File: Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean

full_name: MeasureTheory.eLpNorm'_le_eLpNorm'_mul_rpow_measure_univ
code: theorem eLpNorm'_le_eLpNorm'_mul_rpow_measure_univ {p q : ℝ} (hp0_lt : 0 < p) (hpq : p ≤ q)
    (hf : AEStronglyMeasurable f μ) :
    eLpNorm' f p μ ≤ eLpNorm' f q μ * μ Set.univ ^ (1 / p - 1 / q) 
goal: MeasureTheory.eLpNorm' f p μ ≤ MeasureTheory.eLpNorm' f q μ * μ Set.univ ^ (1 / p - 1 / q)

full_name: MeasureTheory.eLpNorm'_le_eLpNormEssSup_mul_rpow_measure_univ
code: theorem eLpNorm'_le_eLpNormEssSup_mul_rpow_measure_univ {q : ℝ} (hq_pos : 0 < q) :
    eLpNorm' f q μ ≤ eLpNormEssSup f μ * μ Set.univ ^ (1 / q) 
goal: MeasureTheory.eLpNorm' f q μ ≤ MeasureTheory.eLpNormEssSup f μ * μ Set.univ ^ (1 / q)

full_name: MeasureTheory.eLpNorm_le_eLpNorm_mul_rpow_measure_univ
code: theorem eLpNorm_le_eLpNorm_mul_rpow_measure_univ {p q : ℝ≥0∞} (hpq : p ≤ q)
    (hf : AEStronglyMeasurable f μ) :
    eLpNorm f p μ ≤ eLpNorm f q μ * μ Set.univ ^ (1 / p.toReal - 1 / q.toReal) 
goal: MeasureTheory.eLpNorm f p μ ≤ MeasureTheory.eLpNorm f q μ * μ Set.univ ^ (1 / p.toReal - 1 / q.toReal)

full_name: MeasureTheory.eLpNorm'_le_eLpNorm'_of_exponent_le
code: theorem eLpNorm'_le_eLpNorm'_of_exponent_le {p q : ℝ} (hp0_lt : 0 < p)
    (hpq : p ≤ q) (μ : Measure α) [IsProbabilityMeasure μ] (hf : AEStronglyMeasurable f μ) :
    eLpNorm' f p μ ≤ eLpNorm' f q μ 
goal: MeasureTheory.eLpNorm' f p μ ≤ MeasureTheory.eLpNorm' f q μ

full_name: MeasureTheory.eLpNorm'_le_eLpNormEssSup
code: theorem eLpNorm'_le_eLpNormEssSup {q : ℝ} (hq_pos : 0 < q) [IsProbabilityMeasure μ] :
    eLpNorm' f q μ ≤ eLpNormEssSup f μ 
goal: MeasureTheory.eLpNorm' f q μ ≤ MeasureTheory.eLpNormEssSup f μ

full_name: MeasureTheory.eLpNorm_le_eLpNorm_of_exponent_le
code: theorem eLpNorm_le_eLpNorm_of_exponent_le {p q : ℝ≥0∞} (hpq : p ≤ q) [IsProbabilityMeasure μ]
    (hf : AEStronglyMeasurable f μ) : eLpNorm f p μ ≤ eLpNorm f q μ 
goal: MeasureTheory.eLpNorm f p μ ≤ MeasureTheory.eLpNorm f q μ

full_name: MeasureTheory.eLpNorm'_lt_top_of_eLpNorm'_lt_top_of_exponent_le
code: theorem eLpNorm'_lt_top_of_eLpNorm'_lt_top_of_exponent_le {p q : ℝ} [IsFiniteMeasure μ]
    (hf : AEStronglyMeasurable f μ) (hfq_lt_top : eLpNorm' f q μ < ∞) (hp_nonneg : 0 ≤ p)
    (hpq : p ≤ q) : eLpNorm' f p μ < ∞ 
goal: MeasureTheory.eLpNorm' f p μ < ⊤

full_name: MeasureTheory.Memℒp.memℒp_of_exponent_le
code: theorem Memℒp.memℒp_of_exponent_le {p q : ℝ≥0∞} [IsFiniteMeasure μ] {f : α → E} (hfq : Memℒp f q μ)
    (hpq : p ≤ q) : Memℒp f p μ 
goal: MeasureTheory.Memℒp f p μ

full_name: MeasureTheory.eLpNorm_le_eLpNorm_top_mul_eLpNorm
code: theorem eLpNorm_le_eLpNorm_top_mul_eLpNorm (p : ℝ≥0∞) (f : α → E) {g : α → F}
    (hg : AEStronglyMeasurable g μ) (b : E → F → G)
    (h : ∀ᵐ x ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊) :
    eLpNorm (fun x => b (f x) (g x)) p μ ≤ eLpNorm f ∞ μ * eLpNorm g p μ 
goal: MeasureTheory.eLpNorm (fun x => b (f x) (g x)) p μ ≤ MeasureTheory.eLpNorm f ⊤ μ * MeasureTheory.eLpNorm g p μ

full_name: MeasureTheory.eLpNorm_le_eLpNorm_mul_eLpNorm_top
code: theorem eLpNorm_le_eLpNorm_mul_eLpNorm_top (p : ℝ≥0∞) {f : α → E} (hf : AEStronglyMeasurable f μ)
    (g : α → F) (b : E → F → G) (h : ∀ᵐ x ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊) :
    eLpNorm (fun x => b (f x) (g x)) p μ ≤ eLpNorm f p μ * eLpNorm g ∞ μ 
goal: MeasureTheory.eLpNorm (fun x => b (f x) (g x)) p μ ≤ MeasureTheory.eLpNorm f p μ * MeasureTheory.eLpNorm g ⊤ μ

full_name: MeasureTheory.eLpNorm'_le_eLpNorm'_mul_eLpNorm'
code: theorem eLpNorm'_le_eLpNorm'_mul_eLpNorm' {p q r : ℝ} (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) (b : E → F → G)
    (h : ∀ᵐ x ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊) (hp0_lt : 0 < p) (hpq : p < q)
    (hpqr : 1 / p = 1 / q + 1 / r) :
    eLpNorm' (fun x => b (f x) (g x)) p μ ≤ eLpNorm' f q μ * eLpNorm' g r μ 
goal: MeasureTheory.eLpNorm' (fun x => b (f x) (g x)) p μ ≤ MeasureTheory.eLpNorm' f q μ * MeasureTheory.eLpNorm' g r μ

full_name: MeasureTheory.eLpNorm_le_eLpNorm_mul_eLpNorm_of_nnnorm
code: /-- Hölder's inequality, as an inequality on the `ℒp` seminorm of an elementwise operation
`fun x => b (f x) (g x)`. -/
theorem eLpNorm_le_eLpNorm_mul_eLpNorm_of_nnnorm {p q r : ℝ≥0∞}
    (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) (b : E → F → G)
    (h : ∀ᵐ x ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊) (hpqr : 1 / p = 1 / q + 1 / r) :
    eLpNorm (fun x => b (f x) (g x)) p μ ≤ eLpNorm f q μ * eLpNorm g r μ 
goal: MeasureTheory.eLpNorm (fun x => b (f x) (g x)) p μ ≤ MeasureTheory.eLpNorm f q μ * MeasureTheory.eLpNorm g r μ

full_name: MeasureTheory.eLpNorm_le_eLpNorm_mul_eLpNorm'_of_norm
code: /-- Hölder's inequality, as an inequality on the `ℒp` seminorm of an elementwise operation
`fun x => b (f x) (g x)`. -/
theorem eLpNorm_le_eLpNorm_mul_eLpNorm'_of_norm {p q r : ℝ≥0∞} (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) (b : E → F → G)
    (h : ∀ᵐ x ∂μ, ‖b (f x) (g x)‖ ≤ ‖f x‖ * ‖g x‖) (hpqr : 1 / p = 1 / q + 1 / r) :
    eLpNorm (fun x => b (f x) (g x)) p μ ≤ eLpNorm f q μ * eLpNorm g r μ 
goal: MeasureTheory.eLpNorm (fun x => b (f x) (g x)) p μ ≤ MeasureTheory.eLpNorm f q μ * MeasureTheory.eLpNorm g r μ

full_name: MeasureTheory.eLpNorm_smul_le_eLpNorm_top_mul_eLpNorm
code: theorem eLpNorm_smul_le_eLpNorm_top_mul_eLpNorm (p : ℝ≥0∞) (hf : AEStronglyMeasurable f μ)
    (φ : α → 𝕜) : eLpNorm (φ • f) p μ ≤ eLpNorm φ ∞ μ * eLpNorm f p μ 
goal: MeasureTheory.eLpNorm (φ • f) p μ ≤ MeasureTheory.eLpNorm φ ⊤ μ * MeasureTheory.eLpNorm f p μ

full_name: MeasureTheory.eLpNorm_smul_le_eLpNorm_mul_eLpNorm_top
code: theorem eLpNorm_smul_le_eLpNorm_mul_eLpNorm_top (p : ℝ≥0∞) (f : α → E) {φ : α → 𝕜}
    (hφ : AEStronglyMeasurable φ μ) : eLpNorm (φ • f) p μ ≤ eLpNorm φ p μ * eLpNorm f ∞ μ 
goal: MeasureTheory.eLpNorm (φ • f) p μ ≤ MeasureTheory.eLpNorm φ p μ * MeasureTheory.eLpNorm f ⊤ μ

full_name: MeasureTheory.eLpNorm'_smul_le_mul_eLpNorm'
code: theorem eLpNorm'_smul_le_mul_eLpNorm' {p q r : ℝ} {f : α → E} (hf : AEStronglyMeasurable f μ)
    {φ : α → 𝕜} (hφ : AEStronglyMeasurable φ μ) (hp0_lt : 0 < p) (hpq : p < q)
    (hpqr : 1 / p = 1 / q + 1 / r) : eLpNorm' (φ • f) p μ ≤ eLpNorm' φ q μ * eLpNorm' f r μ 
goal: MeasureTheory.eLpNorm' (φ • f) p μ ≤ MeasureTheory.eLpNorm' φ q μ * MeasureTheory.eLpNorm' f r μ

full_name: MeasureTheory.eLpNorm_smul_le_mul_eLpNorm
code: /-- Hölder's inequality, as an inequality on the `ℒp` seminorm of a scalar product `φ • f`. -/
theorem eLpNorm_smul_le_mul_eLpNorm {p q r : ℝ≥0∞} {f : α → E} (hf : AEStronglyMeasurable f μ)
    {φ : α → 𝕜} (hφ : AEStronglyMeasurable φ μ) (hpqr : 1 / p = 1 / q + 1 / r) :
    eLpNorm (φ • f) p μ ≤ eLpNorm φ q μ * eLpNorm f r μ 
goal: MeasureTheory.eLpNorm (φ • f) p μ ≤ MeasureTheory.eLpNorm φ q μ * MeasureTheory.eLpNorm f r μ

full_name: MeasureTheory.Memℒp.smul
code: theorem Memℒp.smul {p q r : ℝ≥0∞} {f : α → E} {φ : α → 𝕜} (hf : Memℒp f r μ) (hφ : Memℒp φ q μ)
    (hpqr : 1 / p = 1 / q + 1 / r) : Memℒp (φ • f) p μ 
goal: MeasureTheory.Memℒp (φ • f) p μ

full_name: MeasureTheory.Memℒp.smul_of_top_right
code: theorem Memℒp.smul_of_top_right {p : ℝ≥0∞} {f : α → E} {φ : α → 𝕜} (hf : Memℒp f p μ)
    (hφ : Memℒp φ ∞ μ) : Memℒp (φ • f) p μ 
goal: MeasureTheory.Memℒp (φ • f) p μ

full_name: MeasureTheory.Memℒp.smul_of_top_left
code: theorem Memℒp.smul_of_top_left {p : ℝ≥0∞} {f : α → E} {φ : α → 𝕜} (hf : Memℒp f ∞ μ)
    (hφ : Memℒp φ p μ) : Memℒp (φ • f) p μ 
goal: MeasureTheory.Memℒp (φ • f) p μ