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

Let $p,q\ge 1$, 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 $f_x:B\to \mathbb{R},f_x(y):=f(x,y),f^y:A\to \mathbb{R},f^y(x):=f(x,y)\mathrm{.}$ Their lower and upper section integrals are the everywhere-defined bounded functions ${\mathrm{\ell }}_B(x):=\underset{‾}{{\int }_B}{f}_x,u_B(x):=\overline{{\int }_B}{f}_x,{\mathrm{\ell }}_A(y):=\underset{‾}{{\int }_A}{f}^y,u_A(y):=\overline{{\int }_A}{f}^y,$ 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 ${\mathrm{\ell }}_B\le u_B$ and ${\mathrm{\ell }}_A\le u_A$.

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 ${\int }_A({\int }_{B}f(x,y)dy)dx:={\int }_A(x\mapsto {\int }_B{f}_x)\mathrm{.}$ 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 $h(x)={\int }_B{f}_x$ for $x\mathrm{\notin }N$ 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 $E_x:=\{y\in {\mathbb{R}}^q:(x,y)\in E\},g_x:E_x\to \mathbb{R},g_x(y):=g(x,y)\mathrm{.}$ 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.