For sigma-finite measures, the section-measure functions are measurable

Fix $m\ge 1$ and let ${\mathcal{C}}_m$ consist of product-measurable sets $F$ for which $x\mapsto \nu (F_x\cap Y_m)$ is measurable. If $F$ is a finite disjoint union of rectangles $A_j\times B_j$, its sections are disjoint unions of those $B_j$ with $x\in A_j$. Finite additivity therefore gives $\nu (F_x\cap Y_m)={\sum }_j\nu (B_j\cap Y_m){\mathbf{1}}_{A_j}(x)\mathrm{.}$ All coefficients are finite by [A1]. On each of the finitely many measurable Boolean atoms of the sets $A_j$, this function is constant; hence the inverse image of any Borel set is a finite union of those atoms. The empty union gives the zero function. Thus the entire generating algebra from [L1], not merely its individual rectangles, is contained in ${\mathcal{C}}_m$.

Sections commute with increasing unions and decreasing intersections. For an increasing sequence in ${\mathcal{C}}_m$, [L4] identifies the section-measure function of its union with the pointwise increasing limit. For a decreasing sequence, all restricted sections have measure at most $\nu (Y_m)<\mathrm{\infty }$, so [L4] likewise identifies the function of the intersection with the decreasing limit. In both cases [L3] makes that limit measurable. Thus ${\mathcal{C}}_m$ is a monotone class, and [L1], [L2] and step 1.1 imply that it contains every product-measurable set.

Applying step 2.1 to $E$ shows that $g_m(x):=\nu (E_x\cap Y_m)$ is measurable for every $m$. Since $Y_m\uparrow Y$, [L4] gives $g_m(x)\uparrow \nu (E_x)$ for each $x$, and [L3] proves measurability of $x\mapsto \nu (E_x)$.

Interchanging the two factors and using the finite-measure exhaustion $X_n\uparrow X$ proves measurability of $y\mapsto \mu (E^y)$ by the same argument. Therefore both section-measure functions are measurable.