ae_eq_zero_of_eLpNorm'_eq_zero — Mathlib

English

If eLpNorm'_eq_zero implies a.e. zero, then from eLpNorm' = 0 we get a.e. zero.

Русский

Если eLpNorm'_eq_zero ⇒ a.e. ноль, то eLpNorm'_eq_zero ⇒ a.e. ноль.

LaTeX

$$ae_eq_zero_of_eLpNorm'_eq_zero$$

Lean4

theorem ae_eq_zero_of_eLpNorm'_eq_zero {f : α → ε} (hq0 : 0 ≤ q) (hf : AEStronglyMeasurable f μ)
    (h : eLpNorm' f q μ = 0) : f =ᵐ[μ] 0 :=
  by
  simp only [eLpNorm'_eq_lintegral_enorm, lintegral_eq_zero_iff' (hf.enorm.pow_const q), one_div,
    ENNReal.rpow_eq_zero_iff, inv_pos, inv_neg'', hq0.not_gt, and_false, or_false] at h
  refine h.left.mono fun x hx ↦ ?_
  simp only [Pi.ofNat_apply, ENNReal.rpow_eq_zero_iff, enorm_eq_zero, h.2.not_gt, and_false, or_false] at hx
  simp [hx.1]

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