Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections

Given $\epsilon >0$, [L2] supplies a grid of $A\times B$ with Darboux gap below $\epsilon$. Its coordinate grids form a product grid, and [L1] places the lower and upper Darboux gaps of both ${\mathrm{\ell }}_B$ and $u_B$ inside that same gap.

By [L2], both ${\mathrm{\ell }}_B$ and $u_B$ are integrable. The inequalities in [L1], applied to grids with gaps tending to zero, give ${\int }_A{\mathrm{\ell }}_B\ge {\int }_{A\times B}f$ and ${\int }_A{u}_B\le {\int }_{A\times B}f$; since ${\mathrm{\ell }}_B\le u_B$, all three values are equal. The same argument after exchanging $A$ and $B$ gives the other two equalities.

Suppose $h={\int }_B{f}_x$ outside a content-zero $N$. There $h={\mathrm{\ell }}_B=u_B$. If $M$ bounds $\mathrm{\mid }h\mathrm{\mid },\mathrm{\mid }{\mathrm{\ell }}_B\mathrm{\mid },\mathrm{\mid }{u}_B\mathrm{\mid }$, a finite cube cover of $N$ with arbitrarily small total volume, refined into an outer grid, bounds the upper integral of $\mathrm{\mid }h-{\mathrm{\ell }}_B\mathrm{\mid }$ by $2M$ times that volume. The criterion [L2] therefore makes $h-{\mathrm{\ell }}_B$ integrable with integral $0$, and linearity gives ${\int }_{A}h={\int }_A{\mathrm{\ell }}_B={\int }_{A\times B}f$. The exchanged assertion is identical.