Measure, Integration & Real Analysis · Problem 5C.2

The key is understanding that all cross sections being open doesn't force the set itself to be measurable—you need to exploit the difference between a property holding pointwise versus globally. Build $E$ by taking a non-Borel set $A \subseteq \mathbf{R}$ and carefully construct a set whose vertical and horizontal slices are all open, yet the set's structure prevents it from being in the product $\sigma$-algebra $\mathcal{B}_2$. This shows that openness of cross sections is a weaker condition than measurability of the full set.

The key is understanding that all cross sections being open doesn't force the set itself to be measurable—you need to exploit the difference between a property holding pointwise versus globally. Build $E$ by taking a non-Borel set $A \subseteq \mathbf{R}$ and carefully construct a set whose vertical and horizontal slices are all open, yet the set's structure prevents it from being in the product $\sigma$-algebra $\mathcal{B}_2$. This shows that openness of cross sections is a weaker condition than measurability of the full set.