Continuous t-periodic maps equal almost everywhere on the strip are equal

Proved

CauchyRiemannIndex.eq_of_continuous_periodic_ae_eq

by Mazecto · Nov 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

functional-analysispseudoholomorphic-curvessobolev-spaces

Let $E$ be a Hausdorff space and $g, h : \mathbb{R}\times\mathbb{R} \to E$ continuous and $1$ -periodic in the second variable. If $g = h$ almost everywhere for Lebesgue measure on the strip $\mathbb{R}\times[0,1)$ (cylMeasure), then $g = h$ everywhere.

Role. Uniqueness of continuous representatives of classes in $L^p(\mathbb{R}\times S^1)$ .

Preamble

import Definitions.Def_CauchyRiemannIndex_Setting

Formal statement

namespace CauchyRiemannIndex

open MeasureTheory

/-- Two continuous maps on `ℝ × ℝ`, `1`-periodic in `t`, that agree `cylMeasure`-almost everywhere
are equal. -/
theorem eq_of_continuous_periodic_ae_eq {E : Type*} [TopologicalSpace E] [T2Space E]
    {g h : ℝ × ℝ → E} (hg : Continuous g) (hh : Continuous h)
    (hgp : ∀ s t, g (s, t + 1) = g (s, t)) (hhp : ∀ s t, h (s, t + 1) = h (s, t))
    (hae : g =ᵐ[cylMeasure] h) : g = h := by
  sorry

end CauchyRiemannIndex

Source

Standard Sobolev embedding on the cylinder; setting of C. Wendl, Lectures on Holomorphic Curves in Symplectic and Contact Geometry (arXiv:1011.1690), Ch. 4

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