Sections, lower and upper section integrals, and iterated Riemann integrals on product rectangles and Jordan sets

Let $p,q\ge1$, let $A\subseteq\mathbb R^p$ and $B\subseteq\mathbb R^q$ be nondegenerate closed rectangles, and let $f:A\times B\to\mathbb R$ be bounded. For $x\in A$ and $y\in B$, the sections of $f$ are $f_x:B\to\mathbb R,\quad f_x(y):=f(x,y),\qquad f^y:A\to\mathbb R,\quad f^y(x):=f(x,y).$ Their lower and upper section integrals are the everywhere-defined bounded functions $\ell_B(x):=\underline{\int_B}f_x,\quad u_B(x):=\overline{\int_B}f_x,\qquad \ell_A(y):=\underline{\int_A}f^y,\quad u_A(y):=\overline{\int_A}f^y,$ using The lower and upper Darboux integrals over a nondegenerate rectangle in $\mathbb{R}^m$. They are defined even when the corresponding section is not Riemann integrable, and always satisfy $\ell_B\le u_B$ and $\ell_A\le u_A$.

If every $f_x$ is integrable and the function $x\mapsto\int_B f_x$ is integrable on $A$, define the ordinary iterated integral in the $B$-then-$A$ order by $\int_A\left(\int_B f(x,y)\,dy\right)dx:=\int_A\left(x\mapsto\int_B f_x\right).$ The other order is defined symmetrically. More generally, if the sections are integrable outside a content-zero set $N\subseteq A$, any bounded function $h:A\to\mathbb R$ satisfying $h(x)=\int_Bf_x$ for $x\notin N$ is an exceptionally completed section-integral function. Its integral, when it exists, is independent of its values on $N$.

Let now $E\subseteq\mathbb R^{p+q}$ be a bounded Jordan set and $g:E\to\mathbb R$ be bounded. Its section at $x\in\mathbb R^p$ is $E_x:=\{y\in\mathbb R^q:(x,y)\in E\},\qquad g_x:E_x\to\mathbb R,\quad g_x(y):=g(x,y).$ Empty sections have integral $0$. For a nonempty Jordan section, $\int_{E_x}g_x$ means the Jordan-set integral of The Riemann integral of a bounded function over a bounded Jordan measurable set. Section integrals over a Jordan set and their iterated integrals are defined by first choosing factor rectangles with $E\subseteq A\times B$ and applying the preceding conventions to the zero extension of $g$. Independence of those rectangles is proved in Fubini over a bounded Jordan set when all but a content-zero family of sections are integrable.