Documentation

Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Real-valued Lᵖ norm #

This file proves theorems about MeasureTheory.lpNorm, a real-valued version of MeasureTheory.eLpNorm.

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theorem MeasureTheory.toReal_eLpNorm {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} :
(eLpNorm f p μ).toReal = lpNorm f p μ
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theorem MeasureTheory.ofReal_lpNorm {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} (hf : MemLp f p μ) :
ENNReal.ofReal (lpNorm f p μ) = eLpNorm f p μ
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@[simp]
theorem MeasureTheory.lpNorm_of_not_aestronglyMeasurable {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} (hf : ¬AEStronglyMeasurable f μ) :
lpNorm f p μ = 0
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@[simp]
theorem MeasureTheory.lpNorm_of_not_memLp {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} (hf' : ¬MemLp f p μ) :
lpNorm f p μ = 0
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@[simp]
theorem MeasureTheory.lpNorm_nonneg {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} :
0 ≤ lpNorm f p μ
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theorem MeasureTheory.lpNorm_eq_integral_norm_rpow_toReal {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} (hp₀ : p ≠ 0) (hp : p ≠ ⊤) (hf : AEStronglyMeasurable f μ) :
lpNorm f p μ = (∫ (x : α), ‖f x‖ ^ p.toReal ∂μ) ^ p.toReal⁻¹
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theorem MeasureTheory.lpNorm_nnreal_eq_integral_norm_rpow {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} {p : NNReal} (hp : p ≠ 0) (hf : AEStronglyMeasurable f μ) :
lpNorm f (↑p) μ = (∫ (x : α), ‖f x‖ ^ ↑p ∂μ) ^ (↑p)⁻¹
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theorem MeasureTheory.lpNorm_one_eq_integral_norm {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} (hf : AEStronglyMeasurable f μ) :
lpNorm f 1 μ = ∫ (x : α), ‖f x‖ ∂μ
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@[simp]
theorem MeasureTheory.lpNorm_exponent_zero {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : Measure α} [NormedAddCommGroup E] (f : α → E) :
lpNorm f 0 μ = 0
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@[simp]
theorem MeasureTheory.lpNorm_measure_zero {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} [NormedAddCommGroup E] (f : α → E) :
lpNorm f p 0 = 0
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theorem MeasureTheory.ae_le_lpNorm_exponent_top {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} (hf : MemLp f ⊤ μ) :
∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ lpNorm f ⊤ μ
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theorem MeasureTheory.lpNorm_exponent_top_eq_essSup {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} (hf : MemLp f ⊤ μ) :
lpNorm f ⊤ μ = essSup (fun (x : α) => ‖f x‖) μ
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@[simp]
theorem MeasureTheory.lpNorm_zero {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] (p : ENNReal) (μ : Measure α) :
lpNorm 0 p μ = 0
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@[simp]
theorem MeasureTheory.lpNorm_fun_zero {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] (p : ENNReal) (μ : Measure α) :
lpNorm (fun (x : α) => 0) p μ = 0
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@[simp]
theorem MeasureTheory.lpNorm_eq_zero {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} (hf : MemLp f p μ) (hp : p ≠ 0) :
lpNorm f p μ = 0 ↔ f =ᵐ[μ] 0
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@[simp]
theorem MeasureTheory.lpNorm_of_isEmpty {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : Measure α} [NormedAddCommGroup E] [IsEmpty α] (f : α → E) (p : ENNReal) :
lpNorm f p μ = 0
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@[simp]
theorem MeasureTheory.lpNorm_neg {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] (f : α → E) (p : ENNReal) (μ : Measure α) :
lpNorm (-f) p μ = lpNorm f p μ
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@[simp]
theorem MeasureTheory.lpNorm_fun_neg {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] (f : α → E) (p : ENNReal) (μ : Measure α) :
lpNorm (fun (i : α) => -f i) p μ = lpNorm f p μ

Eta-expanded form of MeasureTheory.lpNorm_neg

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theorem MeasureTheory.lpNorm_sub_comm {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] (f g : α → E) (p : ENNReal) (μ : Measure α) :
lpNorm (f - g) p μ = lpNorm (g - f) p μ
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@[simp]
theorem MeasureTheory.lpNorm_norm {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} (hf : AEStronglyMeasurable f μ) (p : ENNReal) :
lpNorm (fun (x : α) => ‖f x‖) p μ = lpNorm f p μ
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@[simp]
theorem MeasureTheory.lpNorm_abs {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {f : α → ℝ} (hf : AEStronglyMeasurable f μ) (p : ENNReal) :
lpNorm |f| p μ = lpNorm f p μ
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@[simp]
theorem MeasureTheory.lpNorm_fun_abs {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {f : α → ℝ} (hf : AEStronglyMeasurable f μ) (p : ENNReal) :
lpNorm (fun (x : α) => |f x|) p μ = lpNorm f p μ
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@[simp]
theorem MeasureTheory.lpNorm_const {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (hp : p ≠ 0) (hμ : μ ≠ 0) (c : E) :
lpNorm (fun (_x : α) => c) p μ = ‖c‖ * μ.real Set.univ ^ p.toReal⁻¹
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@[simp]
theorem MeasureTheory.lpNorm_const' {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (hp₀ : p ≠ 0) (hp : p ≠ ⊤) (c : E) :
lpNorm (fun (_x : α) => c) p μ = ‖c‖ * μ.real Set.univ ^ p.toReal⁻¹
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@[simp]
theorem MeasureTheory.lpNorm_one {α : Type u_1} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} {𝕜 : Type u_3} [NormedField 𝕜] (hp : p ≠ 0) (hμ : μ ≠ 0) :
lpNorm 1 p μ = μ.real Set.univ ^ p.toReal⁻¹
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@[simp]
theorem MeasureTheory.lpNorm_one' {α : Type u_1} {m : MeasurableSpace α} {p : ENNReal} {𝕜 : Type u_3} [NormedField 𝕜] (hp₀ : p ≠ 0) (hp : p ≠ ⊤) (μ : Measure α) :
lpNorm 1 p μ = μ.real Set.univ ^ p.toReal⁻¹
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theorem MeasureTheory.lpNorm_const_smul {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} [NormedAddCommGroup E] {𝕜 : Type u_3} [NormedField 𝕜] [Module 𝕜 E] [NormSMulClass 𝕜 E] (c : 𝕜) (f : α → E) (μ : Measure α) :
lpNorm (c • f) p μ = ↑‖c‖₊ * lpNorm f p μ
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theorem MeasureTheory.lpNorm_nsmul {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} [NormedAddCommGroup E] [NormedSpace ℝ E] (n : ℕ) (f : α → E) (μ : Measure α) :
lpNorm (n • f) p μ = n • lpNorm f p μ
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theorem MeasureTheory.lpNorm_natCast_mul {α : Type u_1} {m : MeasurableSpace α} {𝕜 : Type u_3} [NormedField 𝕜] [NormedSpace ℝ 𝕜] (n : ℕ) (f : α → 𝕜) (p : ENNReal) (μ : Measure α) :
lpNorm (↑n * f) p μ = ↑n * lpNorm f p μ
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theorem MeasureTheory.lpNorm_fun_natCast_mul {α : Type u_1} {m : MeasurableSpace α} {𝕜 : Type u_3} [NormedField 𝕜] [NormedSpace ℝ 𝕜] (n : ℕ) (f : α → 𝕜) (p : ENNReal) (μ : Measure α) :
lpNorm (fun (x : α) => ↑n * f x) p μ = ↑n * lpNorm f p μ
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theorem MeasureTheory.lpNorm_mul_natCast {α : Type u_1} {m : MeasurableSpace α} {𝕜 : Type u_3} [NormedField 𝕜] [NormedSpace ℝ 𝕜] (f : α → 𝕜) (n : ℕ) (p : ENNReal) (μ : Measure α) :
lpNorm (f * ↑n) p μ = lpNorm f p μ * ↑n
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theorem MeasureTheory.lpNorm_fun_mul_natCast {α : Type u_1} {m : MeasurableSpace α} {𝕜 : Type u_3} [NormedField 𝕜] [NormedSpace ℝ 𝕜] (f : α → 𝕜) (n : ℕ) (p : ENNReal) (μ : Measure α) :
lpNorm (fun (x : α) => f x * ↑n) p μ = lpNorm f p μ * ↑n
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theorem MeasureTheory.lpNorm_div_natCast {α : Type u_1} {m : MeasurableSpace α} {𝕜 : Type u_3} [NormedField 𝕜] [NormedSpace ℝ 𝕜] [CharZero 𝕜] {n : ℕ} (hn : n ≠ 0) (f : α → 𝕜) (p : ENNReal) (μ : Measure α) :
lpNorm (f / ↑n) p μ = lpNorm f p μ / ↑n
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theorem MeasureTheory.lpNorm_fun_div_natCast {α : Type u_1} {m : MeasurableSpace α} {𝕜 : Type u_3} [NormedField 𝕜] [NormedSpace ℝ 𝕜] [CharZero 𝕜] {n : ℕ} (hn : n ≠ 0) (f : α → 𝕜) (p : ENNReal) (μ : Measure α) :
lpNorm (fun (x : α) => f x / ↑n) p μ = lpNorm f p μ / ↑n
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theorem MeasureTheory.lpNorm_add_le {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f g : α → E} (hf : MemLp f p μ) (hp : 1 ≤ p) :
lpNorm (f + g) p μ ≤ lpNorm f p μ + lpNorm g p μ
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theorem MeasureTheory.lpNorm_add_le' {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f g : α → E} (hg : MemLp g p μ) (hp : 1 ≤ p) :
lpNorm (f + g) p μ ≤ lpNorm f p μ + lpNorm g p μ
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theorem MeasureTheory.lpNorm_sub_le {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f g : α → E} (hf : MemLp f p μ) (hp : 1 ≤ p) :
lpNorm (f - g) p μ ≤ lpNorm f p μ + lpNorm g p μ
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theorem MeasureTheory.lpNorm_le_lpNorm_add_lpNorm_sub' {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f g : α → E} (hg : MemLp g p μ) (hp : 1 ≤ p) :
lpNorm f p μ ≤ lpNorm g p μ + lpNorm (f - g) p μ
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theorem MeasureTheory.lpNorm_le_lpNorm_add_lpNorm_sub {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f g : α → E} (hg : MemLp g p μ) (hp : 1 ≤ p) :
lpNorm f p μ ≤ lpNorm g p μ + lpNorm (g - f) p μ
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theorem MeasureTheory.lpNorm_le_add_lpNorm_add {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f g : α → E} (hg : MemLp g p μ) (hp : 1 ≤ p) :
lpNorm f p μ ≤ lpNorm (f + g) p μ + lpNorm g p μ
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theorem MeasureTheory.lpNorm_sub_le_lpNorm_sub_add_lpNorm_sub {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f g h : α → E} (hf : MemLp f p μ) (hg : MemLp g p μ) (hp : 1 ≤ p) :
lpNorm (f - h) p μ ≤ lpNorm (f - g) p μ + lpNorm (g - h) p μ
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theorem MeasureTheory.lpNorm_sum_le {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {ι : Type u_3} {s : Finset ι} {f : ι → α → E} (hf : ∀ i ∈ s, MemLp (f i) p μ) (hp : 1 ≤ p) :
lpNorm (∑ i ∈ s, f i) p μ ≤ ∑ i ∈ s, lpNorm (f i) p μ
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theorem MeasureTheory.lpNorm_expect_le {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] [Module ℚ≥0 E] [NormedSpace ℝ E] {ι : Type u_3} {s : Finset ι} {f : ι → α → E} (hf : ∀ i ∈ s, MemLp (f i) p μ) (hp : 1 ≤ p) :
lpNorm (s.expect fun (i : ι) => f i) p μ ≤ s.expect fun (i : ι) => lpNorm (f i) p μ
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theorem MeasureTheory.lpNorm_mono_real {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} {g : α → ℝ} (hg : MemLp g p μ) (h : ∀ (x : α), ‖f x‖ ≤ g x) :
lpNorm f p μ ≤ lpNorm g p μ
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theorem MeasureTheory.lpNorm_smul_measure_of_ne_zero {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] {f : α → E} {c : NNReal} (hc : c ≠ 0) :
lpNorm f p (c • μ) = c ^ p.toReal⁻¹ • lpNorm f p μ
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theorem MeasureTheory.lpNorm_smul_measure_of_ne_top {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {p : ENNReal} {μ : Measure α} [NormedAddCommGroup E] (hp : p ≠ ⊤) {f : α → E} (c : NNReal) :
lpNorm f p (c • μ) = c ^ p.toReal⁻¹ • lpNorm f p μ
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@[simp]
theorem MeasureTheory.lpNorm_conj {α : Type u_1} {m : MeasurableSpace α} {K : Type u_3} [RCLike K] (f : α → K) (p : ENNReal) (μ : Measure α) :
lpNorm ((starRingEnd (α → K)) f) p μ = lpNorm f p μ