In "Real Analysis" (1999) by Folland, p.105-106, it is stated that a function $F(x) : \mathbb{R} \to \mathbb{R}$ defined by \begin{equation} F(x)=\int^x_{-\infty} f(t)dt \end{equation} for some $f \in L^1(\mathbb{R})$ is absolutely continuous and $F'=f$ a.e.
I wonder if this result (at least partially) generalizes to multivariable cases. That is, suppose that there exists a null set $E_x \subset \mathbb{R}$ for each $x \in \mathbb{R}$ such that a function $F(x,y): \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ has the relation \begin{equation} F(x,y)=\int_{-\infty}^x f(t,y)dt \end{equation} which holds on $\cup_{x \in \mathbb{R}} \text{ }(x) \times \bigl[\mathbb{R}-E_x\bigr]$ for some $f(x,y) \in L^1(\mathbb{R}^2)$.
Then, my questions are:
- Is $F(x,y)$ absolutely continuous with respect to the $x$ variable?
- Or at least, can we say that $F(x,y)$ has a partial derivative w.r.t $x$ almost everywhere on $\mathbb{R}^2$ and $\partial_x F = f$ a.e?
Multivariable cases seem to cause some subtle issues regarding almost everywhere equality. Could anyone please clarify?