eLpNorm_eq_zero_and_zero_of_ae_le_mul_neg

English

If g is EGMEAS, then MemLp is preserved under composition with an equivalence mapping.

Русский

Если g — эквивалентная измеримая функция, то MemLp сохраняется при композиции с эквивалентным отображением.

LaTeX

$$$\operatorname{MemLp}(g,p,\operatorname{map}(f,\mu)) \Rightarrow \operatorname{MemLp}(g\circ f,p,\mu)$$$

Lean4

/-- When `c` is negative, `‖f x‖ ≤ c * ‖g x‖` is nonsense and forces both `f` and `g` to have an
`eLpNorm` of `0`. -/
theorem eLpNorm_eq_zero_and_zero_of_ae_le_mul_neg {f : α → F} {g : α → G} {c : ℝ} (h : ∀ᵐ x ∂μ, ‖f x‖ ≤ c * ‖g x‖)
    (hc : c < 0) (p : ℝ≥0∞) : eLpNorm f p μ = 0 ∧ eLpNorm g p μ = 0 :=
  by
  simp_rw [le_mul_iff_eq_zero_of_nonneg_of_neg_of_nonneg (norm_nonneg _) hc (norm_nonneg _), norm_eq_zero,
    eventually_and] at h
  change f =ᵐ[μ] 0 ∧ g =ᵐ[μ] 0 at h
  simp [eLpNorm_congr_ae h.1, eLpNorm_congr_ae h.2]

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