Lebesgue measurable sections ($B_x=\{y:(x,y)\in B\}\text{ is measurable for a.e. }x$)

For a planar Lebesgue measurable set $B$, its vertical section $B_x$ is a one-dimensional Lebesgue measurable set for $x$ almost everywhere. Write $B△C\subseteq Z$ with $C$ a planar Borel set and $Z$ a Borel Lebesgue null set. Every section $C_x$ and $Z_x$ is Borel; Tonelli's theorem gives $\int m(Z_x)dx=m_2(Z)=0$. Thus $m(Z_x)=0$ for $x$ almost everywhere, and $B_x△C_x\subseteq Z_x$ proves the assertion by Completeness of Lebesgue measure. No finite-measure condition on $B$ is needed.

Exceptional sections need not be measurable: $\{0\}\times A$ is a planar Lebesgue null set for every $A\subseteq \mathbb{R}$, even when $A$ is a non-measurable set. Conversely, if both $A_1$ and $A_2$ are non-measurable, $A_1\times A_2$ cannot be measurable. Otherwise every $x\in A_1$ would have to lie in the null exceptional set, making $A_1$ measurable by Completeness of Lebesgue measure.