Rademacher's Theorem can be interpreted as an almost everywhere little-o improvement principle: if a function admits a uniform pointwise first-order Lipschitz control at every point, then this control improves to a vanishing one at almost every point. In the language of Calderón-Zygmund pointwise spaces, this means that \(f \in T^\infty_1(x) \ \ {\rm for\, all}\,\, x \in \mathbb{R}^d {\implies}\, f \in t^\infty_1(x) \ \ \text{for a.e.}\,\, x \in \mathbb{R}^d.\)
The purpose of this paper is to establish an analogous almost everywhere improvement principle in a refined \(L^p\) setting. We consider pointwise Calderón-Zygmund spaces \(T^p_{\phi}(x)\) defined via polynomial approximation in \(L^p\) with a function parameter \(\phi\) , allowing for fractional regularity indices and logarithmic corrections through Boyd functions. We prove that, under natural assumptions on \(\phi\) , the uniform membership \(f \in T^p_{\phi}(x) \quad {\rm for\,\, all}\,\, x \in E\) on a measurable set \(E \subset \mathbb{R}^d\) implies an almost everywhere improvement to a vanishing approximation rate, namely \(f \in t^p_{\phi,n+1}(x) \quad \text{for a.e. } x \in E,\) where \(n < \underline{b}(\phi) \leq \overline{b}(\phi) < n+1\) .
The proof combines measurability arguments, a generalized Whitney extension theorem, and fine properties of Sobolev spaces. We also show that this result is essentially sharp: in general, one cannot expect almost everywhere membership in \(t^p_{\phi,n}(x)\) for fractional indices, and explicit counterexamples are provided.
