Absolute continuity on almost every coordinate line

• Juha Kinnunen, Sobolev Spaces, Chapter 2 §2.6, Theorem 2.36 (Nikodym, ACL characterization), statement printed p. 55 and proof pp. 56–59. The theorem uses one representative for all coordinate directions and proves its linewise absolute continuity by summable smooth approximations and Fubini. This definition records that representative property on a countable rational-box basis; it does not assert the characterization theorem.
• John K. Hunter, Notes on Partial Differential Equations, Chapter 3 §3.A.2, Definition 3.56, printed pp. 78–79, for absolute continuity on a compact interval.

Assume Countable Choice for the library's completed-product Fubini convention (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open, $n\ge 1$, and $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$. Let $u$ be an almost-everywhere class of Lebesgue-measurable maps $\mathrm{\Omega }\to \mathbb{K}$; in particular, this includes the classes used in $W^{k,p}(\mathrm{\Omega };\mathbb{K})$ under Integer-order Sobolev spaces and their norms.

Let $\mathcal{Q}(\mathrm{\Omega })$ be the countable family of open rational coordinate boxes $Q={\prod }_{j=1}^n(a_j,b_j),a_j,b_j\in \mathbb{Q},a_j<b_j,\bar{Q}\subseteq \mathrm{\Omega }\mathrm{.}$ These boxes cover $\mathrm{\Omega }$, since each point of an open set lies in a rational coordinate box whose closure remains in that set. For $n\ge 2$ and $1\le i\le n$, write $Q_{\widehat{i}}={\prod }_{j\mathrm{\ne }i}(a_j,b_j),I_{i,Q}=(a_i,b_i),$ and let ${\iota }_i(y,t)$ insert $t$ as the $i$th coordinate into the ordered list $y$ of the other $n-1$ coordinates.

A measurable representative $u^{\ast }:\mathrm{\Omega }\to \mathbb{K}$ is absolutely continuous on almost every coordinate line (ACL) if the following holds. For $n\ge 2$, for every coordinate direction $i$ there is a set $N_i\subseteq {\mathbb{R}}^{n-1}$ null for $(n-1)$-dimensional Lebesgue measure such that for every $Q\in \mathcal{Q}(\mathrm{\Omega })$ and every $y\in Q_{\widehat{i}}\setminus N_i$, the function $t⟼u^{\ast }({\iota }_i(y,t))$ is absolutely continuous in the sense of Absolute continuity on a compact interval on every compact interval contained in $I_{i,Q}$. For complex-valued functions, absolute continuity means that the real and imaginary parts are both absolutely continuous. The exceptional set $N_i$ may depend on $i$. For $n=1$, there is one coordinate line and the condition is that $u^{\ast }$ is absolutely continuous on every compact interval contained in $\mathrm{\Omega }$.

The a.e. class $u$ has an ACL representative if one globally defined measurable representative $u^{\ast }$ satisfies this property for all coordinate directions simultaneously. The zero class has the representative $u^{\ast }=0$; its line restrictions are constant and hence ACL. The representative is common to all directions; only the exceptional line sets may differ. The rational boxes give a countable local basis. Equivalently, one may start with a null exceptional set for each direction and box and take their countable union; Countable Choice is sufficient for that step. The completed-product measure convention The Euclidean Lebesgue measure is the completion of the product of the factor Lebesgue measures identifies ${\lambda }_n$ with the completed product used by Tonelli and Fubini for the completed product, with only almost-everywhere section measurability whenever sectional integrals are taken.

If $\mathrm{\Omega }=\mathrm{\varnothing }$, its unique a.e. class has the empty representative and is ACL vacuously. No boundary values or traces at $\mathrm{\partial }\mathrm{\Omega }$ are part of this definition.