLpNorm_lt_top {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α →ₘ[μ] E} : f ∈ Lp E p μ ↔ eLpNorm (↑f) p μ < ⊤

theorem MeasureTheory.Lp.mem_Lp_iff_memℒp {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α →ₘ[μ] E} : f ∈ Lp E p μ ↔ Memℒp (↑f) p μ

theorem MeasureTheory.Lp.antitone {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {μ : Measure α} [NormedAddCommGroup E] [IsFiniteMeasure μ] {p q : ENNReal} (hpq : p ≤ q) : Lp E q μ ≤ Lp E p μ

@[simp]

theorem MeasureTheory.Lp.coeFn_mk {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α →ₘ[μ] E} (hf : eLpNorm (↑f) p μ < ⊤) : ↑↑⟨f, hf⟩ = ↑f

theorem MeasureTheory.Lp.coe_mk {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α →ₘ[μ] E} (hf : eLpNorm (↑f) p μ < ⊤) : ↑⟨f, hf⟩ = f

@[simp]

theorem MeasureTheory.Lp.toLp_coeFn {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) (hf : Memℒp (↑↑f) p μ) : Memℒp.toLp (↑↑f) hf = f

theorem MeasureTheory.Lp.eLpNorm_lt_top {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) : eLpNorm (↑↑f) p μ < ⊤

theorem MeasureTheory.Lp.eLpNorm_ne_top {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) : eLpNorm (↑↑f) p μ ≠ ⊤

theorem MeasureTheory.Lp.stronglyMeasurable {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) : StronglyMeasurable ↑↑f

theorem MeasureTheory.Lp.aestronglyMeasurable {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) : AEStronglyMeasurable (↑↑f) μ

theorem MeasureTheory.Lp.memℒp {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) : Memℒp (↑↑f) p μ

theorem MeasureTheory.Lp.coeFn_zero {α : Type u_1} (E : Type u_4) {m0 : MeasurableSpace α} (p : ENNReal) (μ : Measure α) [NormedAddCommGroup E] : ↑↑0 =ᶠ[ae μ] 0

theorem MeasureTheory.Lp.coeFn_neg {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) : ↑↑(-f) =ᶠ[ae μ] -↑↑f

theorem MeasureTheory.Lp.coeFn_add {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f g : ↥(Lp E p μ)) : ↑↑(f + g) =ᶠ[ae μ] ↑↑f + ↑↑g

theorem MeasureTheory.Lp.coeFn_sub {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f g : ↥(Lp E p μ)) : ↑↑(f - g) =ᶠ[ae μ] ↑↑f - ↑↑g

theorem MeasureTheory.Lp.const_mem_Lp {E : Type u_4} {p : ENNReal} [NormedAddCommGroup E] (α : Type u_7) {x✝ : MeasurableSpace α} (μ : Measure α) (c : E) [IsFiniteMeasure μ] : AEEqFun.const α c ∈ Lp E p μ

instance MeasureTheory.Lp.instNorm {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] : Norm ↥(Lp E p μ)

Equations

• MeasureTheory.Lp.instNorm = { norm := fun (f : ↥(MeasureTheory.Lp E p μ)) => (MeasureTheory.eLpNorm (↑↑f) p μ).toReal }

instance MeasureTheory.Lp.instNNNorm {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] : NNNorm ↥(Lp E p μ)

Equations

• MeasureTheory.Lp.instNNNorm = { nnnorm := fun (f : ↥(MeasureTheory.Lp E p μ)) => ⟨‖f‖, ⋯⟩ }

instance MeasureTheory.Lp.instDist {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] : Dist ↥(Lp E p μ)

Equations

• MeasureTheory.Lp.instDist = { dist := fun (f g : ↥(MeasureTheory.Lp E p μ)) => ‖f - g‖ }

instance MeasureTheory.Lp.instEDist {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] : EDist ↥(Lp E p μ)

Equations

• MeasureTheory.Lp.instEDist = { edist := fun (f g : ↥(MeasureTheory.Lp E p μ)) => MeasureTheory.eLpNorm (↑↑f - ↑↑g) p μ }

theorem MeasureTheory.Lp.norm_def {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) : ‖f‖ = (eLpNorm (↑↑f) p μ).toReal

theorem MeasureTheory.Lp.nnnorm_def {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) : ‖f‖₊ = (eLpNorm (↑↑f) p μ).toNNReal

@[simp]

theorem MeasureTheory.Lp.coe_nnnorm {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) : ↑‖f‖₊ = ‖f‖

@[simp]

theorem MeasureTheory.Lp.enorm_def {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) : ‖f‖ₑ = eLpNorm (↑↑f) p μ

@[deprecated MeasureTheory.Lp.enorm_def (since := "2025-01-20")]

theorem MeasureTheory.Lp.nnnorm_coe_ennreal {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) : ‖f‖ₑ = eLpNorm (↑↑f) p μ

Alias of MeasureTheory.Lp.enorm_def.

@[simp]

theorem MeasureTheory.Lp.norm_toLp {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : α → E) (hf : Memℒp f p μ) : ‖Memℒp.toLp f hf‖ = (eLpNorm f p μ).toReal

@[simp]

theorem MeasureTheory.Lp.nnnorm_toLp {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : α → E) (hf : Memℒp f p μ) : ‖Memℒp.toLp f hf‖₊ = (eLpNorm f p μ).toNNReal

theorem MeasureTheory.Lp.enorm_toLp {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} (hf : Memℒp f p μ) : ‖Memℒp.toLp f hf‖ₑ = eLpNorm f p μ

@[deprecated MeasureTheory.Lp.enorm_toLp (since := "2025-01-20")]

theorem MeasureTheory.Lp.coe_nnnorm_toLp {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} (hf : Memℒp f p μ) : ‖Memℒp.toLp f hf‖ₑ = eLpNorm f p μ

Alias of MeasureTheory.Lp.enorm_toLp.

theorem MeasureTheory.Lp.dist_def {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f g : ↥(Lp E p μ)) : dist f g = (eLpNorm (↑↑f - ↑↑g) p μ).toReal

theorem MeasureTheory.Lp.edist_def {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f g : ↥(Lp E p μ)) : edist f g = eLpNorm (↑↑f - ↑↑g) p μ

theorem MeasureTheory.Lp.edist_dist {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f g : ↥(Lp E p μ)) : edist f g = ENNReal.ofReal (dist f g)

theorem MeasureTheory.Lp.dist_edist {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f g : ↥(Lp E p μ)) : dist f g = (edist f g).toReal

theorem MeasureTheory.Lp.dist_eq_norm {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f g : ↥(Lp E p μ)) : dist f g = ‖f - g‖

@[simp]

theorem MeasureTheory.Lp.edist_toLp_toLp {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f g : α → E) (hf : Memℒp f p μ) (hg : Memℒp g p μ) : edist (Memℒp.toLp f hf) (Memℒp.toLp g hg) = eLpNorm (f - g) p μ

@[simp]

theorem MeasureTheory.Lp.edist_toLp_zero {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : α → E) (hf : Memℒp f p μ) : edist (Memℒp.toLp f hf) 0 = eLpNorm f p μ

@[simp]

theorem MeasureTheory.Lp.nnnorm_zero {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] : ‖0‖₊ = 0

@[simp]

theorem MeasureTheory.Lp.norm_zero {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] : ‖0‖ = 0

@[simp]

theorem MeasureTheory.Lp.norm_measure_zero {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} [NormedAddCommGroup E] (f : ↥(Lp E p 0)) : ‖f‖ = 0

@[simp]

theorem MeasureTheory.Lp.norm_exponent_zero {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E 0 μ)) : ‖f‖ = 0

theorem MeasureTheory.Lp.nnnorm_eq_zero_iff {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : ↥(Lp E p μ)} (hp : 0 < p) : ‖f‖₊ = 0 ↔ f = 0

theorem MeasureTheory.Lp.norm_eq_zero_iff {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : ↥(Lp E p μ)} (hp : 0 < p) : ‖f‖ = 0 ↔ f = 0

theorem MeasureTheory.Lp.eq_zero_iff_ae_eq_zero {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : ↥(Lp E p μ)} : f = 0 ↔ ↑↑f =ᶠ[ae μ] 0

@[simp]

theorem MeasureTheory.Lp.nnnorm_neg {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) : ‖-f‖₊ = ‖f‖₊

@[simp]

theorem MeasureTheory.Lp.norm_neg {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) : ‖-f‖ = ‖f‖

theorem MeasureTheory.Lp.nnnorm_le_mul_nnnorm_of_ae_le_mul {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {c : NNReal} {f : ↥(Lp E p μ)} {g : ↥(Lp F p μ)} (h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ c * ‖↑↑g x‖₊) : ‖f‖₊ ≤ c * ‖g‖₊

theorem MeasureTheory.Lp.norm_le_mul_norm_of_ae_le_mul {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {c : ℝ} {f : ↥(Lp E p μ)} {g : ↥(Lp F p μ)} (h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ c * ‖↑↑g x‖) : ‖f‖ ≤ c * ‖g‖

theorem MeasureTheory.Lp.norm_le_norm_of_ae_le {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {f : ↥(Lp E p μ)} {g : ↥(Lp F p μ)} (h : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ ‖↑↑g x‖) : ‖f‖ ≤ ‖g‖

theorem MeasureTheory.Lp.mem_Lp_of_nnnorm_ae_le_mul {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {c : NNReal} {f : α →ₘ[μ] E} {g : ↥(Lp F p μ)} (h : ∀ᵐ (x : α) ∂μ, ‖↑f x‖₊ ≤ c * ‖↑↑g x‖₊) : f ∈ Lp E p μ

theorem MeasureTheory.Lp.mem_Lp_of_ae_le_mul {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {c : ℝ} {f : α →ₘ[μ] E} {g : ↥(Lp F p μ)} (h : ∀ᵐ (x : α) ∂μ, ‖↑f x‖ ≤ c * ‖↑↑g x‖) : f ∈ Lp E p μ

theorem MeasureTheory.Lp.mem_Lp_of_nnnorm_ae_le {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α →ₘ[μ] E} {g : ↥(Lp F p μ)} (h : ∀ᵐ (x : α) ∂μ, ‖↑f x‖₊ ≤ ‖↑↑g x‖₊) : f ∈ Lp E p μ

theorem MeasureTheory.Lp.mem_Lp_of_ae_le {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α →ₘ[μ] E} {g : ↥(Lp F p μ)} (h : ∀ᵐ (x : α) ∂μ, ‖↑f x‖ ≤ ‖↑↑g x‖) : f ∈ Lp E p μ

theorem MeasureTheory.Lp.mem_Lp_of_ae_nnnorm_bound {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [IsFiniteMeasure μ] {f : α →ₘ[μ] E} (C : NNReal) (hfC : ∀ᵐ (x : α) ∂μ, ‖↑f x‖₊ ≤ C) : f ∈ Lp E p μ

theorem MeasureTheory.Lp.mem_Lp_of_ae_bound {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [IsFiniteMeasure μ] {f : α →ₘ[μ] E} (C : ℝ) (hfC : ∀ᵐ (x : α) ∂μ, ‖↑f x‖ ≤ C) : f ∈ Lp E p μ

theorem MeasureTheory.Lp.nnnorm_le_of_ae_bound {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [IsFiniteMeasure μ] {f : ↥(Lp E p μ)} {C : NNReal} (hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖₊ ≤ C) : ‖f‖₊ ≤ measureUnivNNReal μ ^ p.toReal⁻¹ * C

theorem MeasureTheory.Lp.norm_le_of_ae_bound {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [IsFiniteMeasure μ] {f : ↥(Lp E p μ)} {C : ℝ} (hC : 0 ≤ C) (hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ C) : ‖f‖ ≤ ↑(measureUnivNNReal μ) ^ p.toReal⁻¹ * C

instance MeasureTheory.Lp.instNormedAddCommGroup {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [hp : Fact (1 ≤ p)] : NormedAddCommGroup ↥(Lp E p μ)

Equations

• MeasureTheory.Lp.instNormedAddCommGroup = NormedAddCommGroup.mk ⋯

theorem MeasureTheory.Lp.const_smul_mem_Lp {α : Type u_1} {𝕜 : Type u_2} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] (c : 𝕜) (f : ↥(Lp E p μ)) : c • ↑f ∈ Lp E p μ

def MeasureTheory.Lp.LpSubmodule {α : Type u_1} (𝕜 : Type u_2) (E : Type u_4) {m0 : MeasurableSpace α} (p : ENNReal) (μ : Measure α) [NormedAddCommGroup E] [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] : Submodule 𝕜 (α →ₘ[μ] E)

The 𝕜-submodule of elements of α →ₘ[μ] E whose Lp norm is finite. This is Lp E p μ, with extra structure.

Equations

• MeasureTheory.Lp.LpSubmodule 𝕜 E p μ = { toAddSubmonoid := (MeasureTheory.Lp E p μ).toAddSubmonoid, smul_mem' := ⋯ }

theorem MeasureTheory.Lp.coe_LpSubmodule {α : Type u_1} {𝕜 : Type u_2} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] : (LpSubmodule 𝕜 E p μ).toAddSubgroup = Lp E p μ

instance MeasureTheory.Lp.instModule {α : Type u_1} {𝕜 : Type u_2} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] : Module 𝕜 ↥(Lp E p μ)

Equations

• MeasureTheory.Lp.instModule = Module.mk ⋯ ⋯

theorem MeasureTheory.Lp.coeFn_smul {α : Type u_1} {𝕜 : Type u_2} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] (c : 𝕜) (f : ↥(Lp E p μ)) : ↑↑(c • f) =ᶠ[ae μ] c • ↑↑f

instance MeasureTheory.Lp.instIsCentralScalar {α : Type u_1} {𝕜 : Type u_2} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] [Module 𝕜ᵐᵒᵖ E] [BoundedSMul 𝕜ᵐᵒᵖ E] [IsCentralScalar 𝕜 E] : IsCentralScalar 𝕜 ↥(Lp E p μ)

instance MeasureTheory.Lp.instSMulCommClass {α : Type u_1} {𝕜 : Type u_2} {𝕜' : Type u_3} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [NormedRing 𝕜] [NormedRing 𝕜'] [Module 𝕜 E] [Module 𝕜' E] [BoundedSMul 𝕜 E] [BoundedSMul 𝕜' E] [SMulCommClass 𝕜 𝕜' E] : SMulCommClass 𝕜 𝕜' ↥(Lp E p μ)

instance MeasureTheory.Lp.instIsScalarTower {α : Type u_1} {𝕜 : Type u_2} {𝕜' : Type u_3} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [NormedRing 𝕜] [NormedRing 𝕜'] [Module 𝕜 E] [Module 𝕜' E] [BoundedSMul 𝕜 E] [BoundedSMul 𝕜' E] [SMul 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' E] : IsScalarTower 𝕜 𝕜' ↥(Lp E p μ)

instance MeasureTheory.Lp.instBoundedSMul {α : Type u_1} {𝕜 : Type u_2} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] [Fact (1 ≤ p)] : BoundedSMul 𝕜 ↥(Lp E p μ)

instance MeasureTheory.Lp.instNormedSpace {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {𝕜 : Type u_7} [NormedField 𝕜] [NormedSpace 𝕜 E] [Fact (1 ≤ p)] : NormedSpace 𝕜 ↥(Lp E p μ)

Equations

• MeasureTheory.Lp.instNormedSpace = NormedSpace.mk ⋯

theorem MeasureTheory.Memℒp.toLp_const_smul {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {𝕜 : Type u_7} [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] {f : α → E} (c : 𝕜) (hf : Memℒp f p μ) : toLp (c • f) ⋯ = c • toLp f hf

Indicator of a set as an element of Lᵖ

For a set s with (hs : MeasurableSet s) and (hμs : μ s < ∞), we build indicatorConstLp p hs hμs c, the element of Lp corresponding to s.indicator (fun _ => c).

theorem MeasureTheory.exists_eLpNorm_indicator_le {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (hp : p ≠ ⊤) (c : E) {ε : ENNReal} (hε : ε ≠ 0) : ∃ (η : NNReal), 0 < η ∧ ∀ (s : Set α), μ s ≤ ↑η → eLpNorm (s.indicator fun (x : α) => c) p μ ≤ ε

The ℒ^p norm of the indicator of a set is uniformly small if the set itself has small measure, for any p < ∞. Given here as an existential ∀ ε > 0, ∃ η > 0, ... to avoid later management of ℝ≥0∞-arithmetic.

theorem HasCompactSupport.memℒp_of_bound {E : Type u_4} {p : ENNReal} [NormedAddCommGroup E] {X : Type u_7} [TopologicalSpace X] [MeasurableSpace X] {μ : MeasureTheory.Measure X} [MeasureTheory.IsFiniteMeasureOnCompacts μ] {f : X → E} (hf : HasCompactSupport f) (h2f : MeasureTheory.AEStronglyMeasurable f μ) (C : ℝ) (hfC : ∀ᵐ (x : X) ∂μ, ‖f x‖ ≤ C) : MeasureTheory.Memℒp f p μ

A bounded measurable function with compact support is in L^p.

theorem Continuous.memℒp_of_hasCompactSupport {E : Type u_4} {p : ENNReal} [NormedAddCommGroup E] {X : Type u_7} [TopologicalSpace X] [MeasurableSpace X] {μ : MeasureTheory.Measure X} [MeasureTheory.IsFiniteMeasureOnCompacts μ] [OpensMeasurableSpace X] {f : X → E} (hf : Continuous f) (h'f : HasCompactSupport f) : MeasureTheory.Memℒp f p μ

A continuous function with compact support is in L^p.

def MeasureTheory.indicatorConstLp {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {μ : Measure α} [NormedAddCommGroup E] {s : Set α} (p : ENNReal) (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (c : E) : ↥(Lp E p μ)

Indicator of a set as an element of Lp.

Equations

• MeasureTheory.indicatorConstLp p hs hμs c = MeasureTheory.Memℒp.toLp (s.indicator fun (x : α) => c) ⋯

theorem MeasureTheory.indicatorConstLp_add {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c c' : E} : indicatorConstLp p hs hμs c + indicatorConstLp p hs hμs c' = indicatorConstLp p hs hμs (c + c')

A version of Set.indicator_add for MeasureTheory.indicatorConstLp

theorem MeasureTheory.indicatorConstLp_sub {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c c' : E} : indicatorConstLp p hs hμs c - indicatorConstLp p hs hμs c' = indicatorConstLp p hs hμs (c - c')

A version of Set.indicator_sub for MeasureTheory.indicatorConstLp

theorem MeasureTheory.indicatorConstLp_coeFn {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E} : ↑↑(indicatorConstLp p hs hμs c) =ᶠ[ae μ] s.indicator fun (x : α) => c

theorem MeasureTheory.indicatorConstLp_coeFn_mem {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E} : ∀ᵐ (x : α) ∂μ, x ∈ s → ↑↑(indicatorConstLp p hs hμs c) x = c

theorem MeasureTheory.indicatorConstLp_coeFn_nmem {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E} : ∀ᵐ (x : α) ∂μ, x ∉ s → ↑↑(indicatorConstLp p hs hμs c) x = 0

theorem MeasureTheory.norm_indicatorConstLp {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) : ‖indicatorConstLp p hs hμs c‖ = ‖c‖ * (μ s).toReal ^ (1 / p.toReal)

theorem MeasureTheory.norm_indicatorConstLp_top {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {μ : Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E} (hμs_ne_zero : μ s ≠ 0) : ‖indicatorConstLp ⊤ hs hμs c‖ = ‖c‖

theorem MeasureTheory.norm_indicatorConstLp' {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E} (hp_pos : p ≠ 0) (hμs_pos : μ s ≠ 0) : ‖indicatorConstLp p hs hμs c‖ = ‖c‖ * (μ s).toReal ^ (1 / p.toReal)

theorem MeasureTheory.norm_indicatorConstLp_le {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E} : ‖indicatorConstLp p hs hμs c‖ ≤ ‖c‖ * (μ s).toReal ^ (1 / p.toReal)

theorem MeasureTheory.nnnorm_indicatorConstLp_le {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E} : ‖indicatorConstLp p hs hμs c‖₊ ≤ ‖c‖₊ * (μ s).toNNReal ^ (1 / p.toReal)

theorem MeasureTheory.enorm_indicatorConstLp_le {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E} : ‖indicatorConstLp p hs hμs c‖ₑ ≤ ‖c‖ₑ * μ s ^ (1 / p.toReal)

@[deprecated MeasureTheory.enorm_indicatorConstLp_le (since := "2025-01-20")]

theorem MeasureTheory.ennnorm_indicatorConstLp_le {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E} : ‖indicatorConstLp p hs hμs c‖ₑ ≤ ‖c‖ₑ * μ s ^ (1 / p.toReal)

Alias of MeasureTheory.enorm_indicatorConstLp_le.

theorem MeasureTheory.edist_indicatorConstLp_eq_enorm {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E} {t : Set α} {ht : MeasurableSet t} {hμt : μ t ≠ ⊤} : edist (indicatorConstLp p hs hμs c) (indicatorConstLp p ht hμt c) = ‖indicatorConstLp p ⋯ ⋯ c‖ₑ

@[deprecated MeasureTheory.edist_indicatorConstLp_eq_enorm (since := "2025-01-20")]

theorem MeasureTheory.edist_indicatorConstLp_eq_nnnorm {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E} {t : Set α} {ht : MeasurableSet t} {hμt : μ t ≠ ⊤} : edist (indicatorConstLp p hs hμs c) (indicatorConstLp p ht hμt c) = ‖indicatorConstLp p ⋯ ⋯ c‖ₑ

Alias of MeasureTheory.edist_indicatorConstLp_eq_enorm.

theorem MeasureTheory.dist_indicatorConstLp_eq_norm {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E} {t : Set α} {ht : MeasurableSet t} {hμt : μ t ≠ ⊤} : dist (indicatorConstLp p hs hμs c) (indicatorConstLp p ht hμt c) = ‖indicatorConstLp p ⋯ ⋯ c‖

theorem MeasureTheory.tendsto_indicatorConstLp_set {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {s : Set α} {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {c : E} [hp₁ : Fact (1 ≤ p)] {β : Type u_7} {l : Filter β} {t : β → Set α} {ht : ∀ (b : β), MeasurableSet (t b)} {hμt : ∀ (b : β), μ (t b) ≠ ⊤} (hp : p ≠ ⊤) (h : Filter.Tendsto (fun (b : β) => μ (symmDiff (t b) s)) l (nhds 0)) : Filter.Tendsto (fun (b : β) => indicatorConstLp p ⋯ ⋯ c) l (nhds (indicatorConstLp p hs hμs c))

A family of indicatorConstLp functions tends to an indicatorConstLp, if the underlying sets tend to the set in the sense of the measure of the symmetric difference.

theorem MeasureTheory.continuous_indicatorConstLp_set {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {c : E} [Fact (1 ≤ p)] {X : Type u_7} [TopologicalSpace X] {s : X → Set α} {hs : ∀ (x : X), MeasurableSet (s x)} {hμs : ∀ (x : X), μ (s x) ≠ ⊤} (hp : p ≠ ⊤) (h : ∀ (x : X), Filter.Tendsto (fun (y : X) => μ (symmDiff (s y) (s x))) (nhds x) (nhds 0)) : Continuous fun (x : X) => indicatorConstLp p ⋯ ⋯ c

A family of indicatorConstLp functions is continuous in the parameter, if μ (s y ∆ s x) tends to zero as y tends to x for all x.

@[simp]

theorem MeasureTheory.indicatorConstLp_empty {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {c : E} : indicatorConstLp p ⋯ ⋯ c = 0

theorem MeasureTheory.indicatorConstLp_inj {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {s t : Set α} (hs : MeasurableSet s) (hsμ : μ s ≠ ⊤) (ht : MeasurableSet t) (htμ : μ t ≠ ⊤) {c : E} (hc : c ≠ 0) : indicatorConstLp p hs hsμ c = indicatorConstLp p ht htμ c ↔ s =ᶠ[ae μ] t

theorem MeasureTheory.memℒp_add_of_disjoint {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f g : α → E} (h : Disjoint (Function.support f) (Function.support g)) (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) : Memℒp (f + g) p μ ↔ Memℒp f p μ ∧ Memℒp g p μ

theorem MeasureTheory.indicatorConstLp_disjoint_union {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {s t : Set α} (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ⊤) (hμt : μ t ≠ ⊤) (hst : Disjoint s t) (c : E) : indicatorConstLp p ⋯ ⋯ c = indicatorConstLp p hs hμs c + indicatorConstLp p ht hμt c

The indicator of a disjoint union of two sets is the sum of the indicators of the sets.

def MeasureTheory.Lp.const {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} (p : ENNReal) (μ : Measure α) [NormedAddCommGroup E] [IsFiniteMeasure μ] : E →+ ↥(Lp E p μ)

Constant function as an element of MeasureTheory.Lp for a finite measure.

Equations

• MeasureTheory.Lp.const p μ = { toFun := fun (c : E) => ⟨MeasureTheory.AEEqFun.const α c, ⋯⟩, map_zero' := ⋯, map_add' := ⋯ }

theorem MeasureTheory.Lp.coeFn_const {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} (p : ENNReal) (μ : Measure α) [NormedAddCommGroup E] [IsFiniteMeasure μ] (c : E) : ↑↑((Lp.const p μ) c) =ᶠ[ae μ] Function.const α c

@[simp]

theorem MeasureTheory.Lp.const_val {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} (p : ENNReal) (μ : Measure α) [NormedAddCommGroup E] [IsFiniteMeasure μ] (c : E) : ↑((Lp.const p μ) c) = AEEqFun.const α c

@[simp]

theorem MeasureTheory.Memℒp.toLp_const {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} (p : ENNReal) (μ : Measure α) [NormedAddCommGroup E] [IsFiniteMeasure μ] (c : E) : toLp (fun (x : α) => c) ⋯ = (Lp.const p μ) c

@[simp]

theorem MeasureTheory.indicatorConstLp_univ {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} (p : ENNReal) (μ : Measure α) [NormedAddCommGroup E] [IsFiniteMeasure μ] (c : E) : indicatorConstLp p ⋯ ⋯ c = (Lp.const p μ) c

theorem MeasureTheory.Lp.norm_const {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} (p : ENNReal) (μ : Measure α) [NormedAddCommGroup E] [IsFiniteMeasure μ] (c : E) [NeZero μ] (hp_zero : p ≠ 0) : ‖(Lp.const p μ) c‖ = ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal)

theorem MeasureTheory.Lp.norm_const' {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} (p : ENNReal) (μ : Measure α) [NormedAddCommGroup E] [IsFiniteMeasure μ] (c : E) (hp_zero : p ≠ 0) (hp_top : p ≠ ⊤) : ‖(Lp.const p μ) c‖ = ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal)

theorem MeasureTheory.Lp.norm_const_le {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} (p : ENNReal) (μ : Measure α) [NormedAddCommGroup E] [IsFiniteMeasure μ] (c : E) : ‖(Lp.const p μ) c‖ ≤ ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal)

def MeasureTheory.Lp.constₗ {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} (p : ENNReal) (μ : Measure α) [NormedAddCommGroup E] [IsFiniteMeasure μ] (𝕜 : Type u_7) [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] : E →ₗ[𝕜] ↥(Lp E p μ)

MeasureTheory.Lp.const as a LinearMap.

Equations

• MeasureTheory.Lp.constₗ p μ 𝕜 = { toFun := ⇑(MeasureTheory.Lp.const p μ), map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem MeasureTheory.Lp.constₗ_apply {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} (p : ENNReal) (μ : Measure α) [NormedAddCommGroup E] [IsFiniteMeasure μ] (𝕜 : Type u_7) [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] (a : E) : (Lp.constₗ p μ 𝕜) a = (Lp.const p μ) a

def MeasureTheory.Lp.constL {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} (p : ENNReal) (μ : Measure α) [NormedAddCommGroup E] [IsFiniteMeasure μ] (𝕜 : Type u_7) [NormedField 𝕜] [NormedSpace 𝕜 E] [Fact (1 ≤ p)] : E →L[𝕜] ↥(Lp E p μ)

Equations

• MeasureTheory.Lp.constL p μ 𝕜 = (MeasureTheory.Lp.constₗ p μ 𝕜).mkContinuous ((μ Set.univ).toReal ^ (1 / p.toReal)) ⋯

@[simp]

theorem MeasureTheory.Lp.constL_apply {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} (p : ENNReal) (μ : Measure α) [NormedAddCommGroup E] [IsFiniteMeasure μ] (𝕜 : Type u_7) [NormedField 𝕜] [NormedSpace 𝕜 E] [Fact (1 ≤ p)] (a : E) : (Lp.constL p μ 𝕜) a = (Lp.const p μ) a

theorem MeasureTheory.Lp.norm_constL_le {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} (p : ENNReal) (μ : Measure α) [NormedAddCommGroup E] [IsFiniteMeasure μ] (𝕜 : Type u_7) [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [Fact (1 ≤ p)] : ‖Lp.constL p μ 𝕜‖ ≤ (μ Set.univ).toReal ^ (1 / p.toReal)

theorem MeasureTheory.Memℒp.norm_rpow_div {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} (hf : Memℒp f p μ) (q : ENNReal) : Memℒp (fun (x : α) => ‖f x‖ ^ q.toReal) (p / q) μ

theorem MeasureTheory.memℒp_norm_rpow_iff {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {q : ENNReal} {f : α → E} (hf : AEStronglyMeasurable f μ) (q_zero : q ≠ 0) (q_top : q ≠ ⊤) : Memℒp (fun (x : α) => ‖f x‖ ^ q.toReal) (p / q) μ ↔ Memℒp f p μ

theorem MeasureTheory.Memℒp.norm_rpow {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} (hf : Memℒp f p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) : Memℒp (fun (x : α) => ‖f x‖ ^ p.toReal) 1 μ

theorem MeasureTheory.AEEqFun.compMeasurePreserving_mem_Lp {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {β : Type u_7} [MeasurableSpace β] {μb : Measure β} {g : β →ₘ[μb] E} (hg : g ∈ Lp E p μb) {f : α → β} (hf : MeasurePreserving f μ μb) : g.compMeasurePreserving f hf ∈ Lp E p μ

Composition with a measure preserving function

def MeasureTheory.Lp.compMeasurePreserving {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {β : Type u_7} [MeasurableSpace β] {μb : Measure β} (f : α → β) (hf : MeasurePreserving f μ μb) : ↥(Lp E p μb) →+ ↥(Lp E p μ)

Composition of an L^p function with a measure preserving function is an L^p function.

Equations

• MeasureTheory.Lp.compMeasurePreserving f hf = { toFun := fun (g : ↥(MeasureTheory.Lp E p μb)) => ⟨(↑g).compMeasurePreserving f hf, ⋯⟩, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem MeasureTheory.Lp.compMeasurePreserving_val {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {β : Type u_7} [MeasurableSpace β] {μb : Measure β} {f : α → β} (g : ↥(Lp E p μb)) (hf : MeasurePreserving f μ μb) : ↑((compMeasurePreserving f hf) g) = (↑g).compMeasurePreserving f hf

theorem MeasureTheory.Lp.coeFn_compMeasurePreserving {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {β : Type u_7} [MeasurableSpace β] {μb : Measure β} {f : α → β} (g : ↥(Lp E p μb)) (hf : MeasurePreserving f μ μb) : ↑↑((compMeasurePreserving f hf) g) =ᶠ[ae μ] ↑↑g ∘ f

@[simp]

theorem MeasureTheory.Lp.norm_compMeasurePreserving {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {β : Type u_7} [MeasurableSpace β] {μb : Measure β} {f : α → β} (g : ↥(Lp E p μb)) (hf : MeasurePreserving f μ μb) : ‖(compMeasurePreserving f hf) g‖ = ‖g‖

theorem MeasureTheory.Lp.isometry_compMeasurePreserving {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {β : Type u_7} [MeasurableSpace β] {μb : Measure β} {f : α → β} [Fact (1 ≤ p)] (hf : MeasurePreserving f μ μb) : Isometry ⇑(compMeasurePreserving f hf)

theorem MeasureTheory.Lp.toLp_compMeasurePreserving {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {β : Type u_7} [MeasurableSpace β] {μb : Measure β} {f : α → β} {g : β → E} (hg : Memℒp g p μb) (hf : MeasurePreserving f μ μb) : (compMeasurePreserving f hf) (Memℒp.toLp g hg) = Memℒp.toLp (g ∘ f) ⋯

theorem MeasureTheory.Lp.indicatorConstLp_compMeasurePreserving {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {β : Type u_7} [MeasurableSpace β] {μb : Measure β} {f : α → β} {s : Set β} (hs : MeasurableSet s) (hμs : μb s ≠ ⊤) (c : E) (hf : MeasurePreserving f μ μb) : (compMeasurePreserving f hf) (indicatorConstLp p hs hμs c) = indicatorConstLp p ⋯ ⋯ c

def MeasureTheory.Lp.compMeasurePreservingₗ {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {β : Type u_7} [MeasurableSpace β] {μb : Measure β} (𝕜 : Type u_8) [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] (f : α → β) (hf : MeasurePreserving f μ μb) : ↥(Lp E p μb) →ₗ[𝕜] ↥(Lp E p μ)

MeasureTheory.Lp.compMeasurePreserving as a linear map.

Equations

• MeasureTheory.Lp.compMeasurePreservingₗ 𝕜 f hf = { toFun := (↑(MeasureTheory.Lp.compMeasurePreserving f hf)).toFun, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem MeasureTheory.Lp.compMeasurePreservingₗ_apply {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {β : Type u_7} [MeasurableSpace β] {μb : Measure β} (𝕜 : Type u_8) [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] (f : α → β) (hf : MeasurePreserving f μ μb) (a✝ : ↥(Lp E p μb)) : (compMeasurePreservingₗ 𝕜 f hf) a✝ = (↑(compMeasurePreserving f hf)).toFun a✝

def MeasureTheory.Lp.compMeasurePreservingₗᵢ {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {β : Type u_7} [MeasurableSpace β] {μb : Measure β} (𝕜 : Type u_8) [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] [Fact (1 ≤ p)] (f : α → β) (hf : MeasurePreserving f μ μb) : ↥(Lp E p μb) →ₗᵢ[𝕜] ↥(Lp E p μ)

MeasureTheory.Lp.compMeasurePreserving as a linear isometry.

Equations

• MeasureTheory.Lp.compMeasurePreservingₗᵢ 𝕜 f hf = { toLinearMap := MeasureTheory.Lp.compMeasurePreservingₗ 𝕜 f hf, norm_map' := ⋯ }

@[simp]

theorem MeasureTheory.Lp.compMeasurePreservingₗᵢ_apply_coe {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {β : Type u_7} [MeasurableSpace β] {μb : Measure β} (𝕜 : Type u_8) [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] [Fact (1 ≤ p)] (f : α → β) (hf : MeasurePreserving f μ μb) (a✝ : ↥(Lp E p μb)) : ↑((compMeasurePreservingₗᵢ 𝕜 f hf) a✝) = (↑a✝).compMeasurePreserving f hf

Composition on L^p

We show that Lipschitz functions vanishing at zero act by composition on L^p, and specialize this to the composition with continuous linear maps, and to the definition of the positive part of an L^p function.

theorem LipschitzWith.comp_memℒp {p : ENNReal} {α : Type u_7} {E : Type u_8} {F : Type u_9} {K : NNReal} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : E → F} (hg : LipschitzWith K g) (g0 : g 0 = 0) (hL : MeasureTheory.Memℒp f p μ) : MeasureTheory.Memℒp (g ∘ f) p μ

theorem MeasureTheory.Memℒp.of_comp_antilipschitzWith {p : ENNReal} {α : Type u_7} {E : Type u_8} {F : Type u_9} {K' : NNReal} [MeasurableSpace α] {μ : Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : E → F} (hL : Memℒp (g ∘ f) p μ) (hg : UniformContinuous g) (hg' : AntilipschitzWith K' g) (g0 : g 0 = 0) : Memℒp f p μ

theorem LipschitzWith.memℒp_comp_iff_of_antilipschitz {p : ENNReal} {α : Type u_7} {E : Type u_8} {F : Type u_9} {K K' : NNReal} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : E → F} (hg : LipschitzWith K g) (hg' : AntilipschitzWith K' g) (g0 : g 0 = 0) : MeasureTheory.Memℒp (g ∘ f) p μ ↔ MeasureTheory.Memℒp f p μ

def LipschitzWith.compLp {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {g : E → F} {c : NNReal} (hg : LipschitzWith c g) (g0 : g 0 = 0) (f : ↥(MeasureTheory.Lp E p μ)) : ↥(MeasureTheory.Lp F p μ)

When g is a Lipschitz function sending 0 to 0 and f is in Lp, then g ∘ f is well defined as an element of Lp.

Equations

• hg.compLp g0 f = ⟨MeasureTheory.AEEqFun.comp g ⋯ ↑f, ⋯⟩

theorem LipschitzWith.coeFn_compLp {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {g : E → F} {c : NNReal} (hg : LipschitzWith c g) (g0 : g 0 = 0) (f : ↥(MeasureTheory.Lp E p μ)) : ↑↑(hg.compLp g0 f) =ᵐ[μ] g ∘ ↑↑f

@[simp]

theorem LipschitzWith.compLp_zero {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {g : E → F} {c : NNReal} (hg : LipschitzWith c g) (g0 : g 0 = 0) : hg.compLp g0 0 = 0

theorem LipschitzWith.norm_compLp_sub_le {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {g : E → F} {c : NNReal} (hg : LipschitzWith c g) (g0 : g 0 = 0) (f f' : ↥(MeasureTheory.Lp E p μ)) : ‖hg.compLp g0 f - hg.compLp g0 f'‖ ≤ ↑c * ‖f - f'‖

theorem LipschitzWith.norm_compLp_le {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {g : E → F} {c : NNReal} (hg : LipschitzWith c g) (g0 : g 0 = 0) (f : ↥(MeasureTheory.Lp E p μ)) : ‖hg.compLp g0 f‖ ≤ ↑c * ‖f‖

theorem LipschitzWith.lipschitzWith_compLp {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {g : E → F} {c : NNReal} [Fact (1 ≤ p)] (hg : LipschitzWith c g) (g0 : g 0 = 0) : LipschitzWith c (hg.compLp g0)

theorem LipschitzWith.continuous_compLp {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {g : E → F} {c : NNReal} [Fact (1 ≤ p)] (hg : LipschitzWith c g) (g0 : g 0 = 0) : Continuous (hg.compLp g0)

def ContinuousLinearMap.compLp {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_7} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (L : E →L[𝕜] F) (f : ↥(MeasureTheory.Lp E p μ)) : ↥(MeasureTheory.Lp F p μ)

Composing f : Lp with L : E →L[𝕜] F.

Equations

• L.compLp f = ⋯.compLp ⋯ f

theorem ContinuousLinearMap.coeFn_compLp {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_7} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (L : E →L[𝕜] F) (f : ↥(MeasureTheory.Lp E p μ)) : ∀ᵐ (a : α) ∂μ, ↑↑(L.compLp f) a = L (↑↑f a)

theorem ContinuousLinearMap.coeFn_compLp' {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_7} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (L : E →L[𝕜] F) (f : ↥(MeasureTheory.Lp E p μ)) : ↑↑(L.compLp f) =ᵐ[μ] fun (a : α) => L (↑↑f a)

theorem ContinuousLinearMap.comp_memℒp {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_7} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (L : E →L[𝕜] F) (f : ↥(MeasureTheory.Lp E p μ)) : MeasureTheory.Memℒp (⇑L ∘ ↑↑f) p μ

theorem ContinuousLinearMap.comp_memℒp' {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_7} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (L : E →L[𝕜] F) {f : α → E} (hf : MeasureTheory.Memℒp f p μ) : MeasureTheory.Memℒp (⇑L ∘ f) p μ

theorem MeasureTheory.Memℒp.ofReal {α : Type u_1} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} {K : Type u_8} [RCLike K] {f : α → ℝ} (hf : Memℒp f p μ) : Memℒp (fun (x : α) => ↑(f x)) p μ

theorem MeasureTheory.memℒp_re_im_iff {α : Type u_1} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} {K : Type u_8} [RCLike K] {f : α → K} : Memℒp (fun (x : α) => RCLike.re (f x)) p μ ∧ Memℒp (fun (x : α) => RCLike.im (f x)) p μ ↔ Memℒp f p μ

theorem ContinuousLinearMap.add_compLp {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_7} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (L L' : E →L[𝕜] F) (f : ↥(MeasureTheory.Lp E p μ)) : (L + L').compLp f = L.compLp f + L'.compLp f

theorem ContinuousLinearMap.smul_compLp {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_7} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {𝕜' : Type u_8} [NormedRing 𝕜'] [Module 𝕜' F] [BoundedSMul 𝕜' F] [SMulCommClass 𝕜 𝕜' F] (c : 𝕜') (L : E →L[𝕜] F) (f : ↥(MeasureTheory.Lp E p μ)) : (c • L).compLp f = c • L.compLp f

theorem ContinuousLinearMap.norm_compLp_le {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_7} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (L : E →L[𝕜] F) (f : ↥(MeasureTheory.Lp E p μ)) : ‖L.compLp f‖ ≤ ‖L‖ * ‖f‖

def ContinuousLinearMap.compLpₗ {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} (p : ENNReal) (μ : MeasureTheory.Measure α) [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_7} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] (L : E →L[𝕜] F) : ↥(MeasureTheory.Lp E p μ) →ₗ[𝕜] ↥(MeasureTheory.Lp F p μ)

Composing f : Lp E p μ with L : E →L[𝕜] F, seen as a 𝕜-linear map on Lp E p μ.

Equations

• ContinuousLinearMap.compLpₗ p μ L = { toFun := fun (f : ↥(MeasureTheory.Lp E p μ)) => L.compLp f, map_add' := ⋯, map_smul' := ⋯ }

def ContinuousLinearMap.compLpL {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} (p : ENNReal) (μ : MeasureTheory.Measure α) [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_7} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [Fact (1 ≤ p)] (L : E →L[𝕜] F) : ↥(MeasureTheory.Lp E p μ) →L[𝕜] ↥(MeasureTheory.Lp F p μ)

Composing f : Lp E p μ with L : E →L[𝕜] F, seen as a continuous 𝕜-linear map on Lp E p μ. See also the similar

• LinearMap.compLeft for functions,
• ContinuousLinearMap.compLeftContinuous for continuous functions,
• ContinuousLinearMap.compLeftContinuousBounded for bounded continuous functions,
• ContinuousLinearMap.compLeftContinuousCompact for continuous functions on compact spaces.

Equations

• ContinuousLinearMap.compLpL p μ L = (ContinuousLinearMap.compLpₗ p μ L).mkContinuous ‖L‖ ⋯

theorem ContinuousLinearMap.coeFn_compLpL {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_7} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [Fact (1 ≤ p)] (L : E →L[𝕜] F) (f : ↥(MeasureTheory.Lp E p μ)) : ↑↑((compLpL p μ L) f) =ᵐ[μ] fun (a : α) => L (↑↑f a)

theorem ContinuousLinearMap.add_compLpL {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_7} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [Fact (1 ≤ p)] (L L' : E →L[𝕜] F) : compLpL p μ (L + L') = compLpL p μ L + compLpL p μ L'

theorem ContinuousLinearMap.smul_compLpL {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_7} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [Fact (1 ≤ p)] {𝕜' : Type u_8} [NormedRing 𝕜'] [Module 𝕜' F] [BoundedSMul 𝕜' F] [SMulCommClass 𝕜 𝕜' F] (c : 𝕜') (L : E →L[𝕜] F) : compLpL p μ (c • L) = c • compLpL p μ L

theorem ContinuousLinearMap.norm_compLpL_le {α : Type u_1} {E : Type u_4} {F : Type u_5} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedAddCommGroup F] {𝕜 : Type u_7} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [Fact (1 ≤ p)] (L : E →L[𝕜] F) : ‖compLpL p μ L‖ ≤ ‖L‖

theorem MeasureTheory.indicatorConstLp_eq_toSpanSingleton_compLp {α : Type u_1} {F : Type u_5} {m0 : MeasurableSpace α} {μ : Measure α} [NormedAddCommGroup F] {s : Set α} [NormedSpace ℝ F] (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : F) : indicatorConstLp 2 hs hμs x = (ContinuousLinearMap.toSpanSingleton ℝ x).compLp (indicatorConstLp 2 hs hμs 1)

theorem MeasureTheory.Lp.lipschitzWith_pos_part : LipschitzWith 1 fun (x : ℝ) => x ⊔ 0

theorem MeasureTheory.Memℒp.pos_part {α : Type u_1} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} {f : α → ℝ} (hf : Memℒp f p μ) : Memℒp (fun (x : α) => f x ⊔ 0) p μ

theorem MeasureTheory.Memℒp.neg_part {α : Type u_1} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} {f : α → ℝ} (hf : Memℒp f p μ) : Memℒp (fun (x : α) => -f x ⊔ 0) p μ

def MeasureTheory.Lp.posPart {α : Type u_1} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} (f : ↥(Lp ℝ p μ)) : ↥(Lp ℝ p μ)

Positive part of a function in L^p.

Equations

• MeasureTheory.Lp.posPart f = MeasureTheory.Lp.lipschitzWith_pos_part.compLp MeasureTheory.Lp.posPart.proof_2 f

def MeasureTheory.Lp.negPart {α : Type u_1} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} (f : ↥(Lp ℝ p μ)) : ↥(Lp ℝ p μ)

Negative part of a function in L^p.

Equations

• MeasureTheory.Lp.negPart f = MeasureTheory.Lp.posPart (-f)

theorem MeasureTheory.Lp.coe_posPart {α : Type u_1} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} (f : ↥(Lp ℝ p μ)) : ↑(posPart f) = (↑f).posPart

theorem MeasureTheory.Lp.coeFn_posPart {α : Type u_1} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} (f : ↥(Lp ℝ p μ)) : ↑↑(posPart f) =ᶠ[ae μ] fun (a : α) => ↑↑f a ⊔ 0

theorem MeasureTheory.Lp.coeFn_negPart_eq_max {α : Type u_1} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} (f : ↥(Lp ℝ p μ)) : ∀ᵐ (a : α) ∂μ, ↑↑(negPart f) a = -↑↑f a ⊔ 0

theorem MeasureTheory.Lp.coeFn_negPart {α : Type u_1} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} (f : ↥(Lp ℝ p μ)) : ∀ᵐ (a : α) ∂μ, ↑↑(negPart f) a = -(↑↑f a ⊓ 0)

theorem MeasureTheory.Lp.continuous_posPart {α : Type u_1} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [Fact (1 ≤ p)] : Continuous fun (f : ↥(Lp ℝ p μ)) => posPart f

theorem MeasureTheory.Lp.continuous_negPart {α : Type u_1} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [Fact (1 ≤ p)] : Continuous fun (f : ↥(Lp ℝ p μ)) => negPart f

L^p is a complete space

We show that L^p is a complete space for 1 ≤ p.

theorem MeasureTheory.Lp.eLpNorm'_lim_eq_lintegral_liminf {α : Type u_1} {G : Type u_6} {m0 : MeasurableSpace α} {μ : Measure α} [NormedAddCommGroup G] {ι : Type u_7} [Nonempty ι] [LinearOrder ι] {f : ι → α → G} {p : ℝ} {f_lim : α → G} (h_lim : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun (n : ι) => f n x) Filter.atTop (nhds (f_lim x))) : eLpNorm' f_lim p μ = (∫⁻ (a : α), Filter.liminf (fun (x : ι) => ‖f x a‖ₑ ^ p) Filter.atTop ∂μ) ^ (1 / p)

theorem MeasureTheory.Lp.eLpNorm'_lim_le_liminf_eLpNorm' {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α} {E : Type u_7} [NormedAddCommGroup E] {f : ℕ → α → E} {p : ℝ} (hp_pos : 0 < p) (hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ) {f_lim : α → E} (h_lim : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun (n : ℕ) => f n x) Filter.atTop (nhds (f_lim x))) : eLpNorm' f_lim p μ ≤ Filter.liminf (fun (n : ℕ) => eLpNorm' (f n) p μ) Filter.atTop

theorem MeasureTheory.Lp.eLpNorm_exponent_top_lim_eq_essSup_liminf {α : Type u_1} {G : Type u_6} {m0 : MeasurableSpace α} {μ : Measure α} [NormedAddCommGroup G] {ι : Type u_7} [Nonempty ι] [LinearOrder ι] {f : ι → α → G} {f_lim : α → G} (h_lim : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun (n : ι) => f n x) Filter.atTop (nhds (f_lim x))) : eLpNorm f_lim ⊤ μ = essSup (fun (x : α) => Filter.liminf (fun (m : ι) => ‖f m x‖ₑ) Filter.atTop) μ

theorem MeasureTheory.Lp.eLpNorm_exponent_top_lim_le_liminf_eLpNorm_exponent_top {α : Type u_1} {F : Type u_5} {m0 : MeasurableSpace α} {μ : Measure α} [NormedAddCommGroup F] {ι : Type u_7} [Nonempty ι] [Countable ι] [LinearOrder ι] {f : ι → α → F} {f_lim : α → F} (h_lim : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun (n : ι) => f n x) Filter.atTop (nhds (f_lim x))) : eLpNorm f_lim ⊤ μ ≤ Filter.liminf (fun (n : ι) => eLpNorm (f n) ⊤ μ) Filter.atTop

theorem MeasureTheory.Lp.eLpNorm_lim_le_liminf_eLpNorm {α : Type u_1} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} {E : Type u_7} [NormedAddCommGroup E] {f : ℕ → α → E} (hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ) (f_lim : α → E) (h_lim : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun (n : ℕ) => f n x) Filter.atTop (nhds (f_lim x))) : eLpNorm f_lim p μ ≤ Filter.liminf (fun (n : ℕ) => eLpNorm (f n) p μ) Filter.atTop

Lp is complete iff Cauchy sequences of ℒp have limits in ℒp

theorem MeasureTheory.Lp.tendsto_Lp_iff_tendsto_ℒp' {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {ι : Type u_7} {fi : Filter ι} [Fact (1 ≤ p)] (f : ι → ↥(Lp E p μ)) (f_lim : ↥(Lp E p μ)) : Filter.Tendsto f fi (nhds f_lim) ↔ Filter.Tendsto (fun (n : ι) => eLpNorm (↑↑(f n) - ↑↑f_lim) p μ) fi (nhds 0)

theorem MeasureTheory.Lp.tendsto_Lp_iff_tendsto_ℒp {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {ι : Type u_7} {fi : Filter ι} [Fact (1 ≤ p)] (f : ι → ↥(Lp E p μ)) (f_lim : α → E) (f_lim_ℒp : Memℒp f_lim p μ) : Filter.Tendsto f fi (nhds (Memℒp.toLp f_lim f_lim_ℒp)) ↔ Filter.Tendsto (fun (n : ι) => eLpNorm (↑↑(f n) - f_lim) p μ) fi (nhds 0)

theorem MeasureTheory.Lp.tendsto_Lp_iff_tendsto_ℒp'' {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {ι : Type u_7} {fi : Filter ι} [Fact (1 ≤ p)] (f : ι → α → E) (f_ℒp : ∀ (n : ι), Memℒp (f n) p μ) (f_lim : α → E) (f_lim_ℒp : Memℒp f_lim p μ) : Filter.Tendsto (fun (n : ι) => Memℒp.toLp (f n) ⋯) fi (nhds (Memℒp.toLp f_lim f_lim_ℒp)) ↔ Filter.Tendsto (fun (n : ι) => eLpNorm (f n - f_lim) p μ) fi (nhds 0)

theorem MeasureTheory.Lp.tendsto_Lp_of_tendsto_ℒp {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {ι : Type u_7} {fi : Filter ι} [Fact (1 ≤ p)] {f : ι → ↥(Lp E p μ)} (f_lim : α → E) (f_lim_ℒp : Memℒp f_lim p μ) (h_tendsto : Filter.Tendsto (fun (n : ι) => eLpNorm (↑↑(f n) - f_lim) p μ) fi (nhds 0)) : Filter.Tendsto f fi (nhds (Memℒp.toLp f_lim f_lim_ℒp))

theorem MeasureTheory.Lp.cauchySeq_Lp_iff_cauchySeq_ℒp {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {ι : Type u_7} [Nonempty ι] [SemilatticeSup ι] [hp : Fact (1 ≤ p)] (f : ι → ↥(Lp E p μ)) : CauchySeq f ↔ Filter.Tendsto (fun (n : ι × ι) => eLpNorm (↑↑(f n.1) - ↑↑(f n.2)) p μ) Filter.atTop (nhds 0)

theorem MeasureTheory.Lp.completeSpace_lp_of_cauchy_complete_ℒp {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [hp : Fact (1 ≤ p)] (H : ∀ (f : ℕ → α → E), (∀ (n : ℕ), Memℒp (f n) p μ) → ∀ (B : ℕ → ENNReal), ∑' (i : ℕ), B i < ⊤ → (∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N) → ∃ (f_lim : α → E), Memℒp f_lim p μ ∧ Filter.Tendsto (fun (n : ℕ) => eLpNorm (f n - f_lim) p μ) Filter.atTop (nhds 0)) : CompleteSpace ↥(Lp E p μ)

Prove that controlled Cauchy sequences of ℒp have limits in ℒp

theorem MeasureTheory.Lp.ae_tendsto_of_cauchy_eLpNorm' {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {μ : Measure α} [NormedAddCommGroup E] [CompleteSpace E] {f : ℕ → α → E} {p : ℝ} (hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ) (hp1 : 1 ≤ p) {B : ℕ → ENNReal} (hB : ∑' (i : ℕ), B i ≠ ⊤) (h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm' (f n - f m) p μ < B N) : ∀ᵐ (x : α) ∂μ, ∃ (l : E), Filter.Tendsto (fun (n : ℕ) => f n x) Filter.atTop (nhds l)

theorem MeasureTheory.Lp.ae_tendsto_of_cauchy_eLpNorm {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [CompleteSpace E] {f : ℕ → α → E} (hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ENNReal} (hB : ∑' (i : ℕ), B i ≠ ⊤) (h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N) : ∀ᵐ (x : α) ∂μ, ∃ (l : E), Filter.Tendsto (fun (n : ℕ) => f n x) Filter.atTop (nhds l)

theorem MeasureTheory.Lp.cauchy_tendsto_of_tendsto {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : ℕ → α → E} (hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ) (f_lim : α → E) {B : ℕ → ENNReal} (hB : ∑' (i : ℕ), B i ≠ ⊤) (h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N) (h_lim : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun (n : ℕ) => f n x) Filter.atTop (nhds (f_lim x))) : Filter.Tendsto (fun (n : ℕ) => eLpNorm (f n - f_lim) p μ) Filter.atTop (nhds 0)

theorem MeasureTheory.Lp.memℒp_of_cauchy_tendsto {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (hp : 1 ≤ p) {f : ℕ → α → E} (hf : ∀ (n : ℕ), Memℒp (f n) p μ) (f_lim : α → E) (h_lim_meas : AEStronglyMeasurable f_lim μ) (h_tendsto : Filter.Tendsto (fun (n : ℕ) => eLpNorm (f n - f_lim) p μ) Filter.atTop (nhds 0)) : Memℒp f_lim p μ

theorem MeasureTheory.Lp.cauchy_complete_ℒp {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [CompleteSpace E] (hp : 1 ≤ p) {f : ℕ → α → E} (hf : ∀ (n : ℕ), Memℒp (f n) p μ) {B : ℕ → ENNReal} (hB : ∑' (i : ℕ), B i ≠ ⊤) (h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N) : ∃ (f_lim : α → E), Memℒp f_lim p μ ∧ Filter.Tendsto (fun (n : ℕ) => eLpNorm (f n - f_lim) p μ) Filter.atTop (nhds 0)

Lp is complete for 1 ≤ p

instance MeasureTheory.Lp.instCompleteSpace {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [CompleteSpace E] [hp : Fact (1 ≤ p)] : CompleteSpace ↥(Lp E p μ)

theorem MeasureTheory.Lp.pow_mul_meas_ge_le_enorm {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (ε : ENNReal) : (ε * μ {x : α | ε ≤ ‖↑↑f x‖ₑ ^ p.toReal}) ^ (1 / p.toReal) ≤ ENNReal.ofReal ‖f‖

A version of Markov's inequality with elements of Lp.

theorem MeasureTheory.Lp.mul_meas_ge_le_pow_enorm {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (ε : ENNReal) : ε * μ {x : α | ε ≤ ‖↑↑f x‖ₑ ^ p.toReal} ≤ ENNReal.ofReal ‖f‖ ^ p.toReal

A version of Markov's inequality with elements of Lp.

theorem MeasureTheory.Lp.mul_meas_ge_le_pow_enorm' {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (ε : ENNReal) : ε ^ p.toReal * μ {x : α | ε ≤ ↑‖↑↑f x‖₊} ≤ ENNReal.ofReal ‖f‖ ^ p.toReal

A version of Markov's inequality with elements of Lp.

theorem MeasureTheory.Lp.meas_ge_le_mul_pow_enorm {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) {ε : ENNReal} (hε : ε ≠ 0) : μ {x : α | ε ≤ ↑‖↑↑f x‖₊} ≤ ε⁻¹ ^ p.toReal * ENNReal.ofReal ‖f‖ ^ p.toReal

A version of Markov's inequality with elements of Lp.

@[deprecated MeasureTheory.Lp.pow_mul_meas_ge_le_enorm (since := "2025-01-20")]

theorem MeasureTheory.Lp.pow_mul_meas_ge_le_norm {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (ε : ENNReal) : (ε * μ {x : α | ε ≤ ‖↑↑f x‖ₑ ^ p.toReal}) ^ (1 / p.toReal) ≤ ENNReal.ofReal ‖f‖

Alias of MeasureTheory.Lp.pow_mul_meas_ge_le_enorm.

---

A version of Markov's inequality with elements of Lp.

@[deprecated MeasureTheory.Lp.mul_meas_ge_le_pow_enorm (since := "2025-01-20")]

theorem MeasureTheory.Lp.mul_meas_ge_le_pow_norm {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (ε : ENNReal) : ε * μ {x : α | ε ≤ ‖↑↑f x‖ₑ ^ p.toReal} ≤ ENNReal.ofReal ‖f‖ ^ p.toReal

Alias of MeasureTheory.Lp.mul_meas_ge_le_pow_enorm.

---

A version of Markov's inequality with elements of Lp.

@[deprecated MeasureTheory.Lp.mul_meas_ge_le_pow_enorm' (since := "2025-01-20")]

theorem MeasureTheory.Lp.mul_meas_ge_le_pow_norm' {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (ε : ENNReal) : ε ^ p.toReal * μ {x : α | ε ≤ ↑‖↑↑f x‖₊} ≤ ENNReal.ofReal ‖f‖ ^ p.toReal

Alias of MeasureTheory.Lp.mul_meas_ge_le_pow_enorm'.

---

A version of Markov's inequality with elements of Lp.

@[deprecated MeasureTheory.Lp.meas_ge_le_mul_pow_enorm (since := "2025-01-20")]

theorem MeasureTheory.Lp.meas_ge_le_mul_pow_norm {α : Type u_1} {E : Type u_4} {m0 : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (f : ↥(Lp E p μ)) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) {ε : ENNReal} (hε : ε ≠ 0) : μ {x : α | ε ≤ ↑‖↑↑f x‖₊} ≤ ε⁻¹ ^ p.toReal * ENNReal.ofReal ‖f‖ ^ p.toReal

Alias of MeasureTheory.Lp.meas_ge_le_mul_pow_enorm.

---

A version of Markov's inequality with elements of Lp.