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

full_name: MeasureTheory.eLpNorm'_add_le
code: theorem eLpNorm'_add_le (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
    (hq1 : 1 ≤ q) : eLpNorm' (f + g) q μ ≤ eLpNorm' f q μ + eLpNorm' g q μ 
goal: MeasureTheory.eLpNorm' (f + g) q μ ≤ MeasureTheory.eLpNorm' f q μ + MeasureTheory.eLpNorm' g q μ

full_name: MeasureTheory.eLpNorm'_add_le_of_le_one
code: theorem eLpNorm'_add_le_of_le_one (hf : AEStronglyMeasurable f μ) (hq0 : 0 ≤ q) (hq1 : q ≤ 1) :
    eLpNorm' (f + g) q μ ≤ (2 : ℝ≥0∞) ^ (1 / q - 1) * (eLpNorm' f q μ + eLpNorm' g q μ) 
goal: MeasureTheory.eLpNorm' (f + g) q μ ≤ 2 ^ (1 / q - 1) * (MeasureTheory.eLpNorm' f q μ + MeasureTheory.eLpNorm' g q μ)

full_name: MeasureTheory.eLpNormEssSup_add_le
code: theorem eLpNormEssSup_add_le {f g : α → E} :
    eLpNormEssSup (f + g) μ ≤ eLpNormEssSup f μ + eLpNormEssSup g μ 
goal: MeasureTheory.eLpNormEssSup (f + g) μ ≤ MeasureTheory.eLpNormEssSup f μ + MeasureTheory.eLpNormEssSup g μ

full_name: MeasureTheory.eLpNorm_add_le
code: theorem eLpNorm_add_le {f g : α → E} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
    (hp1 : 1 ≤ p) : eLpNorm (f + g) p μ ≤ eLpNorm f p μ + eLpNorm g p μ 
goal: MeasureTheory.eLpNorm (f + g) p μ ≤ MeasureTheory.eLpNorm f p μ + MeasureTheory.eLpNorm g p μ

full_name: MeasureTheory.LpAddConst
code: /-- A constant for the inequality `‖f + g‖_{L^p} ≤ C * (‖f‖_{L^p} + ‖g‖_{L^p})`. It is equal to `1`
for `p ≥ 1` or `p = 0`, and `2^(1/p-1)` in the more tricky interval `(0, 1)`. -/
noncomputable def LpAddConst (p : ℝ≥0∞) : ℝ≥0∞ 
goal: ℝ≥0∞

full_name: MeasureTheory.LpAddConst_of_one_le
code: theorem LpAddConst_of_one_le {p : ℝ≥0∞} (hp : 1 ≤ p) : LpAddConst p = 1 
goal: MeasureTheory.LpAddConst p = 1

full_name: MeasureTheory.LpAddConst_zero
code: theorem LpAddConst_zero : LpAddConst 0 = 1 
goal: MeasureTheory.LpAddConst 0 = 1

full_name: MeasureTheory.LpAddConst_lt_top
code: theorem LpAddConst_lt_top (p : ℝ≥0∞) : LpAddConst p < ∞ 
goal: MeasureTheory.LpAddConst p < ⊤

full_name: MeasureTheory.eLpNorm_add_le'
code: theorem eLpNorm_add_le' (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
    (p : ℝ≥0∞) : eLpNorm (f + g) p μ ≤ LpAddConst p * (eLpNorm f p μ + eLpNorm g p μ) 
goal: MeasureTheory.eLpNorm (f + g) p μ ≤
  MeasureTheory.LpAddConst p * (MeasureTheory.eLpNorm f p μ + MeasureTheory.eLpNorm g p μ)

full_name: MeasureTheory.exists_Lp_half
code: /-- Technical lemma to control the addition of functions in `L^p` even for `p < 1`: Given `δ > 0`,
there exists `η` such that two functions bounded by `η` in `L^p` have a sum bounded by `δ`. One
could take `η = δ / 2` for `p ≥ 1`, but the point of the lemma is that it works also for `p < 1`.
-/
theorem exists_Lp_half (p : ℝ≥0∞) {δ : ℝ≥0∞} (hδ : δ ≠ 0) :
    ∃ η : ℝ≥0∞,
      0 < η ∧
        ∀ (f g : α → E), AEStronglyMeasurable f μ → AEStronglyMeasurable g μ →
          eLpNorm f p μ ≤ η → eLpNorm g p μ ≤ η → eLpNorm (f + g) p μ < δ 
goal: ∃ η,
  0 < η ∧
    ∀ (f g : α → E),
      MeasureTheory.AEStronglyMeasurable f μ →
        MeasureTheory.AEStronglyMeasurable g μ →
          MeasureTheory.eLpNorm f p μ ≤ η → MeasureTheory.eLpNorm g p μ ≤ η → MeasureTheory.eLpNorm (f + g) p μ < δ

full_name: MeasureTheory.eLpNorm_sub_le'
code: theorem eLpNorm_sub_le' (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
    (p : ℝ≥0∞) : eLpNorm (f - g) p μ ≤ LpAddConst p * (eLpNorm f p μ + eLpNorm g p μ) 
goal: MeasureTheory.eLpNorm (f - g) p μ ≤
  MeasureTheory.LpAddConst p * (MeasureTheory.eLpNorm f p μ + MeasureTheory.eLpNorm g p μ)

full_name: MeasureTheory.eLpNorm_sub_le
code: theorem eLpNorm_sub_le {f g : α → E} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
    (hp : 1 ≤ p) : eLpNorm (f - g) p μ ≤ eLpNorm f p μ + eLpNorm g p μ 
goal: MeasureTheory.eLpNorm (f - g) p μ ≤ MeasureTheory.eLpNorm f p μ + MeasureTheory.eLpNorm g p μ

full_name: MeasureTheory.eLpNorm_add_lt_top
code: theorem eLpNorm_add_lt_top {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) :
    eLpNorm (f + g) p μ < ∞ 
goal: MeasureTheory.eLpNorm (f + g) p μ < ⊤

full_name: MeasureTheory.eLpNorm'_sum_le
code: theorem eLpNorm'_sum_le {ι} {f : ι → α → E} {s : Finset ι}
    (hfs : ∀ i, i ∈ s → AEStronglyMeasurable (f i) μ) (hq1 : 1 ≤ q) :
    eLpNorm' (∑ i ∈ s, f i) q μ ≤ ∑ i ∈ s, eLpNorm' (f i) q μ 
goal: MeasureTheory.eLpNorm' (∑ i ∈ s, f i) q μ ≤ ∑ i ∈ s, MeasureTheory.eLpNorm' (f i) q μ

full_name: MeasureTheory.eLpNorm_sum_le
code: theorem eLpNorm_sum_le {ι} {f : ι → α → E} {s : Finset ι}
    (hfs : ∀ i, i ∈ s → AEStronglyMeasurable (f i) μ) (hp1 : 1 ≤ p) :
    eLpNorm (∑ i ∈ s, f i) p μ ≤ ∑ i ∈ s, eLpNorm (f i) p μ 
goal: MeasureTheory.eLpNorm (∑ i ∈ s, f i) p μ ≤ ∑ i ∈ s, MeasureTheory.eLpNorm (f i) p μ

full_name: MeasureTheory.Memℒp.add
code: theorem Memℒp.add {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) : Memℒp (f + g) p μ 
goal: MeasureTheory.Memℒp (f + g) p μ

full_name: MeasureTheory.Memℒp.sub
code: theorem Memℒp.sub {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) : Memℒp (f - g) p μ 
goal: MeasureTheory.Memℒp (f - g) p μ

full_name: MeasureTheory.memℒp_finset_sum
code: theorem memℒp_finset_sum {ι} (s : Finset ι) {f : ι → α → E} (hf : ∀ i ∈ s, Memℒp (f i) p μ) :
    Memℒp (fun a => ∑ i ∈ s, f i a) p μ 
goal: MeasureTheory.Memℒp (fun a => ∑ i ∈ s, f i a) p μ

full_name: MeasureTheory.memℒp_finset_sum'
code: theorem memℒp_finset_sum' {ι} (s : Finset ι) {f : ι → α → E} (hf : ∀ i ∈ s, Memℒp (f i) p μ) :
    Memℒp (∑ i ∈ s, f i) p μ 
goal: MeasureTheory.Memℒp (∑ i ∈ s, f i) p μ