= Lebesgue measurable sections
{c}
{title2=$B_x=\{y:(x,y)\in B\}\text{ is measurable for a.e. }x$}
= Almost-everywhere measurability of Lebesgue sections
{c}
{synonym}
For a planar <Lebesgue measurable set> $B$, its vertical section $B_x$ is a one-dimensional <Lebesgue measurable set> for $x$ <almost everywhere>. Write $B\mathbin\triangle C\subseteq Z$ 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 $B_x\mathbin\triangle C_x\subseteq Z_x$ 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>.
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