English
Under suitable conditions on c, m1, m2, the inequality m1 ≤ᶠ c • m2 implies mkMetric m1 ≤ c • mkMetric m2.
Русский
При bestimmten условиях на c, m1, m2, если m1 ≤ᶠ c • m2, то mkMetric m1 ≤ c • mkMetric m2.
LaTeX
$$$\\text{If } c \\neq \\infty, c \\neq 0, \\text{ and } m_1 \\le^\\!\\!_ {\\mathcal{N}} c \\cdot m_2, \\ \\ mkMetric(m_1) \\le c \\cdot mkMetric(m_2)$$$
Lean4
/-- If `c ∉ {0, ∞}` and `m₁ d ≤ c * m₂ d` for `d < ε` for some `ε > 0`
(we use `≤ᶠ[𝓝[≥] 0]` to state this), then `mkMetric m₁ hm₁ ≤ c • mkMetric m₂ hm₂`. -/
theorem mkMetric_mono_smul {m₁ m₂ : ℝ≥0∞ → ℝ≥0∞} {c : ℝ≥0∞} (hc : c ≠ ∞) (h0 : c ≠ 0) (hle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂) :
(mkMetric m₁ : OuterMeasure X) ≤ c • mkMetric m₂ := by
classical
rcases (mem_nhdsGE_iff_exists_Ico_subset' zero_lt_one).1 hle with ⟨r, hr0, hr⟩
refine fun s =>
le_of_tendsto_of_tendsto (mkMetric'.tendsto_pre _ s)
(ENNReal.Tendsto.const_mul (mkMetric'.tendsto_pre _ s) (Or.inr hc))
(mem_of_superset (Ioo_mem_nhdsGT hr0) fun r' hr' => ?_)
simp only [mem_setOf_eq, mkMetric'.pre]
rw [← smul_eq_mul, ← smul_apply, smul_boundedBy hc]
refine le_boundedBy.2 (fun t => (boundedBy_le _).trans ?_) _
simp only [smul_eq_mul, Pi.smul_apply, extend, iInf_eq_if]
split_ifs with ht
· apply hr
exact ⟨zero_le _, ht.trans_lt hr'.2⟩
· simp [h0]
